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Linear Algebra by Georgi E. Shilov

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A published undergraduate linear algebra textbook by Georgi E. Shilov of Moscow University, kept in a folder of downloaded math books. The text covers determinants, linear spaces, systems of linear equations, linear operators, Jordan canonical form, bilinear and quadratic forms, Euclidean and unitary spaces, and finite-dimensional algebras. Each chapter has problems, with hints and answers at the end. Nothing in the extracted text shows notes by Phil.

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LINEAR ALGEBRA GEORGI E.SHILOV Professor ofMathematics Moscow University Revised English Edition Translated andEdited by Richard A.Silverman DOVER PUBLICATIONS, INC., NEW YORK Copyright ©1977 byDover Publications, Inc. Copyright @1971 byRichard A.Silverman. All rights reserved under Pan American and International Copyright Conventions. This Dover edition, ﬁrst published in1977, isan unabridged and corrected republication oftheEng- lish translation originally published byPrentice- Hall, Inc., in1971. International Standard Book Number: 0-486-63518-X Library ofCongress Catalog Card Number:77-075267 Manufactured intheUnited States ofAmerica Dover Publications, Inc. 31East2ndStreet, Mineola, N.Y. 11501 PREFACE This book isintended asatextforundergraduate students majoring in mathematics and physics. Itpresents thematerial ordinarily covered ina course onlinear algebra andsubsequently drawn upon invarious branches of mathematical analysis. However, itshould benoted that theterm “linear algebra” hasforsome time ceased todescribe theactual content ofthecourse, representing asitdoes asynthesis ofvarious ideas from algebra, geometry andanalysis. And although analysis inthestrict sense oftheterm (i.e., the branch ofmathematics concerned with limits, diﬁerentiation, integration, etc.) plays only abackground roleinthisLook, itisinfacttheactual organiz- ingprinciple ofthecourse, since theproblems of“linear algebra” canbe regarded both as“ﬁnite-dimensional projections” andasthe“support” for thebasic problems ofanalysis. The text stems inpart from myprevious book AnIntroduction tothe Theory ofLinear Spaces (Prentice-Hall, 1961), henceforth denoted byLS. Brieﬂy, thediﬁerences between LSandthepresent book arethefollowing: LSisentirely concerned with realspaces, while thisbook considers spaces over anarbitrary number ﬁeld, with therealandcomplex spaces being con- sidered asclosely related special cases ofthegeneral theory. Achapter has been introduced ontheJordan canonical form ofthematrix ofalinear operator inarealorcomplex space. Moreover, wealso study thecanonical form ofthematrix ofanormal operator inacomplex space equipped with a scalar product, deducing asspecial cases thecanonical forms ofthematrices ofHermitian, anti-Hermitian andunitary operators andtheir realanalogues. viPREFACE The ﬁnal lengthy chapter inLSonthegeometry ofinﬁnite-dimensional Hilbert space hasbeen omitted, since amore systematic treatment ofthis topic (inafunctional analysis context) isavailable inanumber ofother books. Instead, further new material bearing directly onthebasic content ofthecourse hasbeen added, namely Chapter 11onthestructure ofmatrix algebras (written atmyrequest byA.Y.Khelemski) andanappendix onthe structure ofmatrix categories, based onmyarticle with I.M.Gelfand (Vestnik MGU, Ser.Mat. Mekh., No.4(1963), pp.27-48). Chapter 11and theappendix, although completely elementary inmethod, arenevertheless somewhat higher inlevel than therest ofthebook (asindicated bythe asterisks) and represent advanced developments inthetheory oflinear algebra. Each chapter isequipped with asetofproblems, andhints andanswers to these problems appear attheendofthebook. Toacertain extent, theprob- lems help todevelop necessary technical skill, butthey areprimarily intended toillustrate andamplify thematerial inthetext. Certain groups ofproblems canserve asthebasis forseminar discussions. Thesame istrueofChapter 11 andtheappendix, aswellasofthestarred sections (thelatter contain ancillary material that canbeomitted onﬁrstreading). Itismypleasant duty toacknowledge thepainstaking eﬁorts ofM.S. Agranovich, theeditor ofthebook, andtothank himforanumber ofvalu- able suggestions. Ialso wish tothank I.Y.Dorfman forchecking the solutions toalltheproblems. G.E.S. CONTENTS chapter I DETERMINANTS 1.1. Number Fields 1.2. Problems oftheTheory ofSystems ofLinear Equations 1.3. Determinants ofOrder n 1.4. Properties ofDeterminants 1.5. Cofactors andMinors 1.6. Practical Evaluation ofDeterminants 1.7. Cramer’s Rule 1.8. Minors ofArbitrary Order. Laplace’s Theorem 1.9. Linear Dependence between Columns Problems vii>-->--O\N0ou1Lp>-- 18 20 23 28 viii CONTENTS chapter 2 LINEAR SPACES 3| 2.1. Deﬁnitions 31 2.2. Linear Dependence 36 2.3. Bases, Components, Dimension 38 2.4. Subspaces 42 2.5. Linear Manifolds 49 2.6. Hyperplanes 51 2.7. Morphisms ofLinear Spaces 53 Problems 56 chapter 3 SYSTEMS OFLINEAR EQUATIONS 58 3.1. More ontheRank ofaMatrix 58 3.2. Nontrivial Compatibility ofaHomogeneous Linear System 60 3.3. TheCompatibility Condition foraGeneral Linear System 61 3.4. The General Solution ofaLinear System 63 3.5. Geometric Properties oftheSolution Space 65 3.6. Methods forCalculating theRank ofaMatrix 67 Problems 71 chapter 4 LINEAR FUNCTIONS OFAVECTOR ARGUMENT 75 4.1. Linear Forms 75 4.2. Linear Operators 77 4.3. Sums andProducts ofLinear Operators 82 4.4. Corresponding Operations onMatrices 84 4.5. Further Properties ofMatrix Multiplication 88 4.6. The Range andNull Space ofaLinear Operator 93 4.7. Linear Operators Mapping aSpace KnintoItself 98 4.8. Invariant Subspaces 106 4.9. Eigenvectors andEigenvalues 108 Problems 113 CONTENTS chapter 5 COORDINATE TRANSFORMATIONS 5.1 5.2 5.3 5.4 5.5 *5.6Transformation toaNew Basis Consecutive Transformations Transformation oftheComponents ofaVector Transformation oftheCoefﬁcients ofaLinear Form Transformation oftheMatrix ofaLinear Operator Tensors Problems chapter 6 THE CANONICAL FORM OFTHE MATRIX OFALINEAR OPERATOR 6.1 6.2 6.3 6.4 6.5 6.6 *6.7 *6.sCanonical Form oftheMatrix ofaNilpotent Operator Algebras. TheAlgebra ofPolynomials Canonical Form oftheMatrix ofanArbitrary Operator Elementary Divisors Further Implications The Real Jordan Canonical Form Spectra, JetsandPolynomials Operator Functions andTheir Matrices Problems chapter 7 BILINEAR AND QUADRATIC FORMS 7.1 7.2 7.3 7.4 7.5 7.6 7.7 *7.8 7.9Bilinear Forms Quadratic Forms Reduction ofaQuadratic Form toCanonical Form TheCanonical Basis ofaBilinear Form Construction ofaCanonical Basis byJacobi’s Method Adjoint Linear Operators Isomorphism ofSpaces Equipped with aBilinear Form Multilinear Forms Bilinear andQuadratic Forms inaReal Space Problems XCONTENTS chapter 8 EUCLI DEAN SPACES 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9.Introduction Deﬁnition ofaEuclidean Space Basic Metric Concepts Orthogonal Bases Perpendiculars The Orthogonalization Theorem TheGram Determinant Incompatible Systems andtheMethod ofLeast Squares Adjoint Operators andIsometry Problems chapter 9 UNITARY SPACES 9.1. 9.2. 9.3. 9.4.Hermitian Forms The Scalar Product inaComplex Space Normal Operators Applications toOperator Theory inEuclidean Space Problems chapter I0 QUADRATIC FORMS INEUCLIDEAN AND UNITARY SPACES 10.1 10.2. 10.3 10.4. 10.5 *1o.6 10.7Basic Theorem onQuadratic Forms inaEuclidean Space Extremal Properties ofaQuadratic Form Simultaneous Reduction ofTwo Quadratic Forms Reduction oftheGeneral Equation ofaQuadric Surface Geometric Properties ofaQuadric Surface Analysis ofaQuadric Surface from ItsGeneral Equation Hermitian Quadratic Forms Problems courems xi *chapter II FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS 11.1. 11.2. 11.3. 11.4. 11.5. 11.6. 11.7. 11.8. 11.9.3I2 312 313 314 315 318 320 323 327 331 332More onAlgebras _ Representations ofAbstract Algebras Irreducible Representations andSchur’s Lemma Basic Types ofFinite-Dimensional Algebras TheLeft Regular Representation ofaSimple Algebra Structure ofSimple Algebras Structure ofSemisimple Algebras Representations ofSimple andSemisimple Algebras Some Further Results Problems *Appendix CATEGORIES OFFINITE-DIMENSIONAL SPACES A.1. A.2. A.3. A.4. A.5. A.6. HINTS AND ANSWERS BIBLIOGRAPHY INDEX335 335 338 340 345 353 357Introduction TheCase ofComplete Algebras TheCase ofOne-Dimensional Algebras TheCase ofSimple Algebras TheCase ofComplete Algebras ofDiagonal Matrices Categories andDirect Sums 36I 379 38I chapter I DETERMINANTS |.l.Number Fields 1.11. Like most ofmathematics, linear algebra makes useofnumber systems (number ﬁelds). Byanumber ﬁeld wemean anysetKofobjects, called “numbers,” which, when subjected tothefour arithmetic operations again give elements ofK.More exactly, these operations have thefollowing properties (ﬁeld axioms): a.Toevery pair ofnumbers onandBinKthere corresponds a(unique) number on+BinK,called thesumofonandB,where l)on+B=B+onforevery onandBinK(addition iscommutative); 2)(on+B)+Y=on+(B+Y)forevery on,B,YinK(addition is associative); 3)There exists anumber 0(zero) inKsuch that 0+on=onforevery on inK; 4)Forevery oninKthere exists anumber (negative element) YinKsuch that on+Y=0. Thesolvability oftheequation on+Y=0forevery onallows ustocarry outtheoperation ofsubtraction, bydeﬁning thediﬂerence B—onasthesum ofthenumber Bandthesolution Yoftheequation on+Y=0. b.Toevery pair ofnumbers onandBinKthere corresponds a(unique) number on-B(orocﬁ)inK,called theproduct ofonandB,where 5)<10=Botforevery onandBinK(multiplication iscommutative); 6)(ocB)Y =¢z(BY) forevery on,B,YinK(multiplication isassociative); 2DETERMINANTS cl-IAP, 1 7)There exists anumber l(¢0)inKsuch that l-on=onforevery on inK; 8)Forevery on¢0inKthere exists anumber (reciprocal element) Yin Ksuch that or.Y=l. c.Multiplication isdistributive over addition, i.e., 9)oc(I3+Y)=<10+ocYforevery on,B,YinK.T Thesolvability oftheequation ocY:Iforevery onvé0allows ustocarry outtheoperation ofdivision, bydeﬁning thequotient B/onastheproduct of thenumber Bandthesolution Yoftheequation or.Y==l. Thenumbers 1,1+1=2,2+1=3,etc.aresaidtobenatural; itis assumed that none ofthese numbers iszero.I Bytheintegers inaﬁeld Kwe mean thesetofallnatural numbers together with their negatives andthe number zero. Bytherational numbers inaﬁeld Kwemean thesetofall quotients p/q, where pandqareintegers andq¢0. Two ﬁelds KandK’aresaidtobeisomorphic ifwecansetupaone-to-one correspondence between KandK'such thatthenumber associated with every sum (orproduct) ofnumbers inKisthesum (orproduct) ofthecorresponding numbers inK’.The number associated with every diﬁerence (orquotient) ofnumbers inKwillthen bethediﬁerence (orquotient) ofthecorresponding numbers inK’. 1.12. The most commonly encountered concrete examples ofnumber ﬁelds arethefollowing: a.Theﬁeld ofrational numbers, i.e.,ofquotients p/qwhere pandq¢0 aretheordinary integers subject totheordinary operations ofarithmetic. (Itshould benoted that theintegers bythemselves donotform aﬁeld, since they donotsatisfy axiom 8).)Itfollows from theforegoing thatevery ﬁeld Khasasubset (subﬁeld) isomorphic totheﬁeld ofrational numbers. b.Theﬁeld ofrealnumbers, having thesetofallpoints oftherealline asitsgeometric counterpart. Anaxiomatic treatment oftheﬁeld ofreal numbers isachieved bysupplementing axioms l)—9) with theaxioms oforder andtheleast upper bound axiom.§ 1'Note thataxioms 5)and9)alsoimply (oz+[DY1my+by. iGiven twoelements NandE,say,wecanconstruct aﬁeld bytherules N+N=N, N—I—E=E,E+E=N,N-N=N,N-E=N,E-E=E. Then, inkeepingwithour notation, weshould write N=0,E-=1and hence 2=l+1=0.Toexclude such number systems, werequire thatallnatural ﬁeld elements benonzero. §For adetailed treatment ofreal numbers, see, forexample, G.H.Hardy, Pure Mathematics, ninth edition, TheMacmillan Co., New York (1945), Chap. l. SEC. 1.2 PROBLEMS OF Tl-IE THEORY OF SYSTEMS OF LINEAR EQUATIONS 3 c.Theﬁeld ofcomplex numbers oftheform a+ib,where aandbare realnumbers (iisnotarealnumber), equipped with thefollowing operations ofaddition andmultiplication (Hardy, op.cit.,Chap. 3): (a1+1.171) +(a2‘I‘1172) =(a1‘I‘a2)‘I‘i(b1 ‘I‘I72), (a1‘I‘ib1)(a2 ‘I‘ibz) :(a1a2 -b1b2) +i(a1b2 ‘I‘(12171)- Fornumbers oftheform a+i0,these operations reduce tothecorresponding operations forreal numbers; brieﬂy wewrite a+i0=aandcallcomplex numbers ofthisform real. Thus itcanbesaid that theﬁeld ofcomplex numbers hasasubset (subﬁeld) isomorphic totheﬁeld ofreal numbers. Complex numbers oftheform 0+ibaresaid tobe(purely) imaginary and aredesignated brieﬂy byib.Itfollows from themultiplication rulethat i2=i-i=(0—I—il)(0-I-il)=—-l. 1.13. Henceforth wewilldesignate theﬁeld ofreal numbers byRand theﬁeld ofcomplex numbers byC.According tothe“fundamental theorem ofalgebra” (Hardy, op.cit., Appendix II,p.492), wecannotonly carry outthefour arithmetic operations inCbutalsosolve anyalgebraic equation z"—I—a1z"—1—I—---—I—a,,=-0. The ﬁeld Rofrealnumbers does nothave thisproperty. Forexample, the equation 22+l=0hasnosolutions intheﬁeld R. Many ofthesubsequent considerations arevalid foranynumber ﬁeld. Inwhat follows, wewillusetheletter Ktodenote anarbitrary number ﬁeld. Ifsome property istrue fortheﬁeld K,then itisautomatically true forthe ﬁeld Randtheﬁeld C,which arespecial cases ofthegeneral ﬁeld K. l.2.Problems oftheTheory ofSystems ofLinear Equations Inthis and thenext two chapters, weshall study systems oflinear equations. Inthemost general case, such asystem hastheform a11x1 'I‘a12x2 ‘I‘'''‘I‘alnxn :bu a21x1 ‘I‘azzxz +'''‘I‘a2rzxn =172, (l) aux, +akzxz +---+a,,,,x,, =bk. Here x1,x2,...,x,,denote theunknowns (elements oftheﬁeld K)which aretobedetermined. (Note that wedonotnecessarily assume that the number ofunknowns equals thenumber ofequations.) The quantities an,am,...,a,m, taken from theﬁeld K,arecalled thecoeﬂicients ofthe 4DETERMINANTS ¢i-mp, 1 system. Theﬁrstindex ofacoefﬁcient indicates thenumber oftheequation inwhich thecoefﬁcient appears, while thesecond index indicates thenumber oftheunknown with which thecoefﬁcient isassociatedfr Thequantities bl,b2,...,bkappearing intheright-hand side of(l),taken from thesame ﬁeld K,arecalled theconstant terms ofthesystem; likethecoefﬁcients, they areassumed tobeknown. Byasolution ofthesystem (l)wemean anyset ofnumbers cl,C2,...,cnfrom thesame ﬁeld Kwhich, when substituted for theunknowns x1,x2,...,x,, turns alltheequations ofthesystem into identities.1 Not every system oflinear equations oftheform (l)hasasolution. For example, thesystem 2x1+3x2=5, (2) 2x1+3x2=6 obviously has nosolution atall.Indeed, whatever numbers cl,c2we substitute inplace oftheunknowns x1,x2,theleft-hand sides oftheequations ofthesystem (2)arethesame, while theright-hand sides arediﬂerent. There- fore nosuch substitution cansimultaneously convert both equations ofthe system intoidentities. Asystem ofequations oftheform (l)which hasatleast onesolution is called compatible; asystem which does nothave solutions iscalled incom- patible. Acompatible system canhave onesolution orseveral solutions. In thelatter case, wedistinguish thesolutions byindicating thenumber ofthe solution byasuperscript inparentheses; forexample, theﬁrstsolution will bedenoted bycg”,cg“,...,65,1’, thesecond solution bycg”,cf’,...,cffl, and soon.The solutions cg”,cg“,...,c‘n1’ and cf’,cg”,...,cf’are regarded asdistinct ifatleast oneofthenumbers cg"does notcoincide with thecorresponding numbers cf?’(i=l,2,...,n).Forexample, thesystem 2x1+3x2=0, (3) 4x1+6x2=0 hasthedistinct solutions (1l__ l1l__ l2)__ (2)_cl—c2 -0 and cl-3,c2 -—2 (and also inﬁnitely many other solutions). Ifacompatible system hasa unique solution, thesystem iscalled determinate; ifacompatible system has atleast twodiﬂerent solutions, itiscalled indeterminate. 1'Thus, forexample, thesymbol anshould beread as“athree four” andnotas“a thirty-four.” IWeemphasize that thesetofnumbers cl,c,,...,c,,represents onesolution ofthe system andnotnsolutions. SEC. 1.3 DETERMINANTS OF ORDER ll 5 Wecannow formulate thebasic problems which arise instudying the system (1): a)Toascertain whether thesystem (1)iscompatible orincompatible; b)Ifthesystem (l)iscompatible, toascertain whether itisdeterminate; c)Ifthesystem (l)iscompatible and determinate, toﬁnd itsunique solution; d)Ifthesystem (l)iscompatible andindeterminate, todescribe thesetof allitssolutions. The basic mathematical tool forstudying linear systems isthetheory of determinants, which weconsider next. l.3.Determinants ofOrder n 1.31-. Suppose wearegiven asquare matrix, i.e.,anarray ofn2numbers a,-,-(i,j=l,2,... ,n), allelements ofaﬁeld K: an "12 '''am a21 azz '''a2n . . .' (4) anl an2 ''.ann The number ofrows andcolumns ofthematrix (4)iscalled itsorder. The numbers a,-,-arecalled theelements ofthematrix. The ﬁrst index indicates therowandthesecond index thecolumn inwhich a,,-appears. Theelements an,an,...,am,form theprincipal diagonal ofthematrix. Consider anyproduct ofnelements which appear indiﬂerent rows and diﬂerent columns ofthematrix (4),i.e.,aproduct containing justoneelement from each rowandeach column. Such aproduct canbewritten intheform azllaaﬂ ''.aa,,n' Actually, fortheﬁrstfactor wecanalways choose theelement appearing in theﬁrstcolumn ofthematrix (4);then, ifwedenote by<11thenumber ofthe row inwhich theelement appears, theindices oftheelement willbeoq,l. Similarly, forthesecond factor wecanchoose theelement appearing inthe second column; then itsindices will be<12,2,where <12isthenumber of therowinwhich theelement appears, andsoon.Thus, theindices <11,<12, ...,anarethenumbers oftherows inwhich thefactors oftheproduct (5) appear, when weagree towrite thecolumn indices inincreasing order. Since, byhypothesis, theelements aim, anz, ...,aw, appear indiﬂerent rows ofthematrix (4),onefrom each row, then thenumbers <11,<12,...,an arealldiﬂerent andrepresent some permutation ofthenumbers 1,2,...,n. Byaninversion inthesequence <11,<12,...,on",wemean anarrangement 6DETERMINANTS Cl-IAP. l oftwoindices such thatthelarger index comes before thesmaller index. The total number ofinversions will bedenoted byN(oc1, <12,...,on”). For example, inthepermutation 2,l,4,3,there aretwoinversions (2before l, 4before 3),sothat N(2, 1,4, 3)=2. Inthepermutation 4,3,1,2,there areﬁveinversions (4before 3,4before 1, 4before 2,3before l,3before 2),sothat 1v(4,3,1,2)= 5. Ifthenumber ofinversions inthesequence <11,<12,...,aniseven, weputa plus sign before theproduct (5);ifthenumber isodd, weputaminus sign before theproduct. Inother words, weagree towrite infront ofeach product oftheform (5)thesign determined bytheexpression (_1)NI¢1.ﬂz.....ﬂ"). Thetotal number ofproducts oftheform (5)which canbeformed from the elements ofagiven matrix oforder nisequal tothetotal number ofpermuta- tions ofthenumbers 1,2, ...,n.Asiswell known, thisnumber isequal tonl. Wenow introduce thefollowing deﬁnition: Bythedeterminant Dofthematrix (4)ismeant thealgebraic sumofthen! products oftheform (5),each preceded bythesign determined bytherule justgiven, i.e., D:2(_1)N(a1.rz1.....anlaa11an2 ...amﬂw Henceforth, theproducts oftheform (5)will becalled theterms ofthe determinant D.Theelements a,,-ofthematrix (4)willbecalled theelements ofD,andtheorder of(4)willbecalled theorder ofD.Wedenote thedeter- minant Dcorresponding tothematrix (4)byoneofthefollowing symbols: an an '''an a a ---aDI*12* 2"=dc:||¢1.-.|I- (1) anl an2 ann Forexample, weobtain thefollowing expressions forthedeterminants of orders twoandthree: an an 2anazz *a21a12~ (121 azz an an an a a a Ia11a22a33 -I‘a21a32a13 'I'aa1a12a2a 21 22 23 *a31a22a13 '_a21a12a33 *a11aa2a2a- an aaz ass SEC. 1.3 DETERMINANTS OF ORDER ll 7 Wenow indicate therole ofdeterminants insolving systems oflinear equations, byconsidering. theexample ofasystem oftwoequations intwo unknowns: a11x1 ‘I‘a12-x2 :b1» a21x1 ‘I‘azzxz =I72- Eliminating oneoftheunknowns intheusual way, wecaneasily obtain the formulas b1a22 '"b2a12 (111172 *a21b1x1=D, x2I—i-—————, anazz "'a21a12 anazz "'a21a12 assuming thatthese ratios have nonvanishing denominators. Thenumerators and denominators oftheratios can berepresented bythesecond-order determinants an an auazz *a21a12 I a21 ('22 bl a12 b1a22 *b2a12 : 1 I72 azz an bl aubz *a21b1 : a21 I72 Itturns outthat similar formulas hold forthesolutions ofsystems with an arbitrary number ofunknowns (seeSec. 1.7). 1.32. Therulefordetermining thesign ofagiven term of.adeterminant canbeformulated somewhat diﬂerently, ingeometric terms. Corresponding totheenumeration ofelements inthematrix (4),wecandistinguish two natural positive directions: from lefttoright along therows, andfrom topto bottom along thecolumns. Moreover, theslanting lines joining anytwo elements ofthematrix canbefurnished with adirection: weshall saythat theline segment joining theelement a,-,-with theelement ak,haspositive slope ifitsright endpoint lieslower than itsleftendpoint, andthat ithas negative slope ifitsright endpoint lieshigher than itsleftendpoint.T Now imagine that inthematrix (4)wedraw allthesegments with negative slope joining pairs ofelements am, am, ...,aw,oftheproduct (5).Then weput aplus sign before theproduct (5)ifthenumber ofallsuch segments iseven, andaminus sign ifthenumber isodd. TThis deﬁnition of“slope” isnottobeconfused with thegeometric notion with the same name. Infact, thesign convention adopted here istheopposite ofthat used in geometry. 8DETERMINANTS C]-IAP, 1 Forexample, inthecase ofafourth-order matrix, aplus sign must be putbefore theproduct a21a12a4,,a,,4, since there aretwosegments ofnegative slope joining theelements ofthisproduct: an an an a22 a2:: an an aa2 ass at an a42 Q an However, aminus sign must beputbefore theproduct a41a,,2a1,,a24, since in thematrix there areﬁvesegments ofnegative slope joining these elements: an an an an an an a3 an a3, i 2an an Inthese examples, thenumber ofsegments ofnegative slope joining the elements ofagiven term equals thenumber ofinversions intheorder of theﬁrstindices oftheelements appearing intheterm. Intheﬁrstexample, the sequence 2,1,4,3ofﬁrstindices hastwoinversions; inthesecond example, thesequence 4,3,1,2ofﬁrstindices hasﬁveinversions. Wenow show thatthesecond deﬁnition ofthesignofaterm inadeterminant isequivalent totheﬁrst. Toshow this, itsufﬁces toprove thatthenumber of inversions inthesequence ofﬁrst indices ofagiven term (with thesecond indices innatural order) isalways equal tothenumber ofsegments ofnegative slope joining theelements ofthegiven term inthematrix. Butthisisalmost obvious, since thepresence ofasegment ofnegative slope joining theelements aw»andammeans thaton,>on,fori <j,i.e.,there isaninversion intheorder oftheﬁrstindices. l.4.Properties ofDeterminants 1.41. Thetransposition operation. Thedeterminant an a21 "'am a12 a22 ‘'‘an2 (8) alﬂ a2n ... an?! SEC. 1.4 PROPERTIES OF DETERMINANTS 9 obtained from thedeterminant (7)byinterchanging rows andcolumns with thesame indices issaid tobethetranspose ofthedeterminant (7).Wenow show that thetranspose ofadeterminant hasthesame value astheoriginal determinant. Infact, thedeterminants (7)and (8)obviously consist ofthe same terms; therefore itisenough forustoshow that identical terms in thedeterminants (7)and(8)have identical signs. Transposition ofthematrix ofadeterminant isclearly theresult ofrotating it(inspace) through 180° about theprincipal diagonal an,a,,,...,a,m. Asaresult ofthisrotation, every segment with negative slope (e.g., making anangle on<90°with the rows ofthematrix) again becomes asegment with negative slope (i.e., making theangle 90°——atwith therows ofthematrix). Therefore thenumber of segments withnegative slope joining theelements ofagiven term does not change after transposition. Consequently thesignoftheterm does notchange either. Thus thesigns ofalltheterms arepreserved, which means that the value ofthedeterminant remains unchanged. The property just proved establishes theequivalence oftherows and columns ofadeterminant. Therefore further properties ofdeterminants willbestated andproved only forcolumns. 1.42. Theantisymmetry property. Bytheproperty ofbeing antisymmetric with respect tocolumns, wemean thefactthat adeterminant changes sign when twoofitscolumns areinterchanged. Weconsider ﬁrstthecase where two adjacent columns areinterchanged, forexample columns jandj+l. The determinant which isobtained after these columns areinterchanged obviously still consists ofthesame terms astheoriginal determinant. Consider anyoftheterms oftheoriginal determinant. Such aterm contains anelement ofthejthcolumn andanelement ofthe(j+1)th column. If thesegment joining these twoelements originally hadnegative slope, then after theinterchange ofcolumns, itsslope becomes positive, andconversely. Asfortheother segments joining pairs ofelements oftheterm inquestion, each ofthese segments does notchange thecharacter ofitsslope after the column interchange. Consequently thenumber ofsegments with negative slope joining theelements ofthegiven term changes byonewhen thetwo columns areinterchanged; therefore each term ofthedeterminant, andhence thedeterminant itself, changes signwhen thecolumns areinterchanged. Suppose now that two nonadjacent columns areinterchanged, e.g., column jandcolumn k(j<k),where there aremother columns between. This interchange canbeaccomplished bysuccessive interchanges ofadjacent columns asfollows: First column jisinterchanged with column j+l,then with columns j+2,j+3,...,k.Then thecolumn k—1soobtained (which wasformerly column k)isinterchanged with columns k—2,k—3, ...,j.Inall,m+1+m=2m+1interchanges ofadjacent columns are required, eachofwhich, according towhat hasjustbeenproved, changes the I0DETERMINANTS CHAP. 1 sign ofthedeterminant. Therefore, attheend oftheprocess, thedeter- minant willhave asign opposite toitsoriginal sign (since foranyinteger m, thenumber 2m+lisodd). 1.43. COROLLARY. Adeterminant withtwoidentical columns vanishes. Proof. Interchanging thecolumns does notchange thedeterminant D. Ontheother hand, asjust proved, thedeterminant must change itssign. Thus D==——D, which implies that D=0.IT 1.44. Thelinear property ofdeterminants. This property canbeformulated asfollows: a.THEOREM. Ifalltheelements ofthejthcolumn ofadeterminant D are“linear combinations” oftwocolumns ofnumbers, i.e., a,-,-=7\b,»—I--p.c,- (i=l,2,...,n) where 71anditareﬁxed numbers, then Disequal toalinear combination of twodeterminants: 1)=xi),+pLD2. (9) Here both determinants D,andD2have thesame columns asthedeterminant Dexcept forthejthcolumn; thejthcolumn ofD1consists ofthenumbers bi, while thejthcolumn ofD2consists ofthenumbers c,». Proof. Every term ofthedeterminant Dcanberepresented intheform aa11aa,2 '''ac,-l I.Iaaﬂn =aa11aa22 III(Abel +F1701!) IIIaaﬂn =)‘aa11aa22 IIIbu; II‘aaﬂn +i‘I'aa11aa22 I.‘ca; .IIaa,,n' Adding upalltheﬁrstterms (with thesigns which thecorresponding terms have intheoriginal determinant), weclearly obtain thedeterminant D1, multiplied bythenumber 7..Similarly, adding upallthesecond terms, we obtain thedeterminant D2,multiplied bythenumber u.I Itisconvenient towrite thisformula inasomewhat diﬁ‘erent form. Let Dbeanarbitrary ﬁxed determinant. Denote byD,(p,-) thedeterminant which isobtained byreplacing theelements ofthejthcolumn ofDbythe numbers p,»(i=l,2,...,n).Then (9)takes theform D,-WY.‘ -I"Wt)=7\D.(b.-) +i*D.-(¢‘¢)- b.Thelinear property ofdeterminants caneasily beextended tothecase where every element ofthejthcolumn isalinear combination notoftwo terms butofanyother number ofterms, i.e. au=7\bi+I*ci+"'+Tfi~ 1'Thesymbol |means Q.E.D. andindicates theendofaproof. szc. 1.4 PROPERTIES orDETERMINANTS II Inthiscase, Di(aii) =Di(7\b¢ +We+'''‘I‘Tfi) =7\D:(bi) ‘I‘I‘I'Di(ci) ‘I‘'''+TDi'(fi)- (10) 1.45. COROLLARY. Any common factor ofacolumn ofadeterminant can befactored outofthedeterminant. V Proof. Ifa,-,-IAbi,then by(10)wehave Dt(au') =D.-(lb.-) =)\Di(b¢)- I 1.46. COROLLARY. Ifacolumn ofadeterminant consists entirely ofzeros, then thedeterminant vanishes. Proof. Since 0isacommon factor oftheelements ofoneofthecolumns, wecanfactor itoutofthedeterminant, obtaining D,(0) =D,-(0- 1)=0-D,-(1) =0.| 1.47. Addition ofanarbitrary multiple ofonecolumn toanother column. a.THEOREM. The value ofadeterminant isnotchanged byadding the elements ofonecolumn multiplied byanarbitrary number tothecorresponding elements ofanother column. Proof Suppose weaddthekthcolumn multiplied bythenumber Atothe jthcolumn (kqéj). Thejthcolumn oftheresulting determinant consists ofelements oftheform ai,+ha,-k(i=1,2,...,n).By(9)wehave Di(aii ‘I‘hark) =Di(aiJ') ‘I‘7\D,'(aik)- Thejthcolumn ofthesecond determinant consists oftheelements aik,and hence isidentical with thekthcolumn. Itfollows from Corollary 1.43T that D,-(a,-k) =0,sothat Di(aia' 'I'bark) =Di(aii')- I b.Naturally, Theorem 1.47a canbeformulated inthefollowing more general form: Thevalue ofadeterminant isnotchanged byadding tothe elements ofitsjthcolumn ﬁrst thecorresponding elements ofthekthcolumn multiplied byA,next theelements ofthelthcolumn multiplied byu,etc., and ﬁnally theelements ofthepth column multipliedby -r(k¢j, l¢j, ...,p¢j). 1.48. Because oftheinvariance ofdeterminants under transposition (Sec. 1.41), alltheproperties ofdeterminants proved inthis section for columns remain valid forrows aswell. ‘I’Corollary 1.43 refers tothe(unique) corollary inSec. 1.43, Theorem l.47a tothe theorem inSec. l.47a, etc. I2DETERMINANTS cr-1A1>_ l l.5.Cofactors andMinors 1.51. Consider anycolumn, thejthsay, ofthedeterminant D.Leta,-, beanyelement ofthiscolumn. Add upalltheterms containing theelement anappearing intheright-hand sideofequation (6) D= DNIM1.a:2'...,a:,.laa11aa22...a%", andthen factor outtheelement a,,.Thequantity which remains, denoted by A”,iscalled thecofactor oftheelement anofthedeterminant D. Since every term ofthedeterminant Dcontains anelement from thejth column, (6)canbewritten intheform D=an-411 ‘I‘a2;-421 -I""""-I"amAm, (11) called theexpansion ofthedeterminant Dwithrespect tothe(elements ofthe) jthcolumn. Naturally, wecanwrite asimilar formula foranyrow ofthe determinant D.Forexample, fortheithrowwehave theformula D=ai1Ai1 -I"ai2Ai2 -I"''‘-I"amAm- (I2) Thisgives thefollowing THEOREM. Thesum ofalltheproducts oftheelements ofanycolumn (or row) ofthedeterminant Dwith thecorresponding cofactors isequal tothe determinant Ditself. Equations (11)and(12)canbeused tocalculate determinants, butﬁrst wemust know how tocalculate cofactors. Wewillshow how thisisdone in Sec. 1.53. 1.52. Next wenote aconsequence of(11)and(12)which willbeuseful later. Equation (11)isanidentity inthequantities a1,-,a2,»,...,am.There- foreitremains valid ifwe replaceai, (i=1,2,...,n)byanyother quantities. Thequantities A1,-,A2,-,...,AMremain unchanged when such areplacement ismade, since they donotdepend ontheelements a,-,.Suppose that inthe right andleft-hand sides oftheequality (11)wereplace theelements au,a2,-, ...,an,bythecorresponding elements ofanyother column, saythekth. Then thedeterminant intheleft-hand side of(ll) willhave two identical columns andwill therefore vanish, according toCorollary 1.43. Thus we obtain therelation a1kA1i+ "216/121'-I" ‘‘‘-I-ank/4721: 0 (13) fork¢j. Similarly, from (12)weobtain al1Ai1 +ai2A¢2 +"'"+am-4m :0 (14) SEC. 1.5 COFACTORS AND MINORS forl¢i.Thus wehave proved thefollowing THEOREM. Thesum ofalltheproducts oftheelements ofacolumn (orrow) ofthedeterminant Dwith thecofactors _ofthecorresponding elements of another column (orrow) isequal tozero. 1.53. Ifwedelete arowandacolumn from amatrix oforder n,then, of course, theremaining elements form amatrix oforder n-—1.The deter- minant ofthismatrix iscalled aminor oftheoriginal nth-order matrix (and alsoaminor ofitsdeterminant D).Ifwedelete theithrowandthejthcolumn ofD,then theminor soobtained isdenoted byMi,orM,»,(D). Wenow show thattherelation Au=(_1)i+jMii (15) holds, sothat thecalculation ofcofactors reduces tothecalculation ofthe corresponding minors. First weprove (15) forthecase i=1,j=1.We addupalltheterms intheright-hand sideof(6)which contain theelement an,andconsider oneofthese terms. Itisclear that theproduct ofallthe elements ofthisterm except angives aterm coftheminor Mu. Since in thematrix ofthedeterminant D,there arenosegments ofnegative slope joining theelement anwith theother elements oftheterm selected, thesign ascribed totheterm auc ofthedeterminant Disthesame asthesign ascribed totheterm cintheminor Mu. Moreover, bysuitably choosing a term ofthedeterminant Dcontaining anand then deleting an,wecan obtain anyterm oftheminor Mu. Thus thealgebraic sum ofalltheterms ofthedeterminant Dcontaining an,with andeleted, equals theproduct Mu. Butaccording toSec. 1.51, thissum isequal totheproduct Au. There- fore, Au=Muasrequired. Now weprove (15)forarbitrary iandj,making essential useofthefact that theformula isvalid fori=j= 1.Consider theelement a,~,-=a, appearing intheithrow and thejthcolumn ofthedeterminant D.By successively interchanging adjacent rows and columns, wecanmove the element aover totheupper left-hand corner ofthematrix; todothis, we need i—1+j——1=i+j——2 interchanges. Asaresult, weobtain thedeterminant D,with thesame terms asthose oftheoriginal determinant Dmultiplied by (__1)i+i~2 :(__1)i+.'i‘ The minor Mu(D1) ofthedeterminant D,isclearly identical with the minor M,-,-(D) ofthedeterminant D.Bywhat hasbeen proved already, thesum oftheterms ofthedeterminant D,which contain theelement a, with adeleted, isequal toMu(D,). Therefore thesum oftheterms ofthe I4 DETERMINANTS CHAP. l original determinant Dwhich contain theelement a,-,-=a,with adeleted, is equal to<—1)1+iM..<1>.> =<-1)i+iM..<1>>. According toSec. 1.51, thissum isequal toAH. Consequently A.=<—1)*~'M.-,. which completes theproof of(15). 1.54. Formulas (ll) and (12) can now bewritten inthefollowing commonly used variants: D=(_1)1+ja1iM1i ‘I‘(_l)2+ja2iM2i ‘I‘''‘+(“1)”+jan;Mn1, (11!) D=(_T1)i+1ai1Mi1 ‘I‘(__1)i+2ai2Mi2 +‘"'—I—(-—1)i+nainMin_ (121) 1.55. Examples a.Athird-order determinant has sixdistinct expansions, three with respect torows andthree with respect tocolumns. Forexample, theexpansion with respect totheﬁrstrow is an a12 a13 a22 a2a a21 aza (121 (122 (121 a22 a2a =a11 “"a12 draw (132 a33 a31 a33 a31 aa2 a31 aa2 a33 b.Annth-order determinant oftheform an 0 0~~~0 an an 0-'-0 D”==an an an ~--0 anl an2 a7l3 '''ann iscalled triangular. Expanding D”with respect totheﬁrst row, weﬁnd that D,,equals theproduct oftheelement anwith thetriangular determinant an 0...0 aaz ass 0 Dn—1 T: an2 ant! l.‘ ann oforder n——1.Again expanding D,,_, with respect totheﬁrstrow, weﬁnd that D»-1 =a22Dn—2s sec.1.5 COFACTORS ANDMINORS 15 where D,,_2 isatriangular determinant oforder n——2.Continuing inthis way, weﬁnally obtain D=a11a22 '''arm! i.e.,atriangular determinant equals theproduct oftheelements appearing along itsprincipal diagonal. c.Calculate theVandermonde determinant 1 1 ~1 X1 X2 ...X” _ - 2W(x1,..., x,,): xi x; :xn. xp—1 x;t—1 xZ—1 Solution. W(x1, ...,x,,) isapolynomial ofdegree n—1inx,,,with coeﬂicients depending onx1,...,x,,_,. This polynomial vanishes ifxntakes anyofthe values x,,x2,...,x,,_,, since then thedeterminant hastwoidentical columns. Hence, byafamiliar theorem ofelementary algebra, thepoly- nomial W(x1, ...,x,,)isdivisible bytheproduct (x,,—x1)---(x,,—x,,_,), sothat n—1 I4/(Xv --->X791 a(X1, ---5X"-1)H(Xn “Xx)-k=1 The quantity a(x,, ...,x,,_,) istheleading coeﬂicient ofthepolynomial W(x1, ...,x,,). Expanding theVandermonde determinant with respect to thelast column, weseethat this coeﬂicient isjust W(x1,... ,x,,_1). It follows that n—1 W(x,, ...,x,,)—W(x,, ...,xn_1)H(xn ~xk). Similarly, ‘T n—2 W(x,, ...,x,,_1)= W(x,, ...,x,,_2) H(x,,_, —x,-), i=1 W(-‘C1, X2) :W(X1)(X2 _X1), andobviously W(x,) =1. Multiplying allthese equalities together, wegetthedesired result W(X1,---,X.)=TI(X...—X.)- 1<i<rrI<§n Inparticular, thequantities x1,...,x,,arealldistinct, then W(x,, ...,x,,)vi0. I6DETERMINANTS Cl-IAP, 1 l.6.Practical Evaluation ofDeterminants 1.61. Formula (12)takes aparticularly simple form when alltheelements oftheithrowvanish except oneelement, saya,»,,.Inthiscase D=aikAik’ (I6) andthecalculation ofthedeterminant Doforder nreduces atonce tothe calculation ofadeterminant oforder n—1.Ifinaddition toa,-k,there is another nonzero element a,,-intheithrow, then multiplying thekthcolumn by7.=an./a,»,, and subtracting itfrom theithcolumn, weobtain adeter- minant which isequal totheoriginal one(cf.Sec. 1.47) butwhich now hasa zero intheithrow andjthcolumn. Byasequence ofsimilar operations, wechange anydeterminant with anonzero element a,»,,intheithrowinto adeterminant inwhich alltheelements oftheithrowequal zero except a,-,,. This new determinant canthen beevaluated by(16). Ofcourse, similar operations canalso beperformed onthecolumns ofadeterminant. 1.62. Example. Calculate thefollowing determinant oforder ﬁve: —2 5 0—1 3 1 0 3 7—2 D= 3-1 0 5—5 . 2 6—4 1 2 0—3 —1 2 3 Solution. There arealready two zeros inthethird column ofthis determinant. Inorder toobtain twomore zeros inthiscolumn, wemultiply theﬁfth rowby3andaddittothesecond rowandthen multiply theﬁfth row by4and subtract itfrom thefourth row. After performing these operations andexpanding thedeterminant with respect tothethird column, weobtain -2 50-1 3 1-9 0137 1>= 3-1 05-5=(-1)=*+5(-1) 1_91333-1 5-52180-7-10 0-3-1 23 -2 5-1 3 1-913 7 :_ 3-1 5-5' 218-7-10~2 5—1 3 2 18—7—10 SEC. 1.6 PRACTICAL EVALUATION OF DETERMINANTS The simplest thing todonow istoproduce three zeros intheﬁrstcolumn; todothis, weaddtwice thesecond rowtotheﬁrstrow, subtract three times thesecond rowfrom thethird rowandsubtract twice thesecond rowfrom thefourth row: -2 5-1 3 0-13 25 17 D 1-913 7 1-9 13 7 — 3-1 5-5—026-34_26 2 18—7 -10 0 36 —33 —24 -13 25 17 =—(-1)1+‘-* 26-34~26 36-33-24 Tosimplify thecalculation ofthethird-order determinant just obtained, wetrytodecrease theabsolute values ofitselements. Todothis, wefactor thecommon factor 2outofthesecond row, add thesecond row tothe ﬁrstandsubtract twice thesecond rowfrom thethird row: —13 25 17 0 8 4 D=2 13-17 —13 =2 13——17 -13 36 —33 —24 10 1 2 0 2 1 :2-4 13-17 -13 . 10 1 2 There isalready one zero intheﬁrst row. Toobtain still another zero, wesubtract twice thethird column from thesecond column. After this, the evaluation ofthedeterminant iseasily completed. 02 1 00 1 1>=s 13-17-13=8139-13=s(-1)1+B 10 12 0-3 213 9 10-3 133=8-3j10 1[=8-3(-13-30):-s-3-43=-1032. 18DETERMINANTS Cl-IAP. 1 l.7.Cramer’s Rule 1.71. Wearenow inaposition tosolve systems oflinear equations. First weconsider asystem ofthespecial form a11x1 ‘I‘a12x2 -I"‘‘'‘I‘alnxn =bu a21x1 a22x2 +'' a2nxn =b2, (17) anlxl +an2-x2 +'II'I'annxn =bnv i.e.,asystem which hasthesame number ofunknowns andequations. The coeﬂicients a,-,~(i,j=1,2,...,n)form thecoeﬂicient matrix ofthesystem; weassume that thedeterminant ofthismatrix isdiﬁerent from zero. We nowshow thatsuchasystem isalways compatible anddeterminate, andwe obtain aformula which gives theunique solution ofthesystem. Webegin byassuming that cl,c2,...,c,,isasolution of(17), sothat allcl ‘I‘(11252 ‘I‘''''I'alncn :b1’ (12151 ‘I‘(12252 'I''''‘I‘(121151. =b2, (18) anlcl +an2c2 'I'III+anncn =bn' Wemultiply theﬁrstoftheequations (18)bythecofactor Anoftheelement aninthecoeﬂicient matrix, then wemultiply thesecond equation byA21, thethird byA31, andsoon,andﬁnally thelastequation byAM. Then we addalltheequations soobtained. Theresult is (a11A11 ‘I‘(121-421 ‘I‘'''‘I‘an1An1)c1 +(an/411 +a22-421 +''''I'an2An1)c2 'I'''' (19) 'I'(a1nA11 ‘I‘an-421 ‘I‘'''‘I‘annAn1)cn =b1-411 ‘I‘b2-421 ‘I‘'''‘I‘bnAn1' ByTheorem 1.51, thecoeﬂicient ofclin(19)equals thedeterminant Ditself. ByTheorem 1.52, thecoeﬂicients ofalltheother c,(j¢1)vanish. The expression intheright-hand sideof(19)istheexpansion ofthedeterminant bi biz '''but D_b2 a22 '''a2n1_ bn an2 ann with respect toitsﬁrst column. Therefore (19) cannow bewritten inthe fonn Dc,=D1, sec. 1.7 cRAMER’s RULE I9 sothat Dcl=—‘. D Inacompletely analogous way, wecanobtain theexpression 6,-=%’ _(j:l,2,...,n), (20) where an "12 '''(11.1-1 bia1.i+1 '''but _1121 1122 ‘‘‘a2_j—1 b2a2.i+1 '''"2"D7‘— =D1(bt) anl an2 ‘ll an..'i—1 bn an.:i+1 ann isthedeterminant obtained from thedeterminant Dbyreplacing itsjth column bythenumbers bl,b2,...,b,,.Thus weobtain thefollowing result: Ifasolution ofthesystem (17) exists, then (20) expresses thesolution in terms ofthecoeﬂicients ofthesystem andthenumbers intheright-hand side of(17). Inparticular, weﬁnd thatifasolution ofthesystem (17)exists, itis unique. 1.72. Wemust stillshow thatasolution ofthesystem (17)always exists. Consider thequantities D .c,=$’ (]:l,2,...,n), andsubstitute them intothesystem (1.7)inplace oftheunknowns x1,x2,..., x,,.Then thisreduces alltheequations ofthesystem (17) toidentities. In fact, fortheithequation weobtain D D D,, ai1c1+ai2c2+ “Panic”: ai1$1+ai2$2+ +ain$ 1 :'5Iai1(b1A11 ‘I‘b2-'4-21 'I'''''I'bnAn1) +a1'2(b1A12 +b2/422 +'''+bnAn2) +''' +an1(b1-'41” ‘I‘b2-4211 ‘I‘'''‘I‘bﬂAﬂﬂ)] 17% [b1(ai1A11 ‘I‘at-2-412 ‘I‘'''‘I‘ainA1n) ‘I‘''' 'I'l72(ll,-1-421 -I‘an/422 'I''''+am/42“) 'I'''' ‘I‘bn(ai1An1 ‘I‘ai2An2 'I''''‘I‘ainAnn)l' 20 DETERMINANTS cHA1>. 1 ByTheorems 1.51 and 1.52, only oneofthecoeﬂicients ofthequantities bl,b2,...,b,,isdiﬂerent from zero, namely thecoeﬂicient ofbl,which is equal tothedeterminant Ditself. Consequently, theabove expression reduces to Lb,-D=b,~, D i.e.,isidentical with theright-hand side oftheithequation ofthesystem. 1.73. Thus thequantities c,(j=1,...,n)actually constitute asolution ofthesystem (17), andwehave found thefollowing prescription (Cramer’s rule) forobtaining solutions of(17): Ifthedeterminant ofthesystem (17)isdiﬂerent from zero, then (I7)hasa unique solution, namely, forthevalue oftheunknown x,(=1,...,n)we take thefraction whose denominator isthedeterminant Dof(17) andwhose numerator isthedeterminant obtained byreplacing thejthcolumn ofDbythe column consisting oftheconstant terms of(17), i.e.,thenumbers intheright- hand sides ofthesystem. Thus ﬁnding thesolution ofthesystem (17) reduces tocalculating determinants. Ways ofsolving more general systems (with vanishing deter- minants, orwith anumber ofequations diﬁ‘erent from the number of unknowns) willbegiven inthenext twochapters. 1.74. Remark. One sometimes encounters systems oflinear equations whose constant terms arenotnumbers butvectors, e.g., inanalytic geometry orinmechanics. Cramer’s rule and itsproof remain valid inthiscase as well; onemust only bear inmind thatthevalues oftheunknowns xi,x2,..., x,,willthen bevectors rather than numbers. Forexample, thesystem xl'I'x2:I—31, x1—x2=1+5j hastheunique solution c1:i'I'.I, ¢'2=—4.I- l.8.Minors ofArbitrary Order. Laplace's Theorem 1.81. Theorem 1.54 ontheexpansion ofadeterminant with respect to arow oracolumn isaspecial case ofamore general theorem onthe expansion ofadeterminant with respect toawhole setofrows orcolumns. Before formulating thisgeneral theorem (Lap1ace’s theorem), weintroduce some new notation. Suppose thatinasquare matrix oforder nwespecify anyk<ndiﬁerent rows and thesame number ofdiﬂerent columns. The elements appearing SEC. 1.8 MINORS OF ARBITRARY ORDER. LAPLACE’S THEOREM attheintersections ofthese rows andcolumns form asquare matrix oforder k.Thedeterminant ofthismatrix iscalled aminor oforder koftheoriginal matrix oforder n(also aminor oforder kofthedeterminant D);itis denoted by M=M where fl,1'2,...,i,_.arethenumbers ofthedeleted rows, andj1,j2, ...,j,_. arethenumbers ofthedeleted columns. Ifintheoriginal matrix wedelete therows andcolumns which make up theminor M,then theremaining elements again form asquare matrix, this time oforder n—k.The determinant ofthis matrix iscalled thecomple- mentary minor oftheminor M,andisdenoted bythesymbol M Mi1.l2....1L 2 11.52.---.)'i.' Inparticular, iftheoriginal minor isoforder l,i.e.,isjust some element a,-,-ofthedeterminant D,then thecomplementary minor isthesame asthe minor Mi,‘discussed inSec. 1.53. Consider now theminor M.=Ml:§::::iI formed from theﬁrstkrows andtheﬁrstkcolumns ofthedeterminant D; itscomplementary minor is M.=/VI.=/VIl;§:;;:;i.I- lntheright-hand side ofequation (6), p.6group together allthe terms ofthedeterminant whose ﬁrstkelements belong totheminor M,(and thus whose remaining n~kelements belong totheminor M2). Letone ofthese terms bedenoted byc;wenow wish todetermine thesign which must beascribed toc.The ﬁrst kelements ofcbelong toaterm c,ofthe minor M1. Ifwedenote byN,thenumber ofsegments ofnegative slope corresponding tothese elements, then thesignwhich must beputinfront of theterm c,intheminor M,is(—l)‘\'1. The remaining n—kelements of cbelong toaterm c2oftheminor M2;thesign which must beputinfront ofthisterm intheminor M2is(—l)~"'2, where N2isthenumber ofsegments ofnegative slope corresponding tothen—kelements ofc2.Since inthe matrix ofthedeterminant Dthere isnotasingle segment with negative slope joining anelement oftheminor M,with anelement oftheminor M2, thetotal number ofsegments ofnegative slope joining elements ofthe term cequals thesum N,+N2.Therefore thesign which must beputin front oftheterm cisgiven bytheexpression (—1)“'-"2,andhence isequal totheproduct ofthesigns oftheterms c,andc2intheminors M1andM2. Moreover, wenote that theproduct ofanyterm oftheminor M,andany term oftheminor M2gives usoneoftheterms ofthedeterminant Dthat 22 DETERMINANTS Cl-lAP_ 1 have been grouped together. Itfollows that thesum ofalltheterms that wehave grouped together from theexpression forthedeterminant Dgiven by(6)isequal totheproduct oftheminors M1andM2. Next wesolve theanalogous problem foranarbitrary minor M1=M.Il1..i7':2....-.-.-.ilCk’ with complementary minor M2. Bysuccessively interchanging adjacent rows andcolumns, wecanmove theminor M1over totheupper left-hand comer ofthedeterminant D;todoso,weneed atotal of ('.1—1)"I‘(l'2—2)"I""-'1-(1.16-/‘) —l-(1.1-1)-I-(j2—2)-I-"'-I-(ji-/<) interchanges. Asaresult, weobtain adeterminant D1with thesame terms asintheoriginal determinant butmultiplied by(—l)"+", where l=l1‘I‘l2‘I“""I“Ik» l=l1‘I‘j2‘I“""I‘lk- Bywhat hasjustbeen proved, thesum ofalltheterms inthedeterminant D,whose ﬁrst kelemetlts appear intheminor M,isequal totheproduct MYM2. Itfollows from this that thesum ofthecorresponding terms of thedeterminant Disequal totheproduct (—1)‘+iM,M2 =.M,A2, where thequantity A2:(—l)l+jM2 iscalled thecofactor oftheminor M1inthedeterminant D.Sometimes oneuses thenotation A2=/4'iiI'}’.'.'.Z'.'.'i.» where theindices indicate thenumbers ofthedeleted rows andcolumns. Finally, lettherows ofthedeterminant Dwith indices i1,i2,...,ikbe ﬁxed; some elements from these rows appear inevery term ofD.Wegroup together alltheterms ofDsuch that theelements from theﬁxed rows i1,7'2,...,ikbelong tothecolumns with indices j1,j2, ...,j,,.Then, by what hasjustbeen proved, thesum ofallthese terms equals theproduct of theminor Mi1,i¢....,i1 i1.;'g.....i1.- with thecorresponding cofactor. Inthis way, alltheterms ofDcanbe divided intogroups, each ofwhich ischaracterized byspecifying kcolumns. The sum oftheterms ineach group isequal totheproduct ofthecorre- sponding minor anditscofactor. Therefore theentire determinant canbe represented asthesum _ i,.i,.....1). i'1,i1.....il.- D—2Mix-ig.....itA-i1.i2.....ii’ (21) SEC. 1.9 LINEAR DEPENDENCE BETWEEN COLUMNS where theindices ix,i2,...,ik(theindices selected above) areﬁxed, andthe sum isover allpossible values ofthe column indices j1,j2, ...,j,, (1<j1 <j2 <'--<j,, <n).The expansion ofDgiven by(21) iscalled Laplace’s theorem. Clearly, Laplace’s theorem constitutes ageneralization oftheformula forexpanding adeterminant with respect tooneofitsrows (derived inSec. 1.54). There isananalogous formula forexpanding the determinant Dwith respect toaﬁxed setofcolumns. 1.82. Example. Thedeterminant oftheform an '''but 0 ' 0 a21 ''a2k 0 ' 0 D: akk --am. 0 - 0 ak+1.1 '''”lC+1,lC at-+1,k+1 '''ak-I-1,n am '''arllt amt.-+1 '''ann such that alltheelements appearing inboth theﬁrst krows and thelast n—kcolumns vanish, iscalled quasi-triangular. Tocalculate thedeter- minant, weexpand itwith respect totheﬁrst krows byusing Laplace’s theorem. Only oneterm survives inthesum (21), andweobtain an but ak+1.k+1 ak+1_n 1): .. .>< . . _ am '''akk an.k+1 '''ann 1.9.Linear Dependence between Columns 1.91. Suppose wearegiven n7columns ofnumbers with nnumbers in each: an a12 alm a21 a22 a2m A1: -, A2= -,..., Am: -. anl an2 anm Wemultiply every element oftheﬁrst column bysome number 711,every element ofthesecond column by7.2,etc., andﬁnally every element ofthe last(mth) column byAm;wethen addcorresponding elements ofthecolumns. 24 DETERMINANTS cHA1>. l Asaresult, wegetanewcolumn ofnumbers, whose elements wedenote by c1,c2,...,c,,.We can represent allthese operations schematically as follows: I I IIa11 a21I Ialm C1 a21 a22 a2'm C2 )\1 . +)\2 . +...+)\m .‘= .’ anl an2 anm cn ormore brieﬂy as X1141 'I')\2A2'I' "'’I‘)\mAm :C, where Cdenotes thecolumn whose elements arec1,c2,...,c,,.Thecolumn Ciscalled alinear combination ofthecolumns A1,A2,...,A,,,, and the numbers 71,72,...,71,,arecalled thecoeﬂicients ofthelinear combination. Asspecial cases ofthelinear combination C,wehave thesumofthecolumns if71=72:---=7.,"=land theproduct ofacolumn byanumber if m=1. Suppose now that ourcolumns arenotchosen independently, butrather make upadeterminant Doforder n.Then wehave thefollowing THEOREM. Ifoneofthecolumns ofthedeterminant Disalinear combi- nation oftheother columns, then D:0. Proof. Suppose, forexample, that theqthcolumn ofthedeterminant D isalinear combination ofthejth,kth, ...,pthcolumns ofD,with coeﬂi- cients 7.,-,71,,,...,71D, respectively. Then, according toSec. 1.47, bysub- tracting from theqthcolumn ﬁrstthejth column multiplied by7.,-,then the kthcolumn multiplied by71,,etc., andﬁnally thepthcolumn multiplied by 711,,wedonotchange thevalue ofthedeterminant D.However, asaresult, theqthcolumn consists ofzeros only, from which itfollows that D:0.I Itisremarkable thattheconverse isalso true, i.e., agiven determinant Disequal tozero, then (atleast) oneofitscolumns isalinear combination oftheother columns. The proof ofthistheorem requires some preliminary considerations, towhich wenow turn. 1.92. Again suppose wehave mcolumns ofnumbers with nelements in each. Wecanwrite them intheform ofamatrix an a12 alm A: a21 a22 a27n anl an2 iiianm SEC. 1.9 LINEAR DEPENDENCE BETWEEN COLUMNS 25 with nrows and mcolumns. Ifkcolumns andkrows ofthis matrix are held ﬁxed, then theelements appearing attheintersections ofthese columns and rows form asquare matrix oforder k,whose determinant isaminor oforder koftheoriginal matrix A(seep.21); thisdeterminant may either bevanishing ornonvanishing. If,asweshall always assume, notallofthe aikarezero, then wecanalways ﬁnd aninteger rwhich hasthefollowing twoproperties: - 1)The matrix Ahasaminor oforder rwhich does notvanish; 2)Every minor ofthematrix Aoforder r+1and higher (ifsuch actually exist) vanishes. The number rwhich hasthese properties iscalled therank ofthematrix A.Ifalltheaikvanish, then therank ofthematrix Aisconsidered tobe zero (r=0).Henceforth weshall assume that r>0.The minor oforder rwhich isdiﬁerent from zero iscalled thebasis minor ofthematrix A. (Ofcourse, Acanhave several basis minors, butthey allhave thesame order r.)The columns which contain thebasis minor arecalled thebasis columns. 1.93. Concerning thebasis columns, wehave thefollowing important THEOREM (Basis minor theorem). Any column ofthematrix Aisalinear combination ofitsbasis columns. Proof Tobeexplicit, weassume that thebasis minor ofthematrix is located intheﬁrstrrows andﬁrstrcolumns ofA.Letsbeanyinteger from 1tom,letkbeanyinteger from 1ton,andconsider thedeterminant an a12 '''alr an a21 a22 '''a27' a2.- D= . . .. . arl ar2 III arr ars am ak2 '''akr aks oforder r+1.Ifk<r,thedeterminant Disobviously zero, since it then hastwo identical rows. Similarly, D=0fors<r.Ifk>rand s>r,then thedeterminant Disalso equal tozero, since itisthen aminor oforder r+1ofamatrix ofrank r.Consequently D=0foranyvalues ofkands. Wenow expand Dwith respect toitslastrow, obtaining therelation ak1Ak1 +ak2-'41.-2 'I'''''I'akTAkr +at.-_<-41¢; =05 (22) where thenumbers A11, A12, ...,A1,, A1,denote thecofactors ofthe elements a,,1,a,,2,...,ll)”,(1),,appearing inthelastrowofD.These cofactors 26 DETERMINANTS Cl-IAP. l donotdepend onthenumber k,since they areformed byusing elements a,-1with i<r.Therefore wecanintroduce thenotation Akl :C1,Ak2 :C2:---7Akr :'Cr’Aka :cs- Substituting thevalues k=1,2, ...,ninturn into (22), weobtain the system ofequations ¢'1a11 ‘I‘¢'2a12 ‘I‘'''‘I‘bran ‘I‘csals :0» c1a21 ‘I‘¢‘2a22 *1‘''''I'cram ‘I‘csa2s =0’ (23) clanl -1-c2a'n2 -1-III-1-cranr '1"csans : The number c,=A1,,isdifferent from zero, since A1,isabasis minor ofthe matrix A.Dividing each oftheequations (23)bycs,transposing alltheterms except thelasttotheright-hand side, anddenoting —c,-/c, by1,(j=l,2,..., r),weobtain an=7\1a11 ‘I‘7\2a12 ‘I‘'''+)‘ra1r1 a2.1Z7\1a21 +7\2a22 +'''+)\ra2r1 (24) ans =)\1an1 -1-)‘2an2 -1-III-1-)‘ranr' These equations show that thesthcolumn ofthematrix Aisalinear com- bination oftheﬁrstrcolumns ofthematrix (with coefﬁcients 71,72,...,71,). Theproof ofthetheorem isnow complete, since scanbeanynumber from 1tom. I 1.94. Wearenow inaposition toprove theconverse ofTheorem l.9l (already mentioned attheendofSec. 1.91),: THEOREM. Ifthedeterminant Dvanishes, then ithasatleast onecolumn which isalinear combination oftheother columns. Proof. Consider thematrix ofthedeterminant D.Since D=0,thebasis minor ofthis matrix isoforder r<n.Therefore, after specifying ther basis columns, wecanstillﬁnd atleast onecolumn which isnotoneofthe basis columns. Bythebasis minor theorem, this column isalinear combination ofthebasis columns. Thus wehave found acolumn ofthe determinant Dwhich isalinear combination oftheother columns. I Note thatwecaninclude alltheremaining columns ofthedeterminant D inthislinear combination byassigning them zero coefﬁcients (say). 1.95. Theresults justobtained canbeformulated inasomewhat more symmetric way. Ifthecoefﬁcients 711,712,...,71,,ofalinear combination SEC. 1.9 L1NEAR nE1>ENDENcE BETWEEN COLUMNS 27 ofm columns A1,A2,...,Am(seeSec. 1.91) areequal tozero, then obviously thelinear combination isjust thezero column, i.e.,thecolumn consisting entirely ofzeros. Butitmay also bepossible toobtain thezero column from thegiven columns byusing coefﬁcients 71,72,...,71mwhich arenot allequal tozero. Inthiscase, thegiven columns A1,A2,...,Amarecalled linearly dependent. Forexample, thecolumns 1 '2 1 2 4 1 A1= 35 A2: 6 5 A3= 1 4 8 l arelinearly dependent, since thezero column canbeobtained asthelinear combination 2-A1— 1-A2-I-0-A1,. Amore detailed statement ofthedeﬁnition oflinear dependence isthe following: Thecolumns a11 a12 arnl a21 a22 am2 A1: ', A2= ',.Am: ' anl anm arecalled linearly dependent ifthere exist numbers 71,712,...,71m,notall equal tozero, such that thesystem ofequation 7.1a11 -1-7.2a12 -1-----1-7.ma1m =0, 7\1a21 ‘I‘7\2a22 ‘I‘'''‘I‘7\ma2m =0’ 7\1an1 ‘I‘)\2a'n2 ‘I‘'''‘I‘Amanm =0 issatisﬁed, orequivalently such that 711/11 -1-712/12 -1-'''—I—Am/lm =0, where thesymbol 0ontheright-hand sidedenotes thezero column. Ifone ofthecolumns A1,A2,...,Am,(e.g., thelastcolumn) isalinear combination oftheothers, i.e., Am Z)\1A1 -1-)\2A2 -1-iiI-1-)\1n—1Am—1v 28 DETERMINANTS C}-1AP_ 1 then thecolumns A1,A2,...,Amarelinearly dependent. Infact, (25) is equivalent totherelation A1-'41 ‘I‘A2-42 *1‘‘‘'‘I‘)‘m—1Am—1 TAm=0- Consequently, there exists alinear combination ofthecolumns A1,A2,..., Am, whose coefﬁcients arenotequal tozero (e.g., with thelastcoefﬁcient equal to—l)whose sumisthezero column; thisjustmeans thatthecolumns A1,A2,...,Amarelinearly dependent. Conversely, thecolumns A1,A2,...,Amarelinearly dependent, then (at least) oneofthecolumns isalinear combination oftheother columns. Infact, suppose that intherelation A1-41 ‘I‘A2-'42 ‘I‘''''I')\m—1Am—1 ‘I‘)‘mAm =0 (26) expressing thelinear dependence ofthecolumns A1,A2,...,Am, theco- efﬁcient 7.m,say, isnonzero. Then (26)isequivalent totherelation 71 71 71m_ Aﬂl:vﬁlAl—)TiLA2-IIIVTM-1Am—17 which shows that thecolumn Amisalinear combination ofthecolumns A1,A2,...,Am_1. Thus, ﬁnally, thecolumns A1,A2,...,Amarelinearly dependent andonly oneofthecolumns isalinear combination ofthe other columns. 1.96. Theorems l.9l and l.94 show that thedeterminant Dvanishes if andonly ifoneofitscolumns isalinear combination oftheother columns. Using theresults obtained inSec. 1.95, wehave thefollowing THEOREM. The zleterminant Dvanishes andonly there islinear de- pendence between itscolumns. 1.97. Since thevalue ofadeterminant does notchange when itistrans- posed (seeSec. 1.41), andsince transposition changes columns torows, we canchange columns torows inallthestatements made above. Inparticular, thedeterminant Dvanishes andonly there islinear dependence between itsroirs. PROBLEMS 1.With what signdotheterms 3)a2aaa1“42“5s“14a65- biaaea-1a“14a51a66“25 appear inthedeterminant oforder 6‘? 2.Write down alltheterms appearing inthedeterminant oforder four which have aminus signandcontain thefactor 1121,. PROBLEMS 29 3.With what signdoes theterm a1,,a2_,,_1 ---amappear inthedeterminant of Order n‘? 4.Show thatofthen!terms ofadeterminant oforder n,exactly half(n!/2) have aplus signaccording tothedeﬁnition ofSec. 1.3,while theother half have aminus sign. 5.Usethelinear property ofdeterminants (Sec. 1.44) tocalculate A am+bpan+bq cm+dp cn+dq 6.The numbers 20604, 53227_ 25755, 20927 and 78421 aredivisible by17. Show thatthedeterminant 20604 53227 25755 20927 78421 isalsodivisible byl7. 7.Calculate thedeterminants 21111 246 427 327' 13111 A1= 1014 543 443 , Ag=11411. ~342 721 621 11151 11116 8.Calculate thedeterminant 1 1 2 3 12—x2 2 3 PM72315' 2 3 l9—x2 9.Calculate thenth-order determinant xaa nxa A:aax aaa 30 DETERMINANTS CI-IAP. 1 10.Prove that 1 - - 1 1 - 1 x1 x2 ---x,, x1 x2 -x,, X? xg ...xi X? XE ...X2‘ ,,X2= xk.. . . . . . k=1 1t—2 11-2 ___ "-2 n-2 1|—2 ... 1|—2X1 X X" X1 X2 X“ xi» X; ... X: xi»-1 x;t—2 ...x;t—1 11.Solve thesystem ofequations x1+2x2+3x3+4x4+5x5=13, 2x1+x2+2x3+3x4+4x5 =10, 2x1+2x2+x3+2x4+3x5=11, 2x1+2x2+2x3+ x4+2x5= 6, 2x1+2x2+2x3+2x4+x5=3. 12.Formulate andprove thetheorem which bears thesame relation toLaplace’s theorem asTheorem 1.52bears toTheorem 1.51. 13.Construct fourlinearly independent columns offournumbers each. 14.Show thatiftherows ofadeterminant oforder narelinearly dependent, then itscolumns arealsolinearly dependent. chapter 2 LINEAR SPACES 2.l.Deﬁnitions 2.11. Inanalytic geometry andmechanics oneuses vectors (directed li"_° segments) subject tocertain suitably deﬁned operations. The reader I5 undoubtedly already familiar with themeaning ofthesumoftwovectors and theproduct ofavector and areal number, operations obeying theusual laws ofarithmetic.T The concept ofalinear space generalizes that ofthesetofallvect01'5- Thegeneralization consists ﬁrstingetting away from theconcrete nature °f theobjects involved (directed linesegments) without changing thepropertl¢5 oftheoperations ontheobjects, and secondly ingetting away from U16 concrete nature oftheadmissible numerical factors (real numbers). This leads tothefollowing deﬁnition: AsetKiscalled alinear (oraﬂine) space over aﬁeld Kif a)Given anytwoelements x,yeK,there isarule (the addition rl11°) leading toa(unique) element x+y6K,called thesum ofx andy;I TForthetime being, wearenotconcerned with theother vector operations, nan1¢lY scalar andvector products. lnanyevent, these twoproducts cannot play asbasic arole asthatplayed bytheproduct ofavector andarealnumber. Infact, thescalar product oftwovectors isnolonger avector, while theoperation offorming avector prod!-Wt’ although leading toavector, isnoncommutative. itHere andsubsequently, weusesome notation from settheory. Bya6Awemean that theelement abelongs tothesetA;byBCAwemean thatthesetBisasubset ofthe setA (Bmaycoincide with A).Thetworelations BCAandACBareequivalent totheasS¢1" tion that thesetsAandBcoincide. Thesymbols EandCarecalled inclusion relari0"$- ThefactthatasA(orACB)issometimes written A5a(orB3A).By11¢Awemean thattheelement adoes norbelong tothesetA. 3| 32 LINEAR SPACES CHAP. 2 b)Given anyelement xeKandanynumber AeK,there isarule (the ruleformultiplication byanumber) leading toa(unique) element XxeK, called theproduct oftheelement xandthenumber 71; c)These tworules obey theaxioms listed below inSecs. 2.12 and2.13. The elements ofalinear space willbecalled vectors, regardless ofthe factthattheir concrete nature may bequite unlike themore familiar directed line segments. The geometric notions associated with theterm “vector” willhelp usexplain andoften anticipate important results, aswell asﬁnd a direct geometric interpretation (which would otherwise notbeobvious) of various facts from algebra andanalysis. Inparticular, inthenext chapter wewillobtain asimple geometric characterization ofallthesolutions ofa homogeneous ornonhomogeneous system oflinear equations. 2.12. Theaddition rulehasthefollowing properties: l)x+y=y +xforeveryx,y€K; 2)(x+y)+z=x+ (y+z)foreveryx,y,zeK; 3)There exists anelement 0eK(the zero vector) such that x+0=x forevery xeK; 4)For every x6Kthere exists anelement yeK(thenegative element) such thatx+y:0. 2.13. Theruleformultiplication byanumber hasthefollowing properties: 5)l-x :xforeveryx eK; 6)ot({-3x) :(ml-3)x forevery x6Kandevery ex,{-3eK; 7)(ot+l-3)x=ax+ﬁxforevery xeKandevery at,l-36K; 8)v.(x+y)=ax-{-myforevery x,yeKandevery at6K. 2.14. Axioms l)—8) have anumber ofsimple implications: a.THEOREM. Thezero uecror inalinear space isunique. Proof Theexistence ofatleast onezero vector isasserted inaxiom 3). Suppose there aretwozero vectors 01and02inthespace K.Setting x:0,, 0:02inaxiom 3),weobtain 0,+02=0,. Setting x=02,0=0,inthesame axiom, weobtain 02+0,=02. sEc.2.1 DEFINITIONS 33 Comparing theﬁrstofthese relations with thesecond and using axiom 1), weﬁndthat01=02. I b.THEOREM. Every element inalinear space hasaunique negative. Proof. The existence ofatleast one negative element isasserted in axiom 4).Suppose anelement xhastwonegatives yjandyz.Adding yzto both sides oftheequation x+y', =0andusing axioms l)—3), weget y2+(x+yl)=£v2+x) +yi=0+yi=yi, y2+(x+yr)=y2+0=y2, whence y,=y2. I c.THEOREM. Therelation 0-x=0 holds forevery element xinalinear space.T Proof Consider theelement 0-x +1-x. Using axioms 7)and 5),we get 0-x+l-x=(0+ l)-x=l'x=x, 0-x+l-x=0'x+x, whence x=0-x+x. Lety bethenegative ofx,and addy toboth sides ofthelastequation. Then 0:x+y=(0'x+x)+y:0'x+(x+y):0-x+0=0'x, whence 0=0-x. | d.THEOREM. Given anyelement xofalinear space, theelement y:(—l)-x serves asthenegative ofx. Proof Form thesum x+y. Using theaxioms and Theorem 2.l4c, weﬁndthat x-l-y:l-x+(~l)-x:(l—l)-x=0-x:0.I TIntheright-hand sideoftheequation, 0denotes thezero vector, andintheleft-hand sidethenumber 0. 34 LINEAR SPACES CI-IAP. 2 e.The negative ofagiven element xwillnow bedenoted by—x,since Theorem 2.l4d makes thisanatural notation. The presence ofanegative allows ustointroduce theoperation ofsubtraction, i.e.,thedifference x—y isdefined asthesum ofxand —y. This definition agrees with the definition ofsubtraction inarithmetic. 2.15. Alinear space over theﬁeld Rofrealnumbers willbecalled real anddenoted bythesymbol R.Alinear space over theﬁeld Cofcomplex numbers willbecalled complex anddenoted bythesymbol C.Ifthenature oftheelements x,y,2,...andtherules foroperating onthem arespeciﬁed (where axioms l)—8) must besatisﬁed), then wecallthelinear space concrete. Asarule, such spaces willbedenoted bytheir own special symbols. Thefollowing four kinds ofconcrete spaces willbeofparticularimportance later: a.Thespace V3.Theelements ofthisspace arethefreevectors studied inthree-dimensional analytic geometry. Each vector ischaracterized bya length andadirection (with theexception ofthezero vector, whose length iszero andwhose direction isarbitrary). Addition ofvectors isdeﬁned in theusual way bytheparallelogram rule. Multiplication ofavector bya number Aisalso deﬁned intheusual way, i.e.,thelength ofthevector is multiplied by|)tl,while itsdirection remains unchanged ifA>0and is reversed ifA<0.Itiseasily veriﬁed that alltheaxioms l)—8) aresatisﬁed inthiscase. Wedenote theanalogous sets oftwo-dimensional and one- dimensional vectors, which arealsolinear spaces, byV2andV1,respectively; V1,V2andV3arelinear spaces over theﬁeld Rofrealnumbers. b.Thespace Kn.Anelement ofthisspace isanyordered n-tuple x:(€1’€2’---sin) ofn numbers from theﬁeld K.The numbers Z1,Z2,...,2,,arecalled the components oftheelement x.Theoperations ofaddition andmultiplication byanumber AeKarespeciﬁed bythefollowing rules: (€1s€2v'-- ,5,,.)+(‘/11,‘/12,---,m)=(€l+ ‘/11,52+ "12,---, €,.+*/1») (1) )‘(€1s€2v ---1in) = (Ail! )‘€2’ '''2 Itiseasily veriﬁed that axioms l)—8) aresatisﬁed. Inparticular, theelement 0isthen-tuple consisting ofn zeros: 0=(0,0,...,0). Actually, wedealt with elements ofthisspace inSec. 1.9,except that we wrote them there intheform ofcolumns ofnumbers rather than rows of numbers. IfKistheﬁeld Rofreal npmbers, wewrite Rninstead ofKn, while ifKistheﬁeld Cofcomplex numbers, wewrite C"instead ofK". SEC.2.1 DEFINITIONS 35 c.Thespace R(a, b).Anelement ofthisspace isanycontinuous real function x=x(t) deﬁned ontheinterval a<t<b.The operations of addition offunctions and multiplication offunctions byreal numbers are deﬁned bytheusual rules ofanalysis, anditisobvious that axioms l)—8) are satisﬁed. Inthiscase, theelement 0isthefunction which isidentically zero. Thespace R(a, b)isalinear space over theﬁeld Rofrealnumbers. d.Correspondingly, thespace C(a, b)isthespace ofallcontinuous complex-valued functions ontheinterval a<t<b.This space isalinear space over theﬁeld Cofcomplex numbers. 2.16. Wenote thatalltheproperties ofelements ofconcrete spaces (e.g., thevectors ofthespace V3)which arebased only onaxioms l)—8) arealso valid fortheelements ofanarbitrary linear space. Forexample, analyzing theproof ofCramer’s ruleforsolving thesystem oflinear equations auxi +a12-x2 *1‘'''*1‘alnxn :bl: a21x1 +a22x2 +'''+a2nxn =I72, anlxl +an2x2 +I'.+annxn =bn’ weobserve that insofar asthequantities bl,b2,...,b,,areconcerned, the proof isbased only onaxioms l)—8) andthefactthatthese quantities canbe added andmultiplied bynumbers inK.Ashasalready been pointed outin Sec. 1.74, thispermits ustogeneralize Cramer’s- ruletosystems inwhich the quantities bl,b2,...,b,,arevectors (elements ofthespace V3). Furthermore, thispermits ustoassert thatCramer’s ruleisalsovalid forsystems inwhich theelements bl,b2,...,b,,areelements ofanylinear space K.Wenote only that then thevalues oftheunknowns xl,x2,...,x,,arealso elements ofthespace K,andinfactcanbeexpressed linearly interms ofthequantities bl,b2,...,b,,. 2.17. Remark. Inanalytic geometry, itissometimes convenient tocon- sider vectors which arenotfreebuthave their initial points attached tothe origin ofcoordinates. Theconvenience ofthisapproach isthatevery vector isthen associated with apoint ofspace, namely itsendpoint, andevery point ofspace canbespeciﬁed bygiving thecorresponding vector, called the radius vector ofthepoint. With thispicture inmind, wesometimes callthe elements ofalinear space points instead ofvectors.T Ofcourse, thischange interminology isnotaccompanied byanychange whatsoever inthedeﬁnitions, andmerely appeals toourgeometric intuition. TWethen talkofthe“coordinates” ofapoint, rather than ofthe“components” ofa vector. 36 LINEAR SPACES C1-1AP_ 2 2.2.Linear Dependence 2.21. Letxl,x2,...,xl.bevectors ofthelinear space Kover aﬁeld K, andlet<1l,<12,...,<12benumbers from K.Then thevector y:<X1X1+<><2x2+'""+<=<1.x1. iscalled alinear combination ofthevectors xl,x2,...,xl,andthenumbers <1l,<12,...,<12arecalled thecoeﬂicients ofthelinear combination. lf<11=<12=---=<1,:0,then y=0byTheorem 2.140. However, there may exist alinear combination ofthevectors xl,x2,...,xl,which equals thezero vector, even though itscoefﬁcients arenotallzero. Inthis case, thevectors xl,x2,...,xl,arecalled linearly dependent. Inother words, thevectors xl,x2,...,x,,aresaid tobelinearly dependent ifthere exist numbers <1l,<12,...,<12,notallequal tozero, such that °l1x1 +°l2x2 -1‘‘‘‘+akxlc :0- (3) If(3)ispossible only inthecase where <1l=<12= —otk=0, thevectors xl,x2,...,xl,aresaidtobelinearly independent (over K). 2.22. Examples a.Inthelinear space V3,linear dependence oftwovectors means that they areparallel tothesame straight line. Linear dependence ofthree vectors means that they areparallel tothesame plane. Any four vectors arelinearly dependent. b.Wenow explain what ismeant bylinear dependence ofthevectors xl,x2,...,xl,ofthelinear space Kn.Letthevector x,have components El”, El”,...,Z5,”(i=1,2,...,k).Then thelinear dependence expressed by °l1x1 +°l2x2 'i'"""'i'“Ex/.~ :0 means that thenequations (1) (2) t___ (lc)_OHE1 +QQE1 T +“Til v0, V-1:?) 'i'1-2&5?) *1‘‘'‘'i*lkghk) :0, (4) Y(1) t “(2) __, (lc) 7 “1Q,, ?"°12t.tn + +1k€,, —0 hold, where theconstants <1l,<12,...,mlarenotallequal tozero. This is thesame deﬁnition oflinear dependence asthat given inSec. 1.95 for columns ofnumbers. SEC.2.2 LINEAR DEPENDENCE 37 Thus theproblem ofwhether ornotthevectors xl,x2,...,xkare linearly dependent reduces inthegeneral case totheproblem ofwhether or notthere exists anontrivial solution ofthehomogeneous system ofequations (4),T with coefﬁcients equal tothecorresponding components ofthegiven vectors. This problem willbesolved completely inSec. 3.21, where wewill ﬁndaruleallowing ustodecide whether ornotgiven vectors inthespace K,, arelinearly dependent from anexamination oftheir components. c.Insome cases, however, wecaneven now decide whether ornota given system ofvectors islinearly dependent. Forexample, consider then vectors el: (l,0,0,...,0), @2:(0,1,o,...,o), e,,=(0,0,0,...,l) inthespace Kn.Forthese vectors, thesystem (4)hastheform <1l-1+<12~0+<13~0+~-~+<1,,-0:0, otl"0+v.2"l—l—<13'0+'""+m,,"0:0, ml-0+<12'0+<13"0+"""+<1,,-l=O, andobviously hastheunique solution ot1:ot2="“:<1,,:0. Thus thevectors el,e2,...,e,,inthespace K,,arelinearly independent. d.Linear dependence ofthevectors xl=xl(t), x2:x2(t), ...,xl,=x,,(t) inthespace R(a, b)(orC(a, b))means that thefunctions xl(t), x2(t), ..., x,,(t) satisfy arelation oftheform ”~1x1(t) +°l2x2(t) 'i'‘‘"-1‘°l1<-Xk(t) E0, where theconstants <1l,<12,...,<1l.arenotallequal tozero. Forexample, thefunctions xl(t) :cos2 t, x2(t) =sin?t, x3(t) =l arelinearly dependent, since therelation x1(t) +x2(t) —xa(t) 5'0 TConcerning theterms “homogeneous” and“nontrivial,” seeSec.2.42e. 38 LINEAR SPACES Q1-1AP_ 2 holds. Ontheother hand, aswenow show, thefunctions 1,t,t2,...,t" arelinearly independent. Infact, suppose there exists arelation <10-l—l—otlt—l—""'+ot,,t"EO. (5) Then, bysuccessively diﬁerentiating (5)ktimes, weobtain asystem of k+lequations inthequantities <10,<1l,...,<1,,,with adeterminant which isclearly diﬁerent from zero (recall Sec. l.55b). Solving this system by Cramer’s rule (Sec. 1.75), weﬁndthat <10=otl:--~=ot,,=0. Consequently, thefunctions l,t,t2,...,t"arelinearly independent inthe space R(a,b),asasserted. 2.23. Next wenote two simple properties ofsystems ofvectors, both involving thenotion oflinear dependence. a.LEMMA. Ifsome ofthevectors xl,x2,...,x,,arelinearly dependent, then thewhole system xl,x2,...,x,,isalsolinearly dependent. Proof. Without loss ofgenerality, wecan assume that thevectors xl,x2,...,x,»(j<k)arelinearly dependent. Thus there isarelation °‘1x1+ °l2x2-1‘ +09;/2:0, where atleast oneoftheconstants <1l,<12,...,<1,isdiﬁerent from zero. By Theorem 2.14c andaxiom 3)ofSec.2.12, wehave <1lxl+<12x2+---+<1,-x,-+0-x,+l+---+0-x,,=0. But then thevectors xl,x2, ...,x,,arealso linearly dependent, since at least oneoftheconstants <1l,<12,...,<1,~,0,...,0isdiﬁerent from zero. I b.LEMMA. Thevectors xl,x2,...,x,,arelinearly dependent ifandonlyif oneofthevectors canbeexpressed asalinear combination oftheothers. Proof. Asimilar statement hasalready been encountered; infact, itwas proved forcolumns ofnumbers inSec. 1.95. Inspecting theproof given there, weseethatitisbased only onthepossibility ofperforming oncolumns theoperations ofaddition and multiplication byreal numbers. Hence the proof canbecarried through fortheelements ofanylinear space, i.e.,our lemma isvalid foranylinear space. I 2.3. Bases, Components, Dimension 2.31. Bydeﬁnition, asystem oflinearly independent vectors el,e2,..., e,,inalinear space Kover aﬁeld Kiscalled abasis forKif,given anyxeK, sEc.2.3 BASES, COMPONENTS, DIMENSION 39 there exists anexpansion x=€1e1+€2e2+"'+€rie1t (€ieK,j:1,2,---,")- (6) Itiseasy toseethat under these conditions thecoeﬂicients intheexpansion (6)areuniquely determined. Infact, ifwecanwrite twoexpansions x=£131 +€2e2 -1‘ -1‘£1.91.’ x=7l1e1 +7l2e2+‘""+“/11.3,. foravector x,then, subtracting them term byterm, weobtain therelation 0=(£1—"l1)e1 -1"(£2—7l2)e2 -1"'""'1‘(E1:—7l1t)e1n from which, bytheassumption that thevectors el,e2,...,e,,arelinearly independent, weﬁndthat £11‘/l1’€2=7l2»-~ ,€,,:7l,,- Theuniquely deﬁned numbers E1,E2,...,E,,,arecalled thecomponents ofthe vector xwith respect tothebasis el,e2,...,e,,. 2.32. Examples a.Afamiliar basis inthespace V3isformed bythethree orthogonal unit vectors i,j,k.Thecomponents El,E2,E3ofavector xwith respect tothis basis aretheprojections ofXalong thecoordinate axes. b.Anexample ofabasis inthespace K,,isthesystem ofvectors el=(1,0,...,0), e2: (0,1,...,0), e,,:(0,0,...,1), already considered inSec. 2.22c. Indeed itisobvious that therelation x=E1(1,0,...,0)-T-E2(0,1,...,0)-T-~~~+E,,(0,0,...,1) holds forevery vector x:(€1s€2a" -~9€rt)eKn' This fact, together with thelinear independence ofthevectors el,e2,...,e,, already proved, shows that these vectors form abasis inthespace K,,.In particular, weseethat thenumbers El,E2,...,E,,arejustthecomponents ofthevector xwith respect tothebasis el,e2,...,e,,. c.Inthespace R(a, b)there does notexist abasis inthesense deﬁned here. Theproof ofthisstatement willbegiven inSec.2.36c. 2.33. The fundamental signiﬁcance oftheconcept ofabasis foralinear space consists inthefact that when abasis isspeciﬁed, theoriginally ab- stract linear operations inthespace become ordinary linear operations with 40 LINEAR SPACES Q}-1AP_ 2 numbers, i.e.,thecomponents ofthevectors with respect tothegiven basis. Infact, wehave thefollowing THEOREM. When twovectors ofalinear space Kareadded, their components (with respect toanybasis) areadded. When avector ismultiplied byanumber )1,allitscomponents aremultiplied by71. Proof. Let x=£131 'i'£232 'i'"''-1‘E03,,’ y:71131 +71232 41''''+7l1t3tt- Then X-1'y=(£1-1"7l1)31 -1"(£2-1"7l2)32 -1" "'1'(£0'1'7ln)3m xx:X3131 'i'X2232 'i'‘‘‘-1‘7\€n3,,, bytheaxioms ofSecs. 2.12 and2.13. I 2.34. Ifinalinear space Kwecanﬁnd nlinearly independent vectors while every n+1vectors ofthe space arelinearly dependent, then thenumber niscalled thedimension ofthespace Kandthespace Kitself iscalled n- dimensional. Alinear space inwhich wecanﬁndanarbitrarily large number oflinearly independent vectors iscalled inﬁnite-dimensional. THEOREM. Inaspace Kofdimension nthere exists abasis consisting ofn vectors. Moreover, anysetofnlinearly independent vectors ofthespace Kisa basis forthespace. Proof. Letel,e2,...,e,,beasystem ofnlinearly independent vectors ofthegiven n-dimensional space K.Ifxisanyvector ofthespace, then the setofn +lvectors x,el,e2,...,e,, islinearly dependent, i.e.,there exists arelation oftheform ‘box-1‘@131 +@232 'i''''+‘M311 =0: (7) where atleast oneofthecoefﬁcients <10,<1l,. ..,<1,,isdifferent from zero. Clearly <10isdifferent from zero, since otherwise thevectors el,e2,...,e,, would belinearly dependent, contrary tohypothesis. Thus, intheusual way, i.e.,bydividing (7)by<10andtransposing alltheother terms tothe other side, weﬁnd that xcanbeexpressed asalinear combination ofthe vectors el,e2,...,e,,.Since xisanarbitrary vector ofthespace K,wehave shown that thevectors el,e2,...,enform abasis forthespace. I 2.35. Thepreceding theorem hasthefollowing converse: THEOREM. Ifthere isabasis inthespace K,then thedimension ofKequals thenumber ofbasis vectors. SEC_2_3 BASES, COMPONENTS, DIMENSION 41 Proof. Letthevectors el,e2,...,e,,beabasis forK.Bythedeﬁnition ofabasis, thevectors el,e2,...,e,,arelinearly independent; thus wealready have nlinearly independent vectors. Wenow show that anyn+1vectors ofthespace Karelinearly dependent. - Suppose wearegiven n+1vectors ofthespace K: ( (1) x1 Z€i1)e1 +Y€21)e2 +...+in en’ I) 1 ... (2)x2=£1231'i'€22’32 + -1-in311’ _vi+1) (+1) ___ (+1)xn~l>-1 —Q1" 31'i'£2” 32'i' *1‘£11" 311- Writing thecomponents ofeach ofthese vectors asacolumn ofnumbers, we form thematrix £11) £52) ___€:ll"l'1i 6:1) $2) ___ggwl) £11) €(2> ...€(1i+1) 7! YL H with nrows andn+1columns. Thebasis minor ofthematrix A(seeSec. 1.92) isoforder r<n.Ifr:0,thelinear dependence isobvious. Let r>0.After specifying therbasis columns, wecanstillﬁnd atleast one column which isnotoneofthebasis columns. Butthen, according tothe basis minor theorem, this column isalinear combination ofthebasis columns. Thus thecorresponding vector ofthespace Kisalinear combination ofsome other vectors among thegiven xl,x2,...,x,,+l. Butinthiscase, according toLemma 2.23b, the vectors xl,x2,...,x,,+l are linearly dependent. IA: a.The space V0isthree-dimensional, since ithasabasis consisting of thethree vectors i,j,k(seeExample 2.32a). Similarly, V2istwo-dimensional and Vlisone-dimensional. b.Thespace Knisn-dimensional, since itcontains abasis consisting of thenvectors el,e2,...,e,,(seeExample 2.32b). c.Ineach ofthespaces R(a, b)andC(a, b),there isanarbitrarily large number oflinearly independent vectors (seeExample 2.22d), andhence these spaces areinﬁnite-dimensional. Therefore neither space hasabasis, forthe presence ofabasis would contradict Theorem 2.35. d.Every complex linear space Cisobviously arealspace aswell, since thedomain ofcomplex numbers contains thedomain ofreal numbers. However, thedimension ofCasacomplex space does notcoincide with that ofCasarealspace. Infact, ifthevectors el,...,e,,arelinearly independent inCregarded asacomplex space, then thevectors el,iel,...,e,,,ie,,are 42 LINEAR SPACES CHAP. 2 linearly independent inCregarded asarealspace. Hence thedimension of Cregarded asareal space istwice aslarge asthat ofCregarded asa complex space (provided thedimension isﬁnite). 2.4.Subspaces 2.41. Suppose thatasetLofelements ofalinear space Khasthefollowing properties: a)IfxeL,y€L,thenx+y€L; b)IfxeLand7.isanelement oftheﬁeldK,then).x.eL. Thus Lisasetofelements with linear operations deﬁned onthem. Wenow show that thissetisalso alinear space. Todoso,wemust verify that the setLwith theoperations a)andb)satisﬁes theaxioms ofSecs. 2.12 and 2.13, Axioms 1),2)and 5)—8) aresatisﬁed, since they hold quite generally forallelements ofthespace K.Itremains toverify axioms 3)and4).Let xbeanyelement ofL.Then, byhypothesis, 7.x6Lforevery A6K.First wechoose 71:0.Then, since 0-x=0byTheorem 2.l4c, thezero vector belongs tothesetL,i.e., axiom 3)issatisﬁed. Next wechoose A=—1. Then, byTheorem 2.l4d, (—l)xisthenegative oftheelement x.Thus, if anelement xbelongs tothesetx,sodoes thenegative ofx.This means that axiom 4)isalso satisfied, sothat Lisalinear space, asasserted. Consequently, every setLCK with properties a)and b)iscalled alinear subspace (orsimply asubspace) ofthespace K. 2.42. Examples a.Thesetwhose only element isthezero vector ofthe space Kisobviously thesmallest possible subspace ofK. b.Thewhole space Kisthelargest possible subspace ofK. These twosubspaces ofK,thewhole space andtheset{0}consisting of thezero vector alone, aresometimes called trivial subspaces. Alltheother subspaces ofKarethen saidtobenontrivial. c.LetLlandL2betwosubspaces ofthesame linear space K.Then the setofallvectors x6Kbelonging toboth LlandL2forms asubspace called theintersection ofthesubspaces LlandL2.Thesetofallvectors oftheform y+z,wherey eLl,zeL2forms asubspace, denoted byLl+L2andcalled thesumofthesubspaces LlandL2. d.Allthevectors inthespace V0parallel toaplane (oraline) form a subspace. Ifwetalkabout points rather than about vectors, asinSec.2.17, then thesubspaces ofV0arethesetsofpoints lying onsome plane (orline) passing through theorigin ofcoordinates. $EC-2-4 SUBSPACES 43 e.Consider thesetLofallvectors (El,E2,...,E,,)inthespace K,whose coordinates satisfy asystem oflinear equations oftheform a11x1 *1‘a12x2 ‘i*‘‘‘*1‘alnxn =0, a21x1 *1‘a22x2 *1‘''‘*1‘a2'nx'n =0, ....................... (8) aklxl *1‘(112752 *1‘'''*1‘aknxn =0. with coefﬁcients intheﬁeld Kand constant terms equal tozero. Such a system iscalled ahomogeneous linear system. Ahomogeneous linear system isalways compatible, since itobviously hasthe“trivial” solution xl=x2=~--=x,,=0. Letcl",cl“,...,cl,“andcl”,cf’, ...,cf’betwosolutions ofthissystem, andform thenumbers t < <1 (2 _<1» <2»cl=cl“+cl2’,c2=c2’-T-c2’,...,c,,-c,, -T-c,,. Then clearly cl,c2,...,c,,isagain asolution ofthesystem (8).Infact, substituting these numbers into thei'thequation ofthesystem, weobtain ailcl *1‘(11232 *1‘""'*1‘aincn =ai1(3iU ‘l‘312’) ‘l‘ai2(32n ‘i‘322’) *1‘''''1‘am(3itn *1‘31.2’) =(ai13in *1‘(112321) ‘l‘'''‘l‘amcitn) *1‘(@1312, *1‘(112322) *1‘'''‘l‘amc(1t2)) =0’ asasserted; thissolution willbecalled thesumofthesolutions cl“,cl",..., cl,“andcf’,cl”,...,cf’. Similarly, ifcl,c2,...,c,,isanarbitrary solution ofthesystem (8),then thenumbers )tcl,)tc2,...,7tc,,alsoform asolution of (8)forevery ﬁxed AeK;thissolution willbecalled theproduct ofthesolution cl,c2,...,c,,and thenumber 7..Thus solutions ofahomogeneous linear system (8)with coeﬂicients andconstant terms inagiven ﬁeld Kcanbeadded tooneanother andmulti'pli'ed bynumbers from thesame ﬁeld K,withthe result stillasolution of(8).Inother words, thesetLisasubspace ofthe space Kn,andhence alinear space initsown right. WewillcallLthesolution space ofthesystem (8).InSec. 3.41 wewillcalculate thedimension ofthis space andconstruct abasis forit. 2.43. Wenow consider some properties ofsubspaces which arerelated tothedeﬁnitions ofSecs. 2.2and2.3. First ofall,wenote that every linear relation which connects thevectors x,y, ...,zinasubspace Lisalso valid inthewhole space K,andconversely. Inparticular, thefactthatthevectors x,y, ...,zeLarelinearly dependent holds true simultaneously inthe subspace Landinthespace K.Forexample, ifevery setofn+1vectors is 44 LINEAR SPACES Cl-IAP. 2 linearly dependent inthespace K,then thisfactistrue afortiori inthesub- space L.Itfollows that thedimension ofanysubspace Lofann-dimensi'onal space Kdoes notexceed thenumber n.According toTheorem 2.34, inany subspace LCKthere exists abasis with thesame number ofvectors asthe dimension ofL. Ofcourse, ifabasis el,e2,...,e,,ischosen inK,then inthe general case wecannot choose thebasis vectors ofthesubspace Lfrom the vectors el,e2,...,en,because none ofthese vectors may belong toL. However, itcan beasserted that ifabasis fl,f2,. ..,f,ischosen i'nthe subspace L(which, tobeexplicit, isassumed tohave dimension l<n),then additional vectors fl+l,...,f,,canalways bechosen i'nthewhole space Ksuch thatthesystemfl,f2, ...,f,,...,f,,isabasisfor allofK. Toprove this, weargue asfollows: Inthespace Kthere arevectors which cannot beexpressed aslinear combinations offl,ﬁ2, ...,f,.Indeed, if there were nosuch vectors, then thevectorsfl,f2, ...,f,,which arelinearly independent byhypothesis, would constitute abasis forthespace K,andthen byTheorem 2.35 thedimension ofKwould belrather than n.Letf,+l be any ofthevectors that cannot beexpressed asalinear combination of fl,ﬁ2, ...,f,.Then thesystemfl,f2, ...,f,,f,+l islinearly independent. In fact, suppose there were arelation oftheform °‘1f1 ‘i*°‘2f2 ‘i*""'‘i*“if: ‘i*°¢i+1fl+1 :0- Then if<1,+l¢0,thevectorfl+l could beexpressed asalinear combination offl,f2, ...,f,,while if<1,+l=0,thevectorsfl,f2, ...,f,would belinearly dependent. Butboth these results contradict theconstruction. Ifnow every vector ofthespace Kcanbeexpressed asalinear combination offl,f2, ..., f,,f,+l, then thesystemfl,f2, ...,f,,f,+l forms abasis forK (andl +1:n), which concludes ourconstruction. Ifl+1<n,then there isavector f,+2 which cannot beexpressed asalinear combination offl,f2, ...,f,,f,_.rl, and hence wecancontinue theconstruction. Eventually, after n—~lsteps, we obtain abasis forthespace K. 2.44. Wesaythat thevectors gl,....g,.arelinearly independent over thesubspace LCKiftherelation <1lgl+---+<1,,g,,€L (<1l,...,rX;,€K) implies <1l:---:<1,,:0. IfListhesubspace consisting ofthe zero vector alone, then linear independ- ence over Lmeans ordinary linear independence. Linear dependence ofthe vectors gl,...,g,,over thesubspace Lmeans that there exists alinear combination <1lgl +----1-<1,,g,, belonging toL,where atleast oneofthe coefﬁcients <1l,...,<10isnonzero. sEc.2.4 SUBSPACES 45 Thelargest possible number ofvectors ofthespace Kwhich arelinearly independent over thesubspace LCKiscalled thedimension ofKover L. Ifthevectors gl,...,g,,arelinearly independent over thespace LCK andifthevectors fl,...,f,arelinearly independent inthesubspace L,then thevectors gl,....gkifla ...,f,arelinearly independent inthewhole space K.Infact, ifthere were arelation oftheform °‘1f1*l*"'+“ifi*l‘i'31g1‘i*"'"l‘i5kgk:0, orequivalently l51g1+"'" *1‘15181: '(°l1f1*l""*l* ¢;ft)EL1 then l51=""‘=l5i.=0. bytheassumed linear independence ofthevectors gl,...,gkover L.It follows that <1l=--~:<1,:0,bythelinear independence ofthevectors ./1a--- afl' Thevectorsf,+l, ....f,,Constructed inSec.2.43 arelinearly independent over thesubspace L.Infact, ifthere were arelation oftheform °‘i+1fz+1‘tL""l‘°‘1tfn:°‘1f1+"'*l‘°‘tfi with atleast oneofthenumbers <1l+l, ...,<1,notequal tozero, then the vectors fl,...,f,,would belinearly dependent, contrary totheconstruction. Hence thedimension ofthespace Kover Lisnolessthan nvl.Onthe other hand, thisdimension cannot begreater than n—l,since ifn—l+1 vectors hl,...,h,,_,+l, say, were linearly independent over L,then the vectors hl,...,h,,L,+l,fl, ...,f,,ofwhich there aremore than n,would be linearly independent inK.Therefore thedimension ofKover Lisprecisely nYl. 2.45. The direct sum. Wesaythat alinear space Listhedirect sum ofgiven subspaces Ll,...,L,,,CLif a)Forevery xeLthere exists anexpansion X:-\'1+"'-l-Km. where xleLl, ...,x,,,eL,,,; b)This expansion isunique, i.e.,if x:x1‘i*"'"l‘xm=y1‘i"" 'i')/m wherex,-EL,-,y,»€L,-(j= l,...,m),thcn xliylﬁ-"9xm:ym‘ 46 LINEAR SPACES C]-1AP_ 2 However, thevalidity ofcondition b)isaconsequence ofthefollowing simpler condition: b’)If 0=Z1-1-'''-1-Z... where zleLl,...,z,,,eLm, then zl:---=z,,,=O. Infact, given two expansions x:xl+~--+x,,,, x=yl+----T-ym, suppose b’)holds. Then subtracting thesecond expansion from theﬁrst, weget 0:(x1 —)/1) -1-_"+(xm i)/m)» and hence xl=yl,. ..,x,,,=y,,,,because ofb’).Conversely, b’)follows fromb)ifwesetx=0,xl=---=x,,,=0. Itfollows from condition b)that every pair ofsubspaces Ll,...,Lm hasonly theelement 0incommon. Infact, if26L,-andzeLl,then using b)andcomparing thetwoexpansions z=z—l-0, 26L,-, 0€L,,, z=O+z, OEL,-, z€L,,, weﬁndthat z=0. Thus ann-dimensional space K,,isthedirect sum ofthenone-dimensional subspaces determined byanynlinearly independent vectors. Moreover, the space K,,canberepresented invarious ways asadirect sum ofsubspaces not allofdimension 1. 2.46. LetLbeaﬁxed subspace ofann-dimensional space K,,.Then there always exists asubspace MCK,,such that thewhole space K,,isthedirect sum ofL andM.Toprove this, weusethevectors fl+l, ...,f,,constructed inSec. 2.43, which arelinearly independent over thesubspace L.LetMbe thesubspace consisting ofalllinear combinations ofthevectors f,+l, ...,f,,. Then Msatisﬁes thestipulated requirement. Infact, since thevectors fl,. ..,f,,form abasis inK,(see Sec. 2.43), every vector xELhasan expansion oftheform x=°l1f1*1““‘1*011/1*1‘°l1+1fz+1*1‘"‘"*1‘°‘1tfn:)/*1‘Z, where )’:°‘1f1*1"'"‘1*°‘ifieL’ Z=°li+1fl+1*1*"'*1‘°‘1tfn5Nl- Moreover x=0implies <1l= _<1,=0,since thevectors fl,...,f,, arelinearly independent. Therefore conditions a)—b’) ofSec. 2.45 are satisﬁed, sothat K,,isthedirect sum ofLandM. sEc.2.4 SUBSPACES 47 2.47. a.Ifthedimension ofthespace Ll,equals rl,(k=1,...,m)and ifrl,linearly independent vectors fl,l,...,fl,,kareselected ineach space Ll,, then every vector xofthesum L=Ll+--~+Ll,canbeexpressed asa linear combination ofthese vectors. Hence thedimension ofthesum ofthe spaces Ll,...,Ll,does notexceed thesum ofthedimensions oftheseparate spaces. Ifthe sumLl+~~~+Ll,isdirect, then thevectorsfll, ...,fl,1, ..., fl,l,...,fl,,k, ...,f,,,l, ...,f,,,,m arealllinearly independent, sothat inthis case thedimension ofthesumisprecisely thesum ofthedimensions. b.Inthegeneral case, thedimension ofthesum isrelated tothedimen- sions ofthesummands inamore complicated way. Here weconsider only theproblem ofdetermining thedimension ofthesumoftwoﬁnite-dimensional subspaces PandQofthespace K,ofdimensions pandq,respectively. Let Lbetheintersection ofthesubspaces PandQ,andletLhave dimension l. First wechoose abasis el,e2,...,elinL.Then, using theargument of Sec.2.43, weaugment thebasis el,e2,...,elbythevectorsfl+l,fl+2, ...,f,, tomake abasis forthewhole subspace Pandbythevectors gl+l,gl+2, ...,gl, tomake abasis forthewhole subspace Q.Bydeﬁnition, every vector inthe sum P+Qisthesumofavector from Pandavector from Q,andhence can beexpressed asalinear combination ofthevectors el,...,el,f,+l,...,f,,,g,+l,...,g,. (9) Wenow show that these vectors form abasis forthesubspace P+Q.To show this, itremains toverify their linear independence. Assume that there exists alinear relation oftheform @131 ‘1‘'''‘1‘“I31 *1‘151+-1ft+1 ‘1‘''' *1‘151/; *1‘Yi+18v+1 ‘1"" *1‘Yagq =0’(10) where atleast oneofthecoefﬁcients <1l,...,Y,isdifferent from zero. We canthen assert thatatleast oneofthenumbers ylll, ...,Y,isdifferent from zero, since otherwise thevectors 31,---,3I,fi+1» ---Qfp would belinearly dependent, which isimpossible inview ofthefactthatthey form abasis forthesubspace P.Consequently thevector x=Yt+1gi+1‘1"" +Yaga¢0’ (11) forotherwise thevectors gl+l, ...,g,would belinearly dependent. Butit follows from (10)that —x=°l131*1‘"‘"‘1*15ofpeP, while (1l)shows that xeQ.Thus xbelongs toboth PandQ,andhence belongs tothesubspace L.Butthen X=Yi+1Zt+1'1‘ '‘‘"1‘Yaga =A131*1‘‘‘‘‘1‘7231» 48 LINEAR SPACES (jl-1AP_ 2 andsince thevectors 31,[email protected]+1. --.g0 arelinearly independent, wehave Yt+1:""":Y.:°- This contradiction shows thatthevectors (9)areactually linearly independent, and hence form abasis forthesubspace P+Q.Itfollows from Theorem 2.35 that thedimension ofP+Qequals thenumber ofbasis vectors (9). Butthisnumber equals p+qvl.Thus, ﬁnally, thedimension ofthesum oftwosubspaces isequal tothesum oftheir dimensions minus thedimension oftheir intersection. c.COROLLARY. LetR,,andR,betwosubspaces ofdimensions pandq, respectively, ofann-dimensional space Rn,and suppose p+q>n.Then theintersection ofR,,andR0isofdimension nolessthanp+q—n. 2.48. Factor spaces a.Given asubspace Lofalinear space K,anelement x6Kissaidtobe comparable with anelement yeK(more exactly, comparable relative toL) ifxvyeL. Obviously, ifxiscomparable with y,then yiscomparable with x,sothat therelation ofcomparability issymmetric. Every element xeKiscomparable with itself. Moreover, ifxiscomparable with yandy iscomparable with z,then xiscomparable with 2,since x—z=(x—y)+(yv2)€L. b.Thesetofallelements y6Kcomparable withagiven element xeK iscalled aclass, andisdenoted byX.Asjustshown, aclass Xcontains the element xitself, andevery pairofelements yeX,2eXarecomparable with each other. Moreover, ifu¢X,then uisnotcomparable with anyelement ofX.Therefore two classes either have noelements incommon orelse coincide completely. The subspace Litself isaclass. This class isdenoted by0,since itcontains thezero element ofthespace K. c.The whole space Kcanbepartitioned into asetofnonintersecting classes X,Y,....This setofclasses will bedenoted byK/L. Wenow introduce linear operations inK/L asfollows: Given twoclasses X,Yand twoelements <1,Boftheﬁeld K,wewish todeﬁne theclass <1X+l-3Y.Todo this, wechoose arbitrary elements xeX,yeYand ﬁnd theclass Zcon- taining theelement 2=<1x+l-3y.This class isthen denoted by<1X+T-BY. Clearly, <1X+T-BYisuniquely deﬁned. Infact, suppose wechoose another element xloftheclass Xandanother element yloftheclass Y.Then (<1-\'1T511)Y(1-Y+151')I<=<(X1—X)+150/1—Y) sEc.2.5 LINEAR MANIFOLDS 49 belongs tothespace L,since xlYxandyl—yboth belong toL.Itfollows that <1xl+Bylbelongs tothesame class asax+T-3y. Inparticular, theabove prescription deﬁnes addition oftwo classes X andY,aswellasmultiplication ofaclass byanumber <16K.Wenow show that these operations obey theaxioms ofalinear space, enumerated inSecs. 2.12 and2.13. Infact, thevalidity ofaxioms l)and 2)ofSec. 2.12 and axioms 5)—8) ofSec. 2.13 forclasses follows atonce from their validity for elements ofthespace K.Moreover, thezero element ofthespace K/L isthe class 0(consisting ofallelements ofthesubspace L),while theinverse ofthe class Xistheclass consisting ofallinverses ofelements oftheclass X.Thus axioms 3)and4)ofSec. 2.12 arealso satisﬁed forthesetofclasses K/L. Theresulting linear space K/L iscalled thefactor space ofthespace Kwith respect tothesubspace L. 2.49. THEOREM. LetK:K,beaiin-dimensional linear space over the ﬁeld K,andletL:LlCKbeanl-dimensional subspace ofK.Then the factor space K/L isofdimension nYl. Proof. Choose anybasisfl, ...,fleL,andaugment it,asinSec. 2.43, byvectors flll, ...,f,,tomake abasis forthewhole space K.Then the classes Xlll aflll, ...,X,,9f,,form abasis inthespace K/L. Toseethis, wenote that given anyx6K,there isarepresentation 11 x:Z“kfiv_ k=1 andhence arepresentation X:Z<1l,Xl, lc:l+1 fortheclass X9.\".Moreover, theclasses Xl_.l,...,X,,arelinearly indepen- dent. 'Infact, if v.l+lXl_l -1-~~~-1-a,,X,, :06K/L foranyal+l, .,a,inK,then, inparticular, there would bearelation ”'lY1fl+1 ‘1*"‘‘'1‘Vwifn 5L- Butflll, ...,f,,arelinearly independent over L(seeSec. 2.44), and hence <1l+l:--~:an=0,asrequired. Thus thenYlclasses X,,,l, ...,X,, form abasis inK/L. Itfollows from Theorem 2.35 that K/L isofdimension n—l.I 2.5.Linear Manifolds 2.51. Animportant way ofconstructing subspaces istoform thelinear manifold spanned byagiven system ofvectors. Letx,y,z,...beasystem 50 LINEAR SPACES C]-1AP_ 2 ofvectors ofalinear space K.Then bythelinear manifold spanned by x,y,2,...ismeant thesetofall(ﬁnite) linear combinations <1x+l5)/+YZ+""" (12) with coetﬁcients 1,{-3,Y,...intheﬁeld K.Itiseasily veriﬁed thatthissethas properties a)andb)ofSec.2.41. Therefore thelinear manifold spanned bya system x,y,2,...isasubspace ofthespace K.Obviously, every subspace containing thevectors x,y,2,...alsocontains alltheir linear combinations (12). Consequently, thelinear manifold spanned bythevectors x,y,2,...is thesmallest subspace containing these vectors. The linear manifold spanned bythevectors x,y, 2,...isdenoted byL(x,y, 2,...). 2.52. Examples a.The linear manifold spanned bythebasis vectors el,e2,...,e,,ofa space Kisobviously thewhole space K. b.The linear manifold spanned bytwo (noncollinear) vectors ofthe space V0consists ofallthevectors parallel totheplane determined bythe twovectors. c.Thelinear manifold spanned bythesystem offunctions 1,t,t2,...,tk ofthespace K(a, b)(KisRorC)consists ofthesetofallpolynomials int ofdegree nohigher than k.The linear manifold spanned bytheinﬁnite system offunctions 1,t,t2,...consists ofallpolynomials (ofanydegree) in thevariable twith coetﬁcients intheﬁeld K. 2.53. Wenow note twosimple properties oflinear manifolds. a.LEMMA. Ifthevectors x’,y’,...belong tothelinear manifold spanned bythevectors x,y,...,then thelinear manifold L(x, y,...)contains the whole linear manifold L(x', y’,...). Proof. Since thevectors x',y', ...belong tothesubspace L(x,y, ...) then alltheir linear combinations, whose totality constitutes thelinear manifold L(x',y', ...),also belong tothesubspace L(x,y, ...).I b.LEMMA. Every vector ofthesystem x,y,...which islinearly dependent ontheother vectors ofthesystem canbeeliminated without clianging the linear manifold spanned byx,y,.... Proof. Ifthevector x,say,islinearly dependent onthevectors y,2,..., thismeans that xeL(y, z,...).Itfollows from Lemma 2.53a that L(x,y, 2,...)CL(y, 2,...). Ontheother hand, obviously L(y, 2,...) CL(x,y, 2,...). sEc. 2.6 I-IYPERPLANES 51 Together these tworelations imply L(y, 2,...)=L(x,y, 2,...).I 2.54. Wenow pose theproblem ofconstructing abasis foralinear manifold anddetermining thedimension ofalinear manifold Insolving this problem, wewillassume that thenumber ofvectors x,y, ...spanning the linear manifold L(x, y,...)isﬁnite, although some ofourconclusions donot actually require thisassumption. Suppose that among thevectors x,y, ...spanning thelinear manifold L(x,y, ...)wecanﬁnd rlinearly independent vectors xl,x2,...,x,,say, such that every vector ofthesystem x,y,...isalinear combination of xl,x2,...,x,.Then thevectors xl,x2,...,x,form abasis forthespace L(x,y, ...).Indeed, bythevery deﬁnition ofalinear manifold, every vector 2eL(x, y,...)canbeexpressed asalinear combination ofaﬁnite number ofvectors ofthesystem x,y,....But, byhypothesis, each ofthese vectors canbeexpressed asalinear combination ofxl,x2,...,x,. Thus eventually thevector 2canalso beexpressed asalinear combination ofthe vectors xl,x2,...,x,.This, together with theassumption that thevectors xl,x2,...,x,arelinearly independent, shows that xl,x2,...,x,indeed form abasis, asasserted. According toTheorem 2.35, thedimension ofthespace L(x,y, ...)is equal tothenumber r.Since there canbenomore than rlinearly independent vectors inanr-dimensional space, wecandraw thefollowing conclusions: a.Ifthenumber ofvectors x,y, ...spanning L(x,y, ...)islarger than thenumber r,then thevectors x,y,...arelinearly dependent. Ifthenumber ofthese vectors equals r,then thevectors arelinearly independent. b.Every setofr+lvectors from thesystem x,y,...islinearly dependent. c.Thedimension ofthespace L(x, y,...)canbedeﬁned asthemaximum number oflinearly independent vectors inthesystem x,y,.... 2.6.Hyperplanes 2.61. Asalready noted inSec. 2.42d, ifweadopt the“point” rather than the“vector” interpretation inthespace V0,then thegeometric entity corresponding tothenotion ofasubspace isaplane (orastraight line) passing through theorigin ofcoordinates. Butitisalsodesirable toinclude inourscheme ofthings planes andstraight lines which donotpass through theorigin ofcoordinates. Noting that such planes and straight lines are obtained from planes and straight lines passing through theorigin ofco- ordinates bymeans ofaparallel displacement inspace, i.e.,byashift, we areledinanatural waytothefollowing general construction: 52 LINEAR sPAcEs CI-IAP. 2 LetLbeasubspace ofalinear space K,andletx06Kbeaﬁxed vector which ingeneral does notbelong toL.Consider thesetHofallvectors of theform x:x0+y where thevector yranges over thewhole subspace L.Then Hiscalled a hyperplane, more speciﬁcally, theresult ofsliifting thesubspace Lbytlievector x0.Wenote that ingeneral ahyperplane isitself notalinear space. 2.62. Examples a.Inthespace V0thesetofallvectors starting from theorigin ofco- ordinates and terminating onaplane Yforms ahyperplane. Itiseasily veriﬁed that thishyperplane isasubspace ifandonly iftheplane Ypasses through theorigin ofcoordinates. b.Inthespace Knconsider thesetHconsisting ofthevectors x: (El,E2,...,E")whose components satisfy thecompatible nonhomogeneous system oflinear equations 4711-\'I ‘1*1712-Y2 ‘1*'"'1alﬂxll :bl» a21X1 ‘1*4722-"2 1""''1‘a21txn =I72, (13) alrl-X1 ‘1*47112-"2 ‘1*‘''‘1*aim-Yri :bk, and thesetLconsisting ofthevectors yI("r,l,-42,...,'/in)whose com- ponents satisfy thehomogeneous system oflinear equations with thesame coeﬁicients: a11y1 ‘1*l712y2 *1‘‘""‘i‘a1ri}'1i :0» H2111 ‘1'4722)/2 *1‘‘‘‘"1'a2ri}’n :0» (13,) am)/I '1ai.2)’2 ‘1‘''-1‘aicrifri :0- Aswealready know from Example 2.42e, thesetLisasubspace ofthespace Kn.Letx0:(Elm, Eg”,...,Elf”) beasolution ofthesystem (13). Then thesetHisidentical with thesetofallsums x0-1-ywherey ranges over the whole subspace L.Infact, ify:(rll,"/12,...,'rl,l)isasolution ofthe system (l3'), then thevector x=X.+yY(£59+T...55°’+1.... ...555"+1...) isobviously asolution ofthesystem (13), i.e.,belongs tothesetH.Con- versely, if.\"isany vector ofthesetH,then thediﬁerence y=x—x0 certainly satisﬁes thesystem (l3’), i.e.,thevector ybelongs tothesubspace sEc. 2.7 MORPI-IISMS orLINEAR SPACES 53 L.Inview ofthedeﬁnition given above, thesetHisahyperplane, namely theresult ofshifting thespace Lbythevector x0. 2.63. Wecanassign adimension toevery hyperplane, even ifitisnota subspace, i.e.,weconsider thedimension ofthehyperplane Htobeequal to thedimension ofthesubspace Lfrom which Hwas obtained byshifting. Forthisdeﬁnition tobesuitable, wemust show that thegiven hyperplane Hcanbeobtained asashift ofonly onesubspace. Toprove this, suppose H isboth theresult ofshifting thesubspace Lbythevector x0andtheresult of shifting thesubspace L’bythevector x0.Then forany26Hwehave both 2=x0+ywhere yeLand 2:x0+y’where y’6L’.Itfollows that L’ isthesetofvectors oftheform y’:(x0—x0)+ywhere yisanarbitrary vector inL,i.e.,thesubspace L’istheresult ofshifting thesubspace Lby thevector xl=x0—x0.Clearly xlbelongs tothesubspace L.Infact, the zero vector, justlikeanyother element ofthespace L’,canberepresented in theform xl+ylwhere yl6L(since L’isthesubspace Lshifted bythevector xl).Therefore xl:—yl, sothat xleL,asasserted. Butthen every vector y’eL’also belongs tothesubspace L,since y’isthesum ofavector xleL andavectory eL.Itfollows thatL’CL.Because ofthecomplete symmetry ofthehypothesis, wecan prove similarly that LCL’.Together with L’CL,thisimplies L:L’,asrequired. Inwhat follows, hyperplanm ofdimension lwillbecalled straight lines, andhyperplanes ofdimension 2willbecalled planes. 2.7.Morphisms ofLinear Spaces 2.71. Letcobearule which assigns toevery given vector x’ofalinear space K’avector .\"’inalinear space K”.Then coiscalled amorphism (or linear operator)T ifthefollowing twoconditions hold: a)o>(x’ +y’)=o>(x’) +o>(y’) forevery x’,y’eK’; b)m(<1x’) :<1o>(x’) forevery x’6K’andevery <1eK. Amorphism comapping thespace K’onto thewhole space K”iscalled an epimorphism. Amorphism comapping K’onto part (orall)ofK”inaone- to-one fashion (sothat x’¢y' implies o>(x’) ¢m(y’)) iscalled amono- morphism. Amorphism comapping K’onto allofK”inaone-to-one fashion (i.e., amorphism which isboth anepimorphism andamonomorphism) is called anisomorphism, andthespaces K’andK"themselves aresaid tobe isomorphic (more exactly, K-isomorphic). Theusual notation foramorphism is' co:K’YK". TMore exactly, aiiiorphisiii ofK’inloK”(oralinear operator mapping K’intoK”). 54 LINEAR SPACES CI-IAP. 2 2.72. Examples a.LetLbeasubspace ofaspace K.Then themapping cowhich assigns toevery vector xeLthesame vector xeKisamorphism ofLintoK,and infactamonomorphism (but notanepimorphism ifL¢K).This morphism issaid toembedL inK. b.LetLbeasubspace ofaspace K,andletK/L bethefactor space ofK with respect toL(seeSec.2.48). Then themapping cowhich assigns toevery vector xeKtheclass XeK/L containing xisamorphism ofcointo K/L, and infact anepimorphism (but notamonomorphism ifL¢0).This morphism coiscalled thecanonical mapping ofKonto K/L. 2.73. a.Letthespace K’ben-dimensional with basis el,...,e;,,and choose narbitrary vectors el,...,e’;inK”.With every given vector /_ n I x-23:31=1P2‘ inK’weassociate thevector o>(x’) =x”= >1-HI‘/l=J\“(2:N>1-= inK”with thesame components El,(k=1,...,n). Then themapping o>(x’) =x"isamorphism ofthespace K’into thespace K”.Infact, given anytwovectors X’=Ziiei. y’=2711311 k=1 k=1 inK’,itfollows from Theorem 2.33 that x’*1‘Y’= +'4k)3ii~ l:=1 But ‘°(x’) =Z£11311’ ‘°(y’) :2'4k3ii lc=1 k=1 bythedeﬁnition ofthemapping co,andmoreover ‘°(x’ *1‘Y’): *1‘71k)3ii =2€k3ii *1‘2711131: :¢°(x’) *1‘‘°(y’)a k=1 = k=1 sothat condition a)ofSec. 2.71 issatisﬁed. Similarly,Pr‘>- o>(<1x’) =o>(<1é1El,elQ) =o>(g1<1E,,el,) ri 1| =2°l€i¢3ié =“Z2131: :°“°(X') 7,--1 k=1 SEC. 2.7 MORPHISMS OF LINEAR SPACES forevery a€K, sothat condition b)isalso satisﬁed. Therefore misa morphism ofK’intoK",asasserted. b.Obviously, themorphism mjust described isanepimorphism ifand only ifevery vector x"6K"canberepresented intheform Zglcelcia k=1_ i.e.,ifandonly ifK"coincides with thelinear manifold spanned bythevectors ex,...,ex. c.Similarly, ourmorphism misamonomorphism ifand only ifevery pair ofvectors 7| 7| 2gkelii 2ylkeiiIc=1 k=1 diﬁering inatleast onecomponent (i.e., such that Eh¢11,,foratleast one value ofk)aredistinct vectors ofK".But this isequivalent tolinear independence ofthevectors ex,...,ex. Therefore themorphism misa monomorphism ifandonly ifthevectors ex,...,exarelinearly independent. d.Itfollows that themorphism mdmcribed above isanisomorphism if and only ifthevectors ex,...,exarelinearly independent and thelinear manifold spanned bythem coincides with thewhole space K".Inother words, themorphism misanisomorphism ifandonly ifthevectors ex,...,ex form abasis inthespace K". 2.74. THEOREM. Any twon-dimensional spaces K’andK”(over thesame ﬁeld K)areK-isomorphic. Proof. Letex,...,exbeabasis inthespace K’andex,...,exabasis inthespace K”,andusethese twosystems ofvectors toconstruct amorphism mofK’intoK"inthewaydescribed inSec.2.73a. Then misanisomorphism, bySec. 2.73d. | 2.75. COROLLARY. Every n-dimensional linear space over aﬁeld Kis K-isomorphic tothespace K,,ofSec. 2.I5b. Inparticular, every n-dimensional complex space isC-isomorphic tothespace C",andevery n-dimensional real space isR-isomorphic tothespace R". 2.76. Wenow discuss further properties ofepimorphisms and mono- morphisms. a.Given amorphism o>:K’->K”, consider thesetL"ofallvectors m(x’) eK"such that x’6K’.The setL",which isobviously asubspace of K",iscalled therange ofthemorphism co.Itisclear that themapping m 56 LINEAR SPACES CHAP. 2 ofK’intoL”isanepimorphism. Ifthemorphism m:K’—>K"isamonomor- phism, then themorphism co:K’—>L”isanisomorphism. b.Given amorphism o>:K’ —>K",consider thesetL’ofallvectors x’eK’such thatm(x’) =0.ThesetL’,which isobviously asubspace ofK’, iscalled thenullspace (orkernel) ofthemorphism m. Wenow construct thefactor space K’/L’ (seeSec.2.48). Alltheelements x’belonging tothesame class X’eK’/L’ arecarried bythemorphism minto thesame element ofthe space K”.Infact, given twosuch elements x’andy’, wehave x’—y’=2’6L’,andhence <»<x'>—mo’)=we’)=0.<»<x'>=mo’). Suppose that with every class X’GK’/L’ weassociate theelement x”= m(x’) eK”where x’isanarbitrary element ofX’ (asjustshown x”isuniquely determined). Letx”=Q(X’). Then itiseasy toseethat Qisamorphism ofK’/L’ into K”.Moreover Qisamonomorphism, since itfollows from x’asY’,x’ex’,y’eY’that Q04’)—Q<Y'>=<»<><'>—mo’)1we-'—y’)¢0- Thus anymorphism o>:K’ ->K”generates amonomorphism Q:K’/L’ —>K”. Ifthemorphism misanepimorphism, then, obviously, themonomorphism Q isalso anepimorphism, sothat theepimorphism o>:K’ —>K” generates an isomorphism Q:K’/L’ —>K". Wewillcontinue thestudy ofmorphisms inChapter 4. PROBLEMS 1.Consider thesetofvectors intheplane whose initial points arelocated atthe origin ofcoordinates andwhose ﬁnal points liewithin theﬁrstquadrant. Does thissetform alinear space (with theusual operations)? 2.Consider thesetofallvectors intheplane with theexception ofthevectors which areparallel toagiven straight line. Does thissetform alinear space? 3.Consider thesetPconsisting ofthepositive realnumbers only. Weintroduce operations according tothefollowing rules: Bythe“sum” oftwonumbers we mean their product (intheusual sense), andbythe“product” ofanelement r6Pandarealnumber 7.wemean rraised tothepower 7.(intheusual sense). IfPalinear space (with these operations)? 4.Show thatacriterion forthelinear independence ofngiven vectors inthe space K,,isthatthedeterminant formed from thecoordinates ofthevectors does notvanish. 5.Show thatthefunctions t'\,t'*,...,t"=arelinearly independent inthespace K(a, b),where 0<a<bandr1,r2,...,rkaredistinct realnumbers. PROBLEMS 57 6.Thefollowing isknown about asystem ofvectors e1,e2, ...,e,,inalinear space K: a)Every vector xeKhasanexpansion oftheform x=E1e1 +€2e2 —l-H‘+E,,e,,; b)This expansion isunique forsome ﬁxed vector x0€K. Show thatthesystem el,e2,...,e,,forms abasis inK. 7.Does there exist abasis inthespace PofProblem 3? 8.What isthedimension ofthespace PofProblem 3? 9.Find theintersection andsumoftwodistinct two-dimensional subspaces of thespace V3(two distinct planes passing through theorigin ofcoordinates). 10.Prove thatifthedimension ofthesubspace LCKisthesame asthatofthe space K,thenL=K. 11.Istheshift vector x0ﬁguring intheconstruction ofahyperplane uniquely determined bythehyperplane itself? 12.Show thatevery hyperplane HCKhasthefollowing property: Ifx6H, yeH, then ax+(1—oc)yeHforevery element oftheﬁeld K.Conversely, show thatifasubset HCKhasthisproperty, then Hisahyperplane. What geometric characteristic ofahyperplane isexpressed bythisproperty? 13.Thehyperplanes H1andH2have dimensions pandq,respectively. What is the(smallest) dimension which thehyperplane H3must have inorder tobesure tocontain both H1andH2? 14.Solve theanalogous problem forthree hyperplanes H1,H2andH3,with dimensions p,qandr,respectively. 15.According toTheorem 2.74, theone-dimensional spaces R1andP(see Problem 3)areisomorphic. How canoneestablish thisisomorphism inpractice? chapter 3 SYSTEMS OF LINEAR EQUATIONS 3.l.More ontheRank ofaMatrix 3.11. Wehave already touched upon thesubject ofmatrices several times. Inthissection wewillstudy inmore detail those properties ofmatrices which areconnected with theconcept ofrank (seeSec. 1.9). This willallow usto give ageneral solution ofthebasic problems ofthetheory ofsystems of linear equations, posed inSec. 1.2. Webegin byrecalling some basic deﬁnitions from Sec. 1.9.Suppose we have amatrix an a12 alk a21 a22 an anl am amt with nrows andkcolumns, consisting ofthenumbers an.from theﬁeld K, where iistherowindex ranging from ltonandj isthecolumn index ranging from ltok.‘]'Ifwechoose anymrows andmcolumns ofthismatrix, then theelements which appear attheintersections ofthese rows and columns TSometimes theindices ofanelement ofthematrix Awillbewritten differently, i.e., sometimes wewilldenote theelement appearing intheithrowandjth column ofA bythe symbol a§. 58 sEc.3.1 MORE onTHERANK orAMATRIX 59 form asquare matrix oforder m.The determinant ofthismatrix iscalled aminor oforder mofthe matrix A.Theinteger missaidtobetherank ofthe matrix AifAhasanonvanishing minor oforder randallitsminors oforder r+land higher vanish. lfthematrix Ahasrank r>0,then each ofits nonvanishing minors oforder riscalled abasis minor. The columns and rows ofthematrix which intersect attheelements ofthebasis minor are called thebasis columns andbasis rows. The considerations that follow arebased onthepossibility ofregarding anycolumn ofnumbers asageometric object, i.e., asavector inthen- dimensional space KnofSec. 2.l5b. With thisgeometric interpretation, the matrix Aitself corresponds toacertain setofkvectors ofthespace Kn. Letx,-(j=l,...,k)denote thevector corresponding tothejthcolumn of A.Then anylinear relation between thecolumns ofAcanbeinterpreted asthe same linear relation between thecorresponding vectors (seeSec. 2.22b). LetL(x1,x2, ...,xk)bethelinear manifold spanned bythevectors xl,x2,...,xkofKn(seeSec. 2.51). Wenow prove that thevectors corre- sponding tothebasis columns ofthematrix Aform abasis forthislinear manifold. Tobeexplicit, suppose that theﬁrst rcolumns ofAarebasis columns. Then, toprove ourassertion, itsuﬂices toshow ﬁrst that the vectors xl,x2,...,x,arelinearly independent, and secondly that any of theother vectors x,+1, ...,xnisalinear combination oftheﬁrstrvectors (seeSec. 2.54). Toprove theﬁrstassertion, suppose that thevectors xl,x2, ...,x,arelinearly dependent, orequivalently, that theﬁrstrcolumns ofA arelinearly dependent. Then, byTheorem 1.96, anydeterminant oforder r constructed from these columns and any rrows ofAwould vanish. In particular, thebasis minor ofAwould vanish, contrary toitsvery deﬁnition. This contradiction establishes theﬁrst assertion. The second assertion, as applied tocolumns ofthematrix A,hasalready been proved inSec. 1.93 under theguise ofthe“basis minor theorem.” This completes theproof that thevectors xl,x2,...,x,form abasis forthespace L(x,, x2,...,xk). According toTheorem 2.35, thedimension ofthisspace equals thenumber r,i.e.,therank ofthematrix A.Thus wehave established thefollowing important THEOREM. Thedimension ofthelinear manifold spanned bythevectors corresponding tothecolumns ofthematrix Aequals therank ofA.Moreover, thevectors corresponding tothebasis columns ofAform abasis forthis linear manifold. 3.12. Thefollowing propositions areobvious consequences ofconclusions a)—c) ofSec. 2.54: a.THEOREM. Iftherank ofthematrix Aislessthan thenumber ofcolumns inA(r<k),then thecolumns ofAarelinearly dependent. Iftherank ofA 60 SYSTEMS orLINEAR EQUATIONS CHAP. 3 equals thenumber ofcolumns inA(r:k),then thecolumns ofAarelinearly independent. b.THEOREM. Any r+1columns ofthematrix Aarelinearly dependent. c.THEOREM. Therank ofanymatrix Aequals themaximum number of linearly independent columns inA. This lasttheorem isoffundamental importance, since itconstitutes a newdeﬁnition oftherank ofamatrix. 3.13. Suppose wetranspose thematrix A,i.e.,suppose wegoover tothe matrix A’whose rows arethecolumns ofA(cf.Sec. 1.41). Clearly, therank ofthetransposed matrix A’isthesame astherank ofA.Butaccording to Theorem 3.l2c, therank ofA’equals themaximum number oflinearly independent columns inA’,orequivalently, themaximum number of linearly independent rows inA.Thus wearrive atthefollowing somewhat unexpected conclusion: THEOREM. Themaximum number oflinearly independent rows inamatrix Aisthesame asthemaximum number oflinearly independent columns inA. Wenote that thistheorem isnottrivial. Infact, anydirect proof ofthe theorem would require achain ofreasoning equivalent totheproof of Theorems 1.93and3.11. 3.14. Finally wenote thefollowing result, which isaconsequence of Theorem 3.11 andLemma 2.53b: THEOREM. Any column ofthematrix Awhich isalinear combination of theother columns canbedeleted without changing therank ofA. 3.2.Nontrivial Compatibility ofaHomogeneous Linear System 3.21. Suppose wehave ahomogeneous linear system a11x1 +a12x2 +'''+alnxn :or a21x1 +a22x2 +'''+azn-‘(ii :Oi (2) aklxl -l‘ai,-2X2 -l‘'''+aknxn :0- Asweknow, thissystem isalways compatible, since ithasthetrivial solution x1=x2=---=x,,=0. SEC_ 3.3 TI-IE COMPATIBILITY CONDITION FDR AGENERAL LINEAR SYSTEM The basic problem encountered instudying homogeneous linear systems is thefollowing: Under what conditions isahomogeneous li'near system “non- trivially compatible,” i.e., under what conditions does such asystem have solutions other than thetrivial solution? The results ofSec. 3.1allow usto solve thisproblem immediately. Infact, aswehave seen inSec. 2.22b, the existence ofanontrivial solution ofthesystem (2)isequivalent tothe columns ofthematrix an a12 '''am a a '''a A: 21 22 2n akl. an akn being linearly dependent. But, according toTheorem 3.l2a, thisoccurs if andonly iftherank ofthematrix Aislessthan thenumber ofcolumns inA. Thus weobtain thefollowing THEOREM. Thesystem (2)isnontriviall ycompatible, i.e., hasnontrivial solutions ifandonly iftherank ofthematrix Aislessthan n.Iftherank of thematrix Aequals n,thesystem (2)hasnonontrivial solutions. 3.22. Inparticular, ifthenumber ofequations inthesystem (2)isless than thenumber ofunknowns (k<n),therank ofthematrix Aiscertainly lessthan n,andinthiscase nontrivial solutions always exist. Ifk=n,the question ofwhether ornotnontrivial solutions exist depends onthevalue ofdetA.IfdetA ¢0,there arenonontrivial solutions (r:n),while if detA :0,there arenontrivial solutions (r<n).Ifk>n,wehave to examine allpossible determinants oforder nwhich areobtained byﬁxing anynrows ofthematrix A.Ifallthese determinants vanish, then r<nand nontrivial solutions exist. Ifatleast oneofthese determinants isnonvanishing, then r=nandthere isonly thetrivial solution. 3.3.TheCompatibility Condition foraGeneral Linear System 3.31. Suppose wehave ageneral (i.e., nonhomogeneous) system of linear equations a11x1 -l‘a12x2 -l‘'''-l‘alrlxn :bl, a21x1 -l‘azzxe -l‘'''-l‘a27Lx7l :I72, (3) aklxl +a1r2x2 -l‘'''-l‘aknxn :bk- 62 SYSTEMS orLINEAR EQUATIONS CHAP. 3 With thissystem weassociate twomatrices, thematrix a11 a12 aln a21 a22 a2»A= , akl ak2 akn called thecoeﬂicient matrix oftliesystem (3),andthematrix a11 a12 '''a1" a1 a21 a22 '''a2 a2A1: 7’ 9 akl au2 akn bk called theaugmented matrix ofthesystem (3).Regarding thecompatibility ofthesystem (3),wethen have thefollowing basic THEOREM (Kronecker-Capelli). Thesystem (3)iscompatible ifandonly iftherank oftheaugmented matrix ofthesystem equals therank ofthe coeﬂicient matrix. Proof. Assume ﬁrstthatthesystem (3)iscompatible. Then ifcl,c2,..., c,,isasolution ofthesystem, wehave theequations ai.1¢'1 +a12¢'2 ‘l‘'''‘l‘alﬂcﬂ :bl: a21c1 -l‘a22c2 -l‘'''-l‘a2ncn =a2, aklcl +ak2c2 +'''-l‘alcncn =ak- These equations imply that thelastcolumn ofA1isalinear combination of theother columns ofA1(with coeﬂicients cl,c2,...,cn).ByTheorem 3.14, wecandelete thelastcolumn ofA1without changing itsrank. Butwhen thelastcolumn ofA1isdeleted, itbecomes justA.Hence ifthesystem (3) iscompatible, thematrices AandA1have thesame rank. Wenow assume thatthematrices AandA1have thesame rank, andshow that thesystem (3)iscompatible. Letrbetherank ofthematrix A(and consequently also ofthematrix A1). Consider rbasis columns ofA;they willalso bebasis columns ofA1.ByTheorem 1.93, thelastcolumn ofA1 canbewritten asalinear combination ofthebasis columns, and hence itcanbewritten asalinear combination ofallthecolumns ofA.Ifwe SEC. 3.4 THE GENERAL SOLUTION OF ALINEAR SYSTEM denote thecoeﬁicients ofthislinear combination bycl,c2,...,cn,weﬁnd that theequations a11c1 -l‘a12c2 -l‘'''-l‘alncn :bl» a21c1 -l‘a22c2 -l‘'''+a2ncn :az» aklcl -l‘ak2c2 +'''+akncn :bk aresatisﬁed. Thus thevalues x1:c1ax2:c2r---axnzcn satisfy thesystem (3),which istherefore compatible. I 3.4.TheGeneral Solution ofaLinear System 3.41. The Kronecker—Capelli theorem, which gives thegeneral condition forthecompatibility ofalinear system, does notgive amethod forsolving thesystem. Wenow derive aformula which constitutes ageneral solution ofalinear system. Byageneral solution ofthesystem (3)wemean asetofexpressions xJ':.fi(a119-'~>aknab1a~~-abkrqls--->qs) l!"'7n)7 where theright-hand sides arefunctions depending onthecoeﬁicients a,.,»of thesystem (3), theconstant terms b,»of(3)and certain undetermined parameters ql,...,q,,such that 1)The quantities x,-=c,(j=1,...,n)obtained forarbitrary ﬁxed values oftheparameters q,,...,qj.(from theﬁeld K)constitute asolution ofthesystem (3); 2)Any given solution ofthesystem (3)canbeobtained inthisway by suitably choosing thevalues oftheparameters ql,...,q,inK. Asshown inSec. 2.62b, thesetofallsums oftheform x0+y,where x0is any(“particular”) solution ofthesystem (3)andyranges over thesetof allsolutions ofthecorresponding homogeneous system, isjust thesetof allsolutions of(3).This factcannow beexpressed asfollows: Thegeneral solution ofthenonhomogeneous system (3)isthesum ofany particular solution of(3)and thegeneral solution ofthecorresponding homogeneous system (2). Suppose wehave acompatible linear system (3)with acoeﬂicient matrix A=||a,»,-l| ofrank r.Itcanbeassumed thatthebasis minor Mofthematrix Aappears initsupper left-hand corner; otherwise, wecanachieve this conﬁguration byinterchanging rows andcolumns ofA,which corresponds 64 SYSTEMS orLINEAR EQUATIONS CHAP. 3 torenumbering some oftheequations andunknowns inthesystem (3).We take theﬁrstrequations ofthesystem (3)andrewrite them intheform allxl +a12x2 T'''1'"alrxr :at”'a1,1+1xr+1 T'''*alnxns a21x1 +a22-E2 'i‘'''TI“a2rxr :b2*a2,r+1x,-+1 *'''#a2nxn1 (4) arlxl “IIar2x2 +III+arrxr ZbrTar,r+lxr+l TIIIITarnxrr Next weassign theunknowns x,+1, ...,x,, completely arbitrary values c,+1, ...,cn.Then (4)becomes asystem ofrequations intherunknowns xl,x2,...,x,,with adeterminant Mwhich isnonvanishing (abasis minor ofthematrix A).This system canbesolved byusing Cramer’s rule (see Sec. 1.73). Hence there exist numbers cl,c2,...,c,,which, when substituted fortheunknowns xl,x2,...,xnofthesystem (4),reduce alltheequations ofthesystem toidentities. Wenow show that these values cl,c2,...,cn satisfy alltheother equations ofthesystem (3)aswell. The ﬁrst rrows oftheaugmented matrix A1ofthesystem (3)arebasis rows ofthismatrix, since bythecompatibility condition, therank ofthe augmented matrix isr,while byconstruction, thenonvanishing minor M appears intheﬁrstrrows ofA1.ByTheorem 1.93 (applied torows), each ofthelastn~rrows ofA1isalinear combination oftheﬁrstrrows. This means that every equation ofthesystem (3)beginning with the(r+l)st equation isalinear combination oftheﬁrst requations ofthesystem. Therefore, ifthevalues x1=c1,...,.\',,=c,, satisfy theﬁrst requations ofthesystem (3),they also satisfy alltheother equations of(3). 3.42. Towrite anexplicit formula forthesolution ofthesystem (3)just constructed, letM,~(<x,-) denote thedeterminant obtained from thebasis minor Mzdetllaijll (1Ia_]I::la2s-~~sr) byreplacing itsjthcolumn bythecolumn consisting ofthequantities oil,<12,...,ot,.Then, using Cramer’s rule towrite the‘solution ofthe system (4),weobtain 1 ‘Ii:MMi(bi *ai.r+lcr+1 TIIITaillcn) :-1%,iM.(b.>—c...M.(a.....> ----—c,M.-(a..>i (1=1.2.---.r>. <5) sEc.3.5 GEOMETRIC PROPERTIES orTHESOLUTION SPACE 65 These formulas express thevalues oftheunknowns x,-=c,~(j=1,2,...,r) interms ofthecoefﬁcients ofthesystem, theconstant terms andthearbitrary quantities (parameters) Cr-1-1: Cr-I-21 ---7cu‘ Finally, weshow that(5)comprises anysolution ofthesystem (3).Infact, letcf”,cg”,...,cf”,cjfjl, ...,cx”beanarbitrary solution ofthesystem (3). Obviously, itisalso asolution ofthesystem (4).But, using Cramer’s rule tosolve thesystem (4), weobtain unique expressions forthequantities cf’),cg”,...,c§°’interms ofthe quantities cjﬁlll, ...,cx”,namely theformulas (5).Thus, choosing cr+1 :5:21, ---1C7!:Che) in(5),wegetjustthesolution cf”,cg”,...,cx”,asasserted. Thus (5)isthe general solution ofthesystem (3). 3.5.Geometric Properties oftheSolution Space 3.51. Consider ﬁrst thecase ofthehomogeneous linear system (2).As wehave already seen (Sec. 2.42e), thesetofallsolutions ofthissystem forms alinear “solution space,” which wedenote byL.Wenow calculate the dimension ofLandconstruct abasis forL. Forahomogeneous system, theequations (5)become —MCi :cr+1Mj(ai,r+1) -l"'''+cnMi(ain) =I,2,---,7), (6) since M,-(bi) =M,-(0) =0.With every solution cl,c2,...,c,,c,+1, ...,c,, ofthesystem (2)weassociate avector (c,+1, ...,c,,)ofthespace K,,_, (see Sec. 2.l5b). Since thenumbers c,.1,. ..,cncanbechosen arbitrarily and since they uniquely deﬁne asolution ofthesystem (2),thecorrespondence between thespace ofsolutions ofthesystem (2)andthespace K,,_, isone-to- one. This correspondence isanisomorphism, since itpreserves linear operations, asiseasily veriﬁed. Thus thespace Lofsolutions ofahomo- geneous system oflinear equations innunknowns with acoeﬂicient matrix of rank risisomorphic tothespace K,,_,. Inparticular, thedimension ofthe space Lisn‘r. 3.52. Any system ofn—rlinearly independent solutions ofahomo- geneous linear system ofequations (which, byTheorem 2.34, forms abasis inthespace ofallsolutions) iscalled afundamental system ofsolutions. To construct afundamental system ofsolutions, wecanuseanybasis ofthe 66 SYSTEMS orLINEAR EQUATIONS CHAP. 3 space K,,_,. Then, because oftheisomorphism, thecorresponding solutions ofthesystem (2)willform abasis inthespace ofallsolutions ofthesystem. The simplest basis ofthespace K,,_,, consists ofthevectors 91:030.---,0), e2:(0als---yo)’ @,,_,=(0,o,...,1) (see Sec. 2.32c). For example, toobtain thesolution ofthesystem (2) corresponding tothevector el,wesetc,+1=1,c,+2=---=c,,=0inthe formulas (6)anddetermine thecorresponding values c,.=cx.” (i=1,2,...,n). Similarly, weconstruct thesolution corresponding toanyother basis vector e,-(j=2,...,n—r).The setofsolutions ofthesystem (2)constructed inthisway iscalled anormal fundamental system ofsolutions. Ifwedenote these solutions byx“), x‘2l, ...,x‘"_’>, then bythedeﬁnition ofabasis, any solution xisgiven bytheformula X=<=<rx“>+<=<2x‘2’+---+<><,.-.X"‘”’>- (7) Since any solution ofthesystem (2)isaspecial case of(7),this formula gives thegeneral solution of(2). 3.53. Consider now thegeneral case ofanonhomogeneous system (3). Asshown inSec. 2.62b, thegeometric object Hcorresponding tothesetof allsolutions ofanonhomogeneous system isahyperplane inthen-dimensional space Kn.This hyperplane isobtained byshifting thesubspace Lofall solutions ofthecorresponding homogeneous system (Lhasbeen shown tobe isomorphic tothespace K,,_,) byavector x0which isanarbitrary particular solution ofthenonhomogeneous system. From this weconclude that the dimension ofthehyperplane Histhesame asthedimension ofthesubspace L.Moreover, ifristherank ofthecoeﬁ‘icient matrix ofthesystem (3),then anyvector yofthesubspace Lcanberepresented asasum y=alylli +%y(2> +...+%_ry(~-r)’ where y‘1l,y‘2l, ...,y”‘C'> arebasis vectors ofthespace L(afundamental system ofsolutions). Consequently, anyvector xofthehyperplane Hcanbe represented asasum x=x.+y=x.+<=<.y">+<=<.y<2’+---+<=<.._.y‘":"- Inthelanguage appropriate tosolutions ofthesystems (2)and (3),this agrees with theprescription established inSec. 3.41, i.e.,thegeneral solution SEC. 3.6 METHODS FOR CALCULATING THE RANK OF AMATRIX ofthenonhomogeneous system (3)isthesum ofanyparticular solution of(3) andthegeneral solution ofthecorresponding homogeneous system (2). 3.6.Methods forCalculating theRank ofaMatrix 3.61. Tomake practical useofthemethods forsolving systems oflinear equations developed inthepreceding sections, onemust beable tocalculate therank ofamatrix andﬁnd itsbasis minor. Obviously, thedeﬁnition of therank ofamatrix given inSec. 1.92 cannot serve perseasareasonable practical means ofcalculating therank. Forexample, asquare matrix of order ﬁvecontains oneminor oforder ﬁve, 25minors oforder four, 100 minors oforder three, and100minors oforder two. Clearly, itwould bea very laborious task toﬁndtherank ofsuch amatrix bydirect calculation of allitsminors. Inthissection, wewillgive simple methods forcalculating therank ofamatrix anddetermining itsbasis minor. These methods are based onastudy ofcertain operations onrows and columns ofamatrix which donotchange itsrank; these operations willbecalled elementary operations. Since, asalready noted, therank ofamatrix does notchange when itistransposed, wewilldeﬁne these operations only forthecolumns ofamatrix. Inkeeping with this, ourproofs willmake useofthegeometric interpretation ofamatrix with nrows andkcolumns asthematrix formed from thecomponents ofasystem ofkvectors xl,x2,...,x,,inthen- dimensional (real) space Rn.Wewillalso make useofTheorem 3.11, which asserts thattherank ofthismatrix equals thedimension ofthelinear manifold spanned bythevectors xl,x2,...,xk. Wenow study thefollowing elementary operations: a.Permutation ofcolumns. Suppose thecolumns ofthematrix Aare permuted inanyway. This operation does notchange therank ofA.Infact, thedimension ofthelinear manifold spanned bythevectors xl,x2,...,xk does notdepend ontheorder inwhich they arewritten, andhence therank ofthematrix does notdepend ontheorder ofitscolumns. b.Dividing outanonzero common factor oftheelements ofacolumn. Suppose thenumber A¢0being divided outisacommon factor ofthe elements oftheﬁrst column ofthematrix A.This operation isequivalent toreplacing thesystem ofvectors 71x1,x2,...,xkbythesystem xl,x2,..., xk.Butobviously thelinear manifolds spanned bythese two systems have thesame dimension (since thelinear manifolds themselves arethesame). Therefore therank ofthematrix Adoes notchange asaresult ofthiselemen- tary operation. c.Adding anarbitrary multiple ofonecolumn toanother column. Suppose wemultiply themthcolumn ofthematrix Abythenumber Aandadditto 68 SYSTEMS orLINEAR EQUATIONS Cl-IAP_ 3 thejthcolumn. This means that thesystem ofvectors xl,...,x,~,...,xm, ...,xkhasbeen replaced bythesystem x17"'7xj+)\xm7"'7xm7"'7'Xk' Wehave toshow that thelinear manifolds L1andL2spanned bythese two systems arethesame. lntheﬁrstplace, allthevectors ofthesecond system lieinthelinear manifold spanned bythevectors oftheﬁrstsystem. Hence, byLemma 2.53a, wehave L2CL1.Ontheother hand, theequation x.=<x.-+M...)- shows that thevector x,liesinthelinear manifold spanned bythevectors of thesecond system. Since alltheother vectors oftheﬁrst system obviously belong tothislinear manifold, wehave L1CL2.Itfollows that L1:L2. Therefore therank ofAdoes notchange asaresult ofthis elementary operation. d.Deletion ofacolumn consisting entirely ofzeros. Acolumn consisting entirely ofzeros corresponds tothezero vector ofthespace Rn.Obviously, eliminating thezero vector from thesystem xl,x2,...,xkdoes notchange thelinear manifold L(x1, x2,...,xk)and hence does notchange therank ofthematrix A. e.Deletion ofacolumn which isalinear combination oftheother columns. Thelegitimacy ofthiselementary operation wasproved inTheorem 3.14. 3.62. Calculation oftherank ofamatrix anddetermination ofabasis minor. Wenow show how tocalculate therank andﬁndabasis minor ofa given matrix Abyusing theelementary operations justenumerated. Ifthe matrix Aconsists only ofzeros, then itsrank isobviously zero. Suppose A contains anonzero element. Then, bysuitably permuting therows and columns, wecanbring thiselement over totheupper left-hand corner ofthe matrix. Then, subtracting from every column theﬁrst column multiplied byasuitable coeﬁicient, wecanmake alltheother elements oftheﬁrstrow vanish. Weshall make nofurther changes intheﬁrstrowandﬁrstcolumn (except fortherearrangements described below). lfthere arenononzero elements among theremaining elements (i.e., theelements which donot belong totheﬁrstrowandtheﬁrstcolumn), then therank ofthematrix A isobviously 1.Ifthere isanonzero element among theremaining elements, then bysuitably rearranging rows andcolumns, wecanbring thiselement over totheintersection ofthesecond rowandthesecond column andthen make alltheelements following itinthesecond row vanish, just asbefore. (Wenote thatthese operations donotaﬁect theﬁrstrowandtheﬁrstcolumn.) SEC. 3.6 METHODS FOR CALCULATING THE RANK OF AMATRIX Continuing inthisfashion, andassuming that thenumber ofcolumns inA does notexceed thenumber ofrows inA(this canalways beachieved by transposition), wereduce Atooneofthefollowing twoforms: 0&1 0 0--00---0 C21 <12 0--00---0 r,, c32 I13" 00---0 A1= ckl ck2 ck3 '' air 0'''0 clc+l_1 ¢i.+1.: clC+1.3 "' ck+1.k 0"' 0 c,,1 c,,2 c,,3 -c,,,, 0 0 or ix, 0 0--0 c21 <12 0--0 can C32 as ''0 A2: _ _ ___ _ cml cm2 cm3 am cnl ch? cn3 IIIchm Here thenumbers oil,a2,etc.arenonzero. Intheﬁrst case, therank of A1equals kand itsbasis minor (inthetransformed matrix) stands inthe upper left-hand corner. Inthesecond case, therank ofA2equals m(the number ofcolumns) anditsbasis minor (inthetransformed matrix) appears intheﬁrst mrows. This determines therank ofA.The location ofthe basis minor ofAiseasily found byfollowing back inreverse order allthe operations performed onA. 70 SYSTEMS orLINEAR EQUATIONS Cl-IAP_ 3 Asanexample, consider thefollowing matrix with ﬁvecolumns andsix rows: 1 2 6-2 -1 -2 -1 0-5 -1 3 1-1 8 1 -1 0 2-4 -1 -1 -2 -7 3 2 -2 -2 -5 -1 1A= . There isonezero inthesecond row ofA;byusing thegeneral method described above, wecanproduce three more zeros inthisrow. However, forconvenience, weﬁrst interchange theﬁrst and second rows. Then, interchanging theﬁrst and second columns (sothat anelement -1with thesmallest nonzero absolute value again appears intheupper left-hand corner), weobtain‘l' -2-1 0-5-1 -1-2 0-5-1 126-2-1 216-2-1 31-1 81 13-1 81 Ar\4 r\/-1 02-4-1 0-1 2-4-1 -1-2-7 32 -2-1-7 32 -2-2-5-1 1 -2-2-5-1 1 Toobtain three more zeros intheﬁrstrow, wemultiply theﬁrstcolumn by 2,5,and 1,and subtract theresults from thesecond, fourth, and ﬁfth columns, respectively. This gives -1 0000 2-3 6-12-3 11-1 30A~ .0-1 2-4-1 -2 3-7 134 -2 2-5 93 The simplest thing todonext istoproduce additional zeros inthethird row. First weinterchange thisrow with thesecond row. Then wemultiply THere thesymbol ~written between twomatrices means thatthey have thesame rank. PROBLEMS 7| thesecond column by1and -3andaddtheresults tothethird andfourth columns, respectively. Thus wehave -1 0000 -1 0000 11-1 30 11000 2-3 6-12-3 2-3 3-3-3A~ ~ -2,.0-1 2-4-1 0-1 1-1-1 -2 3-7 134 -2 3-4 44 -2 2-5 93 -2 2-3 33 Thefourth andﬁfth columns ofthematrix A1areproportional tothethird column andcanbedeleted. The matrix which isleftobviously hasrank 3, sothat theoriginal matrix Aalso hasrank 3.Moreover, A1hasabasis minor initsﬁrstthree rows andﬁrst three columns. Byreversing thesuc- cessive transformations which ledfrom AtoA1,wecaneasily verify that none ofthetransformations which were carried outhasanyeﬁect onthe absolute value ofthis minor. Therefore theminor appearing intheﬁrst three rows andtheﬁrstthree columns oftheoriginal matrix isalso abasis minor. PROBLEMS 1.Prove thefollowing theorem: Anecessary andsuﬁicient condition fora matrix |la,»,-|| ofordermto haverankr <1isthat there exist numbersal, a2,..., a,,,andbl,b2,...,b,,,such that an'=a¢bi ('I,]I:1,2,---,m)- 2.Letx1,x2, ...,x,,beklinearly independent vectors inann-dimensional space K,,,andletA=||a§.">1| bethematrix made upofthecomponents ofthe vectors xl,x2,...,xkwith respect tosome basis el,e2,...,e,,.Show thatthe linear manifold L(x1, x2,...,xk)isuniquely determined, provided oneknows thevalues ofalltheminors ofAoforder k. 3.Show thatwhen k=n,thesystem (2),p.60hasthesolution ¢'1=Ai1i¢'2=A¢2i---,9»:-4m (l<'I<"), where A,~,~isthecofactor oftheelement a,,-(iﬁxed), provided thattherank of thematrix Aislessthann. 4.Solve thesystem ofequations x1+x2+x3+x4+x5=7, 3x1+2x2+x3+x4 —3x5: -2, x2+2x3+2x4+6x5:23, 5x1+4x2+3x3+3x4—x5:12. 72 SYSTEMS orLINEAR EQUATIONS 5.Study thesolutions ofthesystem Ax+y+z=1, x+Ky+z=7., x+y+7.2=7.2 asafunction of7.. 6.What isthecondition forthethree straight lines a1x+b1y+c1=O, a2x+b2y+c2=O, a3x+b3y+c3=0 topassthrough onepoint? 7.What isthecondition forthenstraight lines a1x+b1y+c1=O, a2x+b2y+c2=O,...,a,,x+b,,y+c,,=O topassthrough onepoint? 8.Find thenormal fundamental system ofsolutions forthesystem ofequations x1+x2+x3+x4+x5= 3x1+2x2+x3+x4-3x5= x2+2x3+2x4+6x5= 5x1+4x2+3x3+3x4-x5: 9.°.°.° 9.Write down thegeneral solution ofthesystem given inProblem 4,using the normal fundamental system ofsolutions ofthecorresponding homogeneous system (found inProblem 8). 10.Determine therank andbasis minor ofthefollowing matrices: 1-2 3-1 —.1 -2 1O1OO 2-I 1 O-2 -2 11OOO A1I -2 -5 8-4 3-1 ,A2=O11OO. 6 O-1 2-7 -5 OO11O -1-11-121 01011 11.Suppose thematrix Ahasanonvanishing minor Moforder r,while every minor oforder r+1containing alltheelements ofMvanishes. Prove thatA hasrank r. 12.Construct amatrix an a12 a13A: a21 a22 a23 such thattheminors an a12 an ala a12 aia=P, 1Q, :R a21 a22 a21 a23 a22 a23 have theindicated values P,QandR.CHAP. 3 PROBLEMS 73 13.Forthesystem ofequations 7! 2a,kxk:b, (j=1,...,n) (8) k=1 with asquare coeﬁicient matrix, prove “Fredholm’s alternative,” which asserts that(8)either hasaunique solution forarbitrary bl,...,b,,orelsethecorre- sponding homogeneous system Za,-kxk:O (j:1,...,n) >.'1- hasanontriw'al solution. 14.Prove thatthesystem ofequations allxl '5''"‘l‘alnxn :bl: an1x1+ III+armxn =bns am-1,1x1 "I""'+a'n+1,nxn :bn-1-la subject tothecondition an IIIam - - '-'5O, anl IIIarm issolvable ifandonly if an 'am bl =0. anl I arm bn an+1,1 "' an+1,n bn-l-1 15(Elimination ofunknowns). Prove thatthesystem aux: '5'''''1‘alnxn Ibllyl +"''5blkyl; ‘FC1’ alllxl II’ armxn :bnlyl +III‘I7bnkyk +cm an+1_1x1 +"'+a1l+l,ﬂxn Ibn+1_1}’1 +"'+bn+l,kyk +¢‘n+1 containing theparameters yl,...,yk,subject tothecondition an ''Iant ' ' '#O anl IIIarm 74 SYSTEMS OF LINEAR EQUATIONS CHAP. 3 issolvable ifandonly iftheparameters yl,...,yksatisfy theequation an '''am an an ''am blk 1....+k yl anl I arm bln y aril I arm brtk an+1_1 ar1+l,l bn+1,1 an+1,1 am-1,11 am-1,1: an am C1 + = anl arm an art+l,l '''ar1+1.ri ‘In+10. chapter 4 LINEAR FUNCTIONS OFAVECTOR ARGUMENT Incourses onmathematical analysis onestudies functions ofoneormore real variables. Such functions can beregarded asfunctions ofavector argument. Forexample, afunction ofthree variables canberegarded asa function whose argument isavector ofthespace V3.This suggests studying functions whose arguments arevectors from anarbitrary linear space. In making thisstudy, wewillforthetime being restrict ourselves tothesimplest functions ofthiskind, namely linear functions. Wewill study both linear numerical functions ofavector argument, i.e.,functions whose values are numbers, and linear vector functions ofavector argument, i.e., functions whose values arevectors. Linear vector functions, otherwise known as linear operators, areofgreat importance inlinear algebra anditsapplications. 4.l.Linear Forms 4.11. Anumerical function L(x) ofavector argument x,deﬁned ona linear space Kover anumber ﬁeld K,iscalled alinearform ifitsatisﬁes the following conditions: a)L(x+y):L(x) +L(y) forevery x,yeK; b)L(otx) :aL(x) forevery xeKandevery ateK. Inother words, alinear form L(x) isamorphism ofthelinear space Kinto theone-dimensional space K1:K(cf.Sec. 2.71). Byusing induction, we easily verify that conditions a)andb)imply theformula L(°‘1x1 +'''‘I7akxk) :°@1L(x1) -I‘'''+a'kL(xk)9 (1) 75 76 LINEAR FUNCTIONS OF AVECTOR ARGUMENT CHAP. 4 where xl,...,xkarearbitrary vectors inKand oil,...,v.2arearbitrary numbers inK. 4.12. Examples a.Suppose abasis ischosen inann-dimensional space K,sothat every vector XEK can bespeciﬁed byitscomponents Z1,Z2,...,Z”.Then L(x) =Z1(theﬁrst component) isobviously alinear form inx. b.Amore general linear form inthesame space isgiven bytheexpression L(x) :Zllcgka k=1 witharbitrary ﬁxed coeﬁicients ll,l2,...,l,,. c.Anexample ofalinear form inthespace K(a, b)(where KisRorC)‘l' istheexpression L(x)=X00). where toisaﬁxed point oftheinterval a<t<b. d.Inthesame space wecanstudy thelinear form I) L(x) :JI l(t)x(t) dt, where l(t)isaﬁxed continuous function. e.Inthespace V3thescalar product (x,x3)ofthevector xwith aﬁxed vector x3eV3isalinear form inx. Linear forms deﬁned oninﬁnite-dimensional spaces areusually called linear functionals. 4.13. We now ﬁnd thegeneral representation ofalinear form L(x) deﬁned onann-dimensional space K,,.Letel,e2,...,enbeanarbitrary basis ofthespace K,,,anddenote thequantity L(e,,) bylk(k=1,2,...,n). Then, by(1),given any 71 x:zlkek, lc=1 wehave L(x) :Lxglgkek) :g1€uL(ek) :kg1li¢€i.-3 i.e.,thevalue ofthelinear form L(x) isalinear combination ofthecom- ponents ofthevector x,with theﬁxed coeﬁicients ll,l2,...,l,,.Thus the TRecall Secs. 2.15c and2.15d. SEC4-2 LINEAR OPERATORS 77 most general representation ofalinear form inann-dimensional linear space hasalready been encountered inExample 4.12b. 4.14. Inacomplex linear space Cwecanalso consider another type of linear form, called alinear form ofthesecond kind (inthis context, the linear form deﬁned inSec. 4.11 iscalled alinear form oftheﬁrst kind). A numerical function L(x) ofavector argument x,deﬁned onacomplex linear space C,iscalled alinear form ofthesecond kind ifitsatisﬁes thefollowing twoconditions: a’)L(x+y)=L(x) +L(y) forevery x,yeC; b’)L(otx) :&L(x) forevery xeCandevery complex number at=ix,+ iot2(here ii:ix,-iot2isthecomplex conjugate ofat). Foralinear form ofthesecond kind, theanalogue offormula (1)becomes L(ot1x1 -1-----1-otkxk) :6't1L(x1) —l—---—l—EtkL(xk), (l’) valid forarbitrary xl,...,xkinCandarbitrary complex numbers oil,...,oak. 4.15. Anexample ofalinear form ofthesecond kind inann-dimensional complex space C”with basis el,...,enisgiven bythefunction L(x)= where ll,...,l,,arearbitrary ﬁxed complex numbers and Z1,...,inare thecomponents ofthevector xwith respect tothebasis el,...,en.More- over, thisformula gives thegeneral representation ofalinear form ofthe second kind deﬁned onthespace C”.Infact, letL(x) beanarbitrary linear form ofthesecond kind, and letll=L(e,), ...,l,,:L(e,,). Then, given anyxeC,,,itfollows from (1')that L(x) :Lxkﬁlgireir) =I§lEkL(ek) Zkﬁlliréiu asrequired. I I 4.2.Linear Operators 4.21. Asjust shown, alinear form L(x) deﬁned onalinear space Kis just amorphism ofKinto theone-dimensional space K1.More generally, wenow consider amorphism A:A(x) ofalinear space Xintoanother linear space Yover thesame ﬁeld K(XandYmay coincide). Asalready noted in Sec.2.71, A(x) isalsocalled alinear operator, mapping XintoY.Instead of A(x), wewilloften write simply Ax. Bythedeﬁnition ofamorphism, A(x) satisﬁes thefollowing conditions: a)A(x+y):Ax+Ayforevery x,yeX; b)A(otx) =<xAx forevery xeXandevery at6K. 78 LINEAR FUNCTIONS OF AVECTOR ARGUMENT CHAP. 4 Justasforlinear forms, conditions a)andb)imply themore general formula A(°‘1x1 +'''+auxk) :°‘1Ax1 rl"'''-I‘ai¢Axi¢ forarbitrary xl,...,xkinXandarbitrary oil,...,akinK. 4.22. Examples a.The operator‘l' associating thezero vector ofthespace Ywith every vector xofthespace Xisobviously alinear operator. This operator is called thezero operator, denoted by0. b.Given anylinear operator Amapping thespace Xinto thespace Y, let Bx:-Ax. Itiseasy toseethat theoperator Bsodeﬁned isalso alinear operator mapping Xinto Y.This operator iscalled thenegative oftheoperator A. c.Letel,...,enbeabasis inthespace X,andletvectorsfl, ...,f,,in thespace Ybeassociated with thevectors el,...,eninanarbitrary way. Then there exists aunique linear operator Amapping XintoYandcarrying every vector ekinto thecorresponding vectorfk (k:1,...,n). Infact, if such anoperator Aexists, then, given anyvector X:2ikek EX, (2) k=l wehave Ax:A<§lZ,,e,,) =,§lZ,,Ae,, :,§lZ,,f,,, thereby proving theuniqueness ofA.Ontheother hand, given anyvector (2),wecanset Ax:ﬁlgkfki bydeﬁnition. The resulting operator, asiseasily veriﬁed, islinear, maps X into Y,andatthesame time carries every vector ekinto thecorresponding vectorﬁ, (k:1,...,n). d.Suppose thatwith every vector xofthespace Xweassociate thesame vector x,thereby obtaining alinear operator E,mapping Xintoitself. Then Eiscalled theidentity operator orunitoperator. 4.23. Matrix representation oflinear operators. LetAbealinear operator mapping aspace Xofdimension ninto aspace Yofdimension m.Let 1'Here weusetheterm operator asasynonym forfunction (mapping onelinear space intoanother). SEC4.2 L1NEAR OPERATORS 79 e1,...,e,,beaﬁxed basis inXandf,,...,f,,,aﬁxed basis inY,The vector e1ismapped byAinto some vector Ae,ofthespace Y,which, like every vector ofY,hasanexpansion Aer =aiI)f1 'i'alanfz 'i''''‘La(7y1|,),fI71|, with respect tothebasis vectors f,,...,f,,,. The operator Ahasasimilar eﬁ‘ect ontheother basis vectors: A32 =aiIIf1 'i'aI2IIf2 'i'‘'''i'a‘.§.’f,,,. Ae.=al"’f.+a§"’f.+---+Mb‘..- These formulas canbewritten more concisely as YllAe,-=§a§"f, (j=l,2,...,n). (3) (=1 The coeﬁicients ax”(i=1,...,m;j :1,...,n)deﬁne anm><nmatrix‘]' aim apt ...apt» apt age» ...apt» A=A(e.f) = , agli air? airs» called thematrix oftheoperator Arelative tothebases {e}={e,,...,en} and{f}={f,,...,f},,}. The components ofthevectors Ael, Ae2, ...,Ae,, with respect tothebasis {f}serve asthecolumns ofthismatrix.I Now, given anyvector FPP..I‘/l=.-J“..rs4..(I)_><X: Y=Ax= "lift- With aview toexpressing thecomponents 111,...,-qmofthevector yin terms ofthecomponents Z1,...,Z"ofthevector x,weobserve that y=§1*/hf, =Ax=A<§1Z,-ej) =§1Z,»Ae,- i1v1=1.“..Il\/ls§Is3= W.=2,(,21a£"’£ l TI.e.,amatrix with mrows andncolumns. 3‘.Note thedistinction between thesymbol A(boldface Roman) foranoperator andthe corresponding symbol A(lightface Italic) forthematrix ofA. 80 LINEAR FUNCTIONS orAVECTOR ARGUMENT CHAP. 4 Comparing coeiﬁcients ofthevector f,-,weﬁndthat Tl ‘q,=zafilij (i=1,2,...,m), (4) j=1 or,inexpanded form I'll=aiII€1 ‘l“ai2I€2 ‘l“'''‘l“ai.III€m 4.=a;"&.+art.+---+a;"’£......................... .. (5) 711,1=aI1hI€1 +aI1?iI€2 +'I'+aI1liII€ri: Therefore from aknowledge ofthematrix oftheoperator Arelative tothe basis e,,e2,...,e,,wecandetermine theresult ofapplying Atoanyvector Tl x:2£191",~=1 ofthespace X.Infact, theequations (5)express thecomponents ofthe vector y=Axaslinear combinations ofthecomponents ofx.Note that thecoeﬂicient matrix ofthesystem ofthe equations (5)isjustthematrix A(,_,,. Next let||a§’I’l| beanarbitrary m><nmatrix, where thesuperscript is thecolumn number andthesubscript istherownumber. Given anyvector 4..,_l\/lEin2'.‘x1 weconstruct thevector Y=12,‘/1.-1‘. with components 111,112,...,1j,,,determined by(5).Itiseasy toseethat theoperator Aeﬁecting thismapping ofthevector xinto thevector yisa linear operator. Wenow construct thematrix oftheoperator Arelative to thebasis e1,e2,...,en.Since thevector e1hascomponents Z,=1,Z3=0, ...,Z,=0,itfollows from (5)that thecomponents ofthevector Ae,will bethenumbers a§‘>,ax”, ...,al“ sothat m1 Aer =aiI)f1 'i'abnfz 'i'''''i'aI1hIfm- Similarly, A@r=¢1I”1’r+¢1§”1‘2+"'+¢1§,’i’i‘.,. (1-1.2.....»). Therefore thematrix oftheoperator Acoincides with theoriginal matrix ﬂax”||.Thus every m><nmatrix isthematrix ofalinear operator Amapping ann-dimensional space Xinto anm-dimensional space Y,with ﬁxed bases el,...,e,,inXandfl, ...,f,,,inY.Thus (3),orequivalently (4),establishes aone-to-one correspondence between linear operators mapping aspace X sEc.4.2 LINEAR OPERATORS 81 (with basis el,...,e,,)into aspace Y(with basis f,,...,f,,,) andm><n matrices made upofnumbers from theﬁeld K.Inparticular, identical operators AandB(i.e., operators such that Ax=Bxforevery x6X)have identical matrices. Finally wenote that (5)canbeused toconstruct theoperator Adirectly (and uniquely) from thematrix A=||af.’I’|l. Infact, Aisjustthecoefﬁcient matrix ofthesystem (5). 4.24. Examples a.Clearly, thematrix ofthezero operator (seeExample 4.22a) relative toanybasis inthespace Xandanybasis inthespace Yconsists entirely of zeros. b.If||a§’I’|| isthematrix ofA,then thematrix ofthenegative operator (seeExample 4.22b) isobviously just-Ilafilll. c.Letm>nandsuppose theoperator Acarries thevectors ofthebasis el,...,e,,ofthespace Xintolinearly independent vectorsfl, ...,f,,ofthe space Y.Weaugment thevectors f,,...,f,,bythevectors f,,+,, ...,f,,,to make abasis forthewhole space Y.Then thematrix oftheoperator A relative tothebases e,,...,e,,andf,,...,f,,,isclearly oftheform M /'*-’\—"-\ H m 00 1 0 000---0 d.lnparticular, thematrix oftheidentity operator E(see Example 4.22d) relative tothebasis e,,...,e,,ofthespace X(thedomain ofE)and thebasis e,,...,enofthesame space (therange ofE)isjust Amatrix ofthisform iscalled theunitmatrix oridentity matrix oforder n. 82 LINEAR FUNCTIONS orAVECTOR ARGUMENT CHAP. 4 4.3.Sums andProducts ofOperators Wenow consider addition ofoperators andmultiplication ofoperators both bynumbers andbyother operators. First wenote that twooperators AandBmapping aspace Xinto aspace Yaresaid tobeequal (written A=B)ifAx =BxforeveryxeX. 4.31. Addition ofoperators. Given twolinear operators AandBmapping aspace Xintoaspace Y,theoperator C=A+Bisdeﬁned bytheformula CxE(A+B)x=Ax+Bx. (6) Obviously, Calso maps thespace Xinto thespace Y.Toverify that Cis again alinear operator, letx=oc,x1 +oc2x2. Then C(°‘1x1 +@2952) =A(a1x1 +@2752) +B(a1x1 +@2952) =ot,Ax, —l—ot2Ax2 —l—ot,Bx, —l—ot2Bx2 =a,(Ax1 +Bx,) +a2(Ax2 +Bx2) :ot1Cx, +<x2Cx2, sothat both conditions-a) and b)ofSec. 4.21 aresatisﬁed. The linear operator Cdeﬁned by(6)iscalled thesum oftheoperators AandB. Itiseasily veriﬁed that A+B=B+A, (A-1-B)+C=A-l-(B-l-C), (7) A+0=A, A+(—A) =0. where A,BandCarearbitrary linear operators, 0isthezero operator (see Example 4.22a), and —A isthenegative oftheoperator A(see Example 4.22b), i.e.,theoperator carrying thevector x6Xinto thevector —Ax. 4.32. Multiplication ofanoperator byanumber. Let Abealinear operator mapping aspace Xinto aspace Y,andlet71beanumber from the ﬁeld K.Then theoperator B=AA,called theproduct oftheoperator Aand thenumber 71,isdeﬁned bytheformula Bx=(7\A)x=).(Ax). Itiseasily veriﬁed (just asinSec. 4.31) that this operator islinear, and moreover that 7\1O\2A) :O\17\2)A, 1-A:A, ,(7)()1,—l—)\2)A =MA —l—712A, ).(A+B)-IA+AB. SEC. 4.3 SUMS AND PRODUCTS OEOPERATORS The relations (7)and (7')show that thesetofalllinear operators mapping alinear space Xintoalinear space Yisitself alinear space. 4.33. Multiplication ofoperators. LetAbealinear operator mapping thespace Xinto thespace YandBalinear operator mapping thespace Y into thespace Z(where allthespaces areover thesame number ﬁeld K). Then theoperator P=BA, called theproduct oftheoperator Bandthe operator A(inthatorder), isdeﬁned astheoperator mapping XintoZsuch that Px=(BA)x =B(Ax) (note that ﬁrsttheoperator Aactsonthevector xandthen theoperator B actsontheresulting vector inthespace Y).The operator Pisagain linear, since P(ot,x, —l—ot2x2) =B[A(ot,x, —l—ot2x2)] =B(ot,Ax, —l—ot2Ax2) =ot,BAx, +<x2BAx2 =ot1Px, +oi2Px2. 4.34. Thefollowing relations areeasily veriﬁed: a)).(BA) =(71B)A forevery number 7.eK and arbitrary operators A mapping thespace Xinto thespace YandBmapping thespace Yinto the space Z; b)(A+B)C =AC+BCforarbitrary operators AandBmapping the space Yinto thespace ZandCmapping thespace Xintothespace Y; c)A(B +C)=AB+ACforarbitrary operators BandCmapping the space Xintothespace YandAmapping thespace Yintothespace Z; d)(AB)C =A(BC) forarbitrary operators Cmapping thespace Xinto thespace Y,Bmapping thespace Yinto thespace Z,andCmapping the space Zintothespace W31‘ For example, toverify d),according tothedeﬁnition ofoperator equality wemust prove theidentity [A(BC)x] =l(AB)ClX forevery xeX.Butbythevery deﬁnition oftheoperator product, wehave [A(BC)x] =A[(BC)x] =A[B(Cx)], l(AB)ClX =(AB)(CX) =A[B(Cx)], which implies therequired formula. Theother formulas areproved similarly. TTheassociative lawforoperator multiplication isexpressed byd),andthedistributive lawbyb)andc). 84 LINEAR FUNCTIONS OF AVECTOR ARGUMENT CHAP. 4 4.4.Corresponding Operations onMatrices Wenow study thematrix analogues ofthealgebraic operations onlinear operators described inSec.4.4. 4.41. Addition ofoperators. LetAandBbetwolinear operators mapping aspace Xwith basis e,,...,e,,into aspace Ywith basis f,,...,f,,,. More- over, letA=llaf.’’l|bethematrix oftheoperator AandB=llbf.”llthematrix oftheoperator B,relative tothese bases. Then Aer=§4i”t.-. Be,=§bib‘. <1"=1.2.....4). i=1 i=1 andhence (A+Bie.=Aer+Be.=__Zy£”+b§”>t.-- Itfollows that thematrix corresponding totheoperator A+Bisjust llax.” —l—b§’I’|l. This matrix iscalled thesum ofthematrices |lax.’I’l| and l|bx.‘I’||- Thus thesum A+Bisdeﬁned forevery pair ofmatrices AandBwith the same number ofrows andthesame number ofcolumns. 4.42. Multiplication ofanoperator byanumber. With thesame notation asbefore, wehave Tl().A)e,- =).(Ae,) =xa,l”f,. Itfollows thatthematrix corresponding totheoperator AAisjustthematrix ||).af.’I’||, obtained bymultiplying alltheelements ofthematrix Ilaf/I’|| by thenumber 71.This matrix iscalled theproduct ofthematrix Ila:/"l| andthe number )1. Since there isaone-to-one correspondence between m><nmatrices and linear operators mapping ann-dimensional space into anm-dimensional space (seeSec.4.22), there isaone-to-one correspondence between algebraic operations involving operators and theanalogous operations involving matrices. Hence, since operators obey therules (7)and(7'), thesame isalso true ofmatrices (ofcourse, thiscaneasily beveriﬁed directly). Thus wesee that thesetofallm><nmatrices isitselfa linear space, which, byitsvery construction, isisomorphic tothelinear space ofalllinear operators mapping ann-dimensional space Xintoanm-dimensional space Y. 4.43. Multiplication ofoperators. LetX,YandZbelinear spaces, and lete,,...,e,, beabasis inX,f,,...,f," abasis inY,andg,,...,g, a SEC. 4.4 CORRESPONDING OPERATIONS ON MATRICES basis inZ.LetBbealinear operator mapping Xinto Ywith m><nmatrix ||b§’I’||, sothat Be.=_§b£”t.- <1-12...»). andletAbealinear operator mapping Yinto Zwith q><mmatrix ||ax’I’||_ sothat G Af,=Z1a,(,'lg,, (i=1,...,m). k= Then fortheproduct P==ABwehave (AB)e, =A(Be,) =Ag b,I’I’f,. = b,l’I’Af,- 7" Q _ G 7!! V _ :2btijizaliciigk :2<2a;:ib1{1i)gk_ tI=1 k=1 =1 i=1 Hence theelements pf,”ofthematrix Poftheoperator P=ABaregiven byPr‘ p§,’I’=21a;,'I’b,l’I’ (j=1,...,n;k=l,...,q). (3) This isthedesired result, which canbeexpressed asfollows: Theelement ofthematrix Pbelonging tothekthrowandjthcolumn equals thesumofthe products oftheelements ofthekthrowofthematrix Awith thecorresponding elements ofthejthcolumn ofthematrix B.The matrix P=1|px’I’|| which is obtained from thematrices A=||a§,”|| andB=||b§”|| inaccordance with formula (8)iscalled theproduct ofthematrices AandB(inthat order). Itshould benoted thatfortheproduct P=ABtomake sense, thenumber ofcolumns inAmust equal thenumber ofrows inB.Then Pwillhave the same number ofrows asAandthesame number ofcolumns asB.Thisfact can beexpressed more strikingly inthe“m><nnotation," namely, the product ABofaq><lmatrix Aandanm><nmatrix Bisdeﬁned ifl=m, inwhich case ABisaq><nmatrix. Both products ABandBAaredeﬁned ifl=mandq=n,inwhich case ABisasquare n><nmatrix while BAisa square m><mmatrix. Moreover, ifl=m=q=n,i.e., ifboth matrices AandBaresquare n><nmatrices, then ABandBAarealson><nmatrices. However, these products need notbeequal. Forexample, 0110 00 ioloo 10 100101 0010_00'4 86 LINEAR FUNCTIONS OE AVECTOR ARGUMENT CHAP. 4 Thus multiplication ofsquare matrices isingeneral noncommutative. As fortheassociative and distributive laws, thesituation ismore favorable. Infact, asshown inSec. 4.34, operator multiplication obeys theassociative and distributive laws, and hence wecan assert that thesame istrue of matrix multiplication, since there isaone-to-one correspondence between operators andmatrices associating sums andproducts ofoperators with the sums andproducts ofthecorresponding matrices. 4.44. Examples Inthefollowing examples, wewrite both indices ofmatrix elements as subscripts, sothat theelement a,,,ofthematrix A=|la,,,|| belongs tothe jthrow andthekthcolumn. Inthisnotation, formula (8)forthematrix product P=ABtakes theform I»...-=.§,“...»1>.-,~ (i=1.---.n;l<=l.---.q)- (8') a.Suppose wemultiply anm><nmatrix A=||a,»,,|| from theleftbyan m><mmatrix B,,:||b,,,l| with allitselements b2,,equal tozero except the single element b,,:1.Then by(8')wegetthem><nmatrix (S) an a12 ant BrsA:(r) 1 as! as2 II asri arnl am2 arnri Z(r) as! as2 IIIa8’!l sothat therthrowofthematrix B,,A consists oftheelements ofthesthrow ofthematrix Awhile allother elements ofB,,A vanish. b.Suppose wemultiply anm><nmatrix A=|la,~,,l| ontheright byan n><nmatrix Cm=llc,-kll with allitselements cjkequal tozero except the SEC.4.4 CORRESPONDING OPERATION$ QNM,\TR1(j[-)5 87 single element cu=1.Then by(8')wegetthem><nmatrix (q) all --. alp -.. ax” I an ah ,2” AC“: _ (P) aml '''am, am, . (q) Q ah 0 : 0 an 0 0 amp 0 sothat theqthcolumn ofthematrix ACM consists oftheelements ofthepth column ofthematrix Awhile allother elements ofAC9, vanish. c.With thesame matrices B”,AandCMwehave (q) B,_,AC,,,:(r) 0 am, 0 0... Q ...0 Thus B,_,AC,,, isanm><nmatrix allofwhose elements vanish with the (possible) exception ofthesingle element, equal toa_,,,,appearing intherth rowandqthcolumn. d.Bywhat m><mmatrix Dmust wemultiply anm><nmatrix Afrom thelefttomake thematrix DAcoincide with thematrix obtained from Aby interchanging itsrthandsthrows? Solution. Example 4.44a shows that thematrix whose rthrowisthesth row ofthematrix Aisobtained bymultiplying Aontheleftbythe m><mmatrix B,_,. Buttheother rows oftheresulting matrix vanish. Itis 88 LINEAR FUNCTIONS orAvEcToR ARGUMENT CHAP. 4 now clear that togettherequired matrix, wemust multiply Afrom the leftbythem><mmatrixI<1)oi 1 1 D=Brs+Bsr+ZBij= Hér JI¢s _ _ 1 .._1 () 1 1 e.Bywhat n><nmatrix Gmust wemultiply anm><nmatrix Afrom the right tomake thematrix AGcoincide with thematrix obtained from Aby interchanging itspthandqthcolumns? Solution. Byanargument likethat inExample 4.44d, wehave G=Cara +Car +zckivan k¢q f.Bywhat m><mmatrix Fmust wemultiply anm><nmatrix Afrom thelefttomake thematrix FAcoincide with thematrix obtained from Aby adding Atimes itssthrowtoitsrthrow? Solution. Using Example 4.44a, weobviously have F=E+AB,, where Eistheunit matrix oforder m. g.Bywhat n><nmatrix Hmust wemultiply anm><nmatrix Afrom theright tomake thematrix AHcoincide with thematrix obtained from Aby adding Atimes itspthcolumn toitsqthcolumn? Solution. Clearly, H=E+1.tC,,, where Eistheunit matrix oforder n. 4.5.Further Properties ofMatrix Multiplication 4.51. Multiplication ofblock matrices. Inmultiplying matrices, itis sometimes convenient topartition thematrices into blocks andafterwards SEC. 4.5 FURTHER PROPERTIES OF MATRIX MULTIPLICATION deal with theblocks asseparate entities. Suppose wearegiven anm><n matrix Aandann><pmatrix B,partitioned into blocks asfollows: '1 P M /im A11 A12 I'' B11 B12 III A A ... B Bx ...A:m 21 22 _,B:n 21 -2 j_ Suppose further that every “block-row” ofthematrix Acontains thesame number ofblocks asevery “block-column” ofthematrix B,andthat the “width” ofevery block A,,,ofthematrix Acoincides with the“height” of every block Bk,ofthematrix B.Then theproducts A,»,,B,,, allmake sense, andinfactarerectangular matrices ofsizedepending ontheindices jands (but notontheindex k).Wethen have thefollowing multiplication rule: Theproduct matrix ABismade upofblocks constructed from theblocks of thematrices AandBinthesame wayastheelements ofABareconstructed from theelements ofA andB,i.e., -411311 ‘l‘-412321 ‘l‘'’‘j-411312 ‘l‘1412322 ‘l‘'" "I _ /421811 +A22B21 +'''/421812 +/422822 +II' III _(9)AB ..-.- ----6 -.... ----6 ... Toprove (9),letibetheindex ofablock-row ofAcontaining thekth ordinary rowofA,andletjbetheindex ofablock-column ofB containing theqthordinary column ofB.Bythegeneral ruleofSec.4.43, theelements oftheproduct matrix P=ABareoftheform PM=aklbla ‘l‘'''‘l‘akribrul :(ak1b1q +III+akaibpq) +III+(akrbrq +III+aknbriq)v where parentheses areinserted inkeeping with thewidths ofblocks ofA (and heights ofblocks ofB).Buttheﬁrstterm inparentheses istheelement inthekthrowandqthcolumn oftheblock A,»1B,,-, thesecond term inparen- theses (not written) istheelement inthekthrowandqthcolumn oftheblock A,-1B2,, andsoon.Thus pk,istheelement inthekthrowandqthcolumn of theblock A,-1B1, +---+A,,B,,, itself theblock intheithrow andjth column ofthematrix P=ABregarded asablock matrix. Theproof of(9) isnow complete. 90 LINEAR FUNCTIONS OF AVECTOR ARGUMENT CHAP. 4 4.52. Multiplication ofquasi-diagonal matrices. Amatrix issaid tobe quasi-diagonal ifitisoftheform AllE A= ' , E where the“oﬁI-diagonal” blocks consist entirely ofzeros. Suppose theblock Akkisanmk><nkmatrix (k=1,...,s),andconsider thequasi-diagonal matrix 7.. —ElB: . 2 |& where theblock Bkkisannk><pkmatrix (k:1,...,s).Then, using the rule ofSec. 4.51 tomultiply thematrices AandB,weimmediately get A..B.. I /422822 AB= I A,,B,, Thus inthiscase thematrix ABisagain aquasi-diagonal matrix, where the block AkkBkk hasmkrows andpkcolumns. 4.53. Multiplication oftransposed matrices. Given anm><nmatrix A=l|a,k||, bythetranspose ofA(cf.Sec. 1.41) ismeant then><mmatrix A’=||a,T,_|| such that a;»k=ak,- (j=1,...,n;k=1,...,m). LetAbeanm><nmatrix andBann><pmatrix. Then theproduct P=AB isdeﬁned andisanm><pmatrix. Moreover, theproduct B’A’ ofthetrans- posed matrices A’andB’isalso deﬁned andisap><mmatrix. Wenow show that B’A’ =(AB)’. (10) SEC.4.5 FURTHER PROPERTIES orMATRIX MULTIPLICATION 91 Lettheelements ofthematrices A,B,P=AB, A’,B’andP’bedenoted by an,b,-,»,p,,-, a,T,.=a,-,,b,T,.=b,,,pg,=p,-,-.Then, bytherule formatrix multiplication, Pvt=Pitt=Ear.-bit. =Zaiiblci :Zblciaiiv j=1 i=1 ]'=1 where thesummation isover theindex jwith theindices iandjheld'f1xed. Thus toform theelement pk,ofthematrix P’,theelements ofthekthrow ofB’aremultiplied bythecorresponding elements oftheithcolumn ofA’ and then added. Inother words, using therule formatrix multiplication once again, weseethatP’istheproduct ofB’andA’(inthatorder), thereby proving (10). 4.54. Minors oftheproduct oftwo matrices. Given anm><nmatrix A=Hajkll andann><pmatrix B=||b,kl|, weconstruct them><pmatrix P=AB=||p,-k||. Fixing therows with indices oak,...,ak(oi,<---<otk) and thecolumns with indices I-3,,...,Bk(B,<---<Bk),where k<m, l<Cp,wenow consider theproblem ofcalculating theminor a111b1B1 +III+alrribriﬁi IIIaallblelt: +III+a°‘1’!lb7|B70 Ia121b1B1 +III+aoieribnﬁi IIIaciglbigk +III+aa2,,b,,B,c aakraier Ti‘I''-I‘a1k’!lb!IB1 IIIa1;_1b1B;_. "I"'''“IFadtjcnbnﬁjc (11) formed from these rows andcolumns. Tomake thiscalculation, weusethe linear property ofdeterminants (Sec. 1.44). Thevthcolumn oftheminor (11) isthesum ofk“elementary columns" with elements oftheform a,‘_,»b,»Bv (where thecolumn indices iandvareﬁxed, andtherowindex jvaries from 1tok).Hence thewhole minor (11)isthesum ofk’~'“elementary determin- ants” consisting only ofelementary columns. Since ineach elementary column thefactor b,Bvdoes notchange aswegodown thecolumn, itcanbe factored outoftheelementary determinant. After this, each elementary determinant takes theform am", aa,t2 ''Iaaktk aaie, aa,i. aeaii ’ (12) b'I1I31bi2B2 IIIbit.-I32 aaki, aakt2 I'Iaakik where i,,i2,...,ikarecertain numbers from 1ton.Ifsome ofthese numbers arethesame, then clearly thecorresponding elementary determinant vanishes. Moreover, thisisalways thecase ifk>n.Therefore ifthematrix ABhas minors oforder k>n,they miist allvanish. 92 LINEAR FUNCTIONS orAVECTOR ARGUMENT CHAP. 4 Returning tothecase k<n,wenote thatitisonly necessary toconsider elementary determinants forwhich theindices i,,i2,...,ikarealldiﬁerent. Inthiscase, thedeterminant aoqil aoi1i2 IIIaoilik a°‘2'I1 a°‘2'I2 IIIa°‘2'Iir aoikil aoikr'2 IIIaoikik isthesame (except possibly forsign) astheminor Mj‘,1"_"_-_',',.°:¢(A) where the indices jl,...,jk(j,<---<jk) aretheindices i1,...,ikrearranged in increasing order. Toﬁnd thesign which must beascribed to(13) toget MjF:j_'_'_'_',i‘x’¢(A), wesuccessively interchange adjacent columns of(13) until we arrive atthenormal arrangement ofthecolumns, i.e.,thearrangement they have inthematrix Aitself. Ateach interchange oftwoadjacent columns, the determinant (13) changes sign and thenumber ofinversions intheper- mutation i,,i2,...,ikchanges byunity. Since intheﬁnal arrangement of thecolumns, thesubscripts areinnatural order (i.e., without inversions), thenumber ofsuccessive changes ofsign isequal tothenumber ofinversions inthepermutation i,,i2,...,ik.‘]'LetN(i) denote thenumber ofsignchanges. Then theexpression (12)takes theform (T7l)NIIIbi1B1 bi2B2 IIIbikBkM1II1I-I'I'I'I'I1IirI(A Toobtain (11),wemust now add upalltheexpressions oftheform (14). First weaddupalltheexpressions with thesame setofindicesjl, ...,jk, taking outthecommon factors M,?‘11_'_'_'_'_',:‘,.'=(A). The remaining expression is then (—l)NIIIbi'1Brbi2B2 IIIbi);;B)¢7 where thesummation isover alldistinct setsofindices i1,i2,...,ik(these indices range from 1ton).Butthisexpression isjusttheminor M$111,-_-_-_-"2’¢k(B) Thus ﬁnally wegettheformula M§::;;:;t:<4B> =2M?::;;;:?:<4>Mi::;;::i:<B>. <15) where the summation isover alldistinct sets ofindices j1,j2, ...,jk (1<j1<j2 <---<jk <n).Thetotal number ofterms inthesum (15)is justthebinomial coefﬁcient C;_ n! Tkl(n-k)!I TItisassumed thatthechange intheindices ik,i2,...,i,,produced byevery column interchange causes asmaller index toappear before alarger index, with theresult thatthe total number ofinversions changes byexactly one. SEC. 4.6 THE RANGE AND NULL SPACE OF ALINEAR OPERATOR 93 Our result canbesummarized inthefollowing THEOREM. Every minor oforder k<nofthematrix ABcanbeexpressed interms oftheminors ofthesame order ofthematrices AandB,intheway given byformula (15). 4.6.TheRange andNull Space ofaLinear Operator 4.61. LetAbealinear operator mapping alinear space Xinto alinear space Y(inthenotation ofSec.2.71, thisisexpressed bywriting A;X ->Y). Letnbethedimension ofXandmthedimension ofY,andchoose anarbitrary basis el,...,e,,inXandfl, ...,f,,,inY.Then, bythemethod ofSec.4.23, wecanassociate theoperator Awith anm><nmatrix A=1|a§"l| (i=1,...,m;j=1,..,,n). LetT(A) betherange ofA,i.e.,thesetofallvectors y=Ax,x6X.We now consider theproblem ofﬁnding thedimension ofthesubspace T(A) from aknowledge ofthematrix A, Writing I,I_l\/I3nx>.~NFx: wehave y=Ax=iiAek.k k=1 Hence therange oftheoperator Acoincides with thelinear manifold spanned bythevectors Ae,, ...,Aek. Asnoted onp.51,thedimension ofthis linear manifold L(Ae1, ...,Ae,,) equals themaximum number oflinearly independent vectors inthesystem Ael, ...,Aek. Weknow from Sec, 4.23 that thecolumns ofthematrix oftheoperator Aconsist ofthecomponents ofthevectors Ael, ...,Ae,,with respect tothebasis el,...,e,,,andhence theproblem ofﬁnding themaximum number oflinearly independent vectors inthesystem Ael, ...,Ae,,reduces atonce tothat ofﬁnding themaximum number oflinearly independent columns ofthematrix A.ButbyTheorem 3.l2c, thelatter quantity isjust therank ofthematrix oftheoperator A. Thus thedimension oftherange ofalinear operator Amapping ann-dimensional space Xintoanm-dimensional space Yequals therank ofthematrix ofA relative toanybasis {e}inXandanybasis {f}inY. Wenote that thechoice ofbases does notmatter here. Therefore the rank ofthematrix ofanoperator Adoes notdepend onthechoice ofbases, i.e.,depends only ontheoperator Aitself. Inwhat follows, therank ofthe matrix oftheoperator A(relative toanybases) willsimply becalled the rank oftheoperator A,denoted byr2. 94 LINEAR FUNCTIONS OE AVECTOR ARGUMENT CHAP. 4 4.62. Next letN(A) bethenullspace oftheoperator A,i.e.,thesetofall vectors xeXsuch that Ax=0,andasbefore letA=||al”|l bethematrix ofA.Wenow consider theproblem ofﬁnding thedimension ofthesubspace N(A) from aknowledge ofthematrix A.Let x:ZZ,»e,»6N(A). i=1 Then thesystem (5),p.80takes theform aIII€1 ‘l“aI12I€2 ‘l“II'‘l“¢1l"’€.. =0, a;"&.+a‘.*’i.+---+a;"’&.=0’ (16) aI1hIE>1 'l'aI1?iI€2 +III'l'aI1iiI€ri =0- Moreover, itisobvious that, conversely, every vector x6Xwhose compo- nents satisfy (16)belongs tothenullspace oftheoperator A.Thus theproblem ofﬁnding thedimension ofthenullspace oftheoperator Aisequivalent to theproblem ofﬁnding thedimension ofthesubspace ofXconsisting ofall solutions ofthesystem (16). Butaccording toSec. 3.51, thedimension n,l ofthissubspace equals n-r,where ristherank ofthecoefﬁcient matrix ofthesystem, orequivalently, therank oftheoperator A.Itfollows that n,l=n-r,l.Thus thedimension ofthenullspace oftheoperator Aequals therank ofthespace X(onwhich Aacts) minus therank oftheoperator A. 4.63. Inparticular, ifthemorphism A:X ->Yisanepimorphism, then T(A) =Yandhence r,l=m.Ifthe morphism A:X —>Yisamonomorphism, then N(A) ={0}and hence r,l=n.The converse assertions arealso true: Iftherank ofthematrix Aequals thenumber mofitsrows, then thedimension ofT(A) coincides with thedimension ofthewhole space Yand hence T(A) =Y.Therefore themorphism Aisanepimorphism ifand only ifr,l =m. Iftherank ofthematrix Aequals thenumber ofitscolumns, then the vectors fl=Ael, ...,f,,=Ae,, arelinearly independent and hence the operator Aisamonomorphism (seeSec. 2.73c). Therefore themorphism A isamonomorphism ifand only ifrjl =n. 4.64. The following proposition istheconverse oftheresults ofSecs. 4.61and4.62: THEOREM. LetXbeann-dimensional linear space and Yanarbitrary linear space. Tlien, given anysiibspaces NCXandTCYthesum ofwhose dimensions equals n,there exists alinear operator A:X->Y such that N(A) =N,T(A) =T. SEC. 4.6 THE RANGE AND NULL SPACE OF ALINEAR OPERATOR Proof. Letthedimensions ofNandTbekandm=n-k,respectively. Moreover, letfl,fl,...,f,,,bemlinearly independent vectors inthesubspace T,andletel,e2,...,e,,beanybasis inthespace Xwhose ﬁrstkvectors lie inthesubspace N(seeSec.2.43). Deﬁning anoperator Abytheconditions Ae,-=0 (i=1,2,...,k), (17) Ae,»+k=f,- (1I=l,2,...,m), wenow show that Asatisﬁes therequirements ofthetheorem. First ofall, itisobvious thatT(A) isthelinear manifold spanned bythevectorsfl,f2, ..., f,,,and hence coincides with thesubspace T.Moreover, by(17), every vector ofthesubspace Nbelongs toN(A), anditremains toshow only that every vector ofN(A) belongs toN.Suppose Ax=0forsome 7|. x=ZZ,e,-. I=I Then, by(17), 0=Ax=A(€1e1 Ti‘I+i.e.) =£164-1/1 Ti‘"'1'" gnfma and hence Zk+l='--=in=0since fl,...,f,,,arelinearly independent. Butthen x= €1e1'i""'i' EIcekeN' I 4.65. Thefollowing theorem ontherank oftheproduct oftwomatrices isaconsequence ofthegeometric notions justintroduced: THEOREM. Therank oftheproduct ABoftwomatrices AandBdoes not exceed therank ofeach ofthefactors. Proof. Naturally, wemust assume that thenumber ofcolumns ofthe matrix Acoincides with thenumber ofrows ofthematrix B,since otherwise theproduct ABcould notbeformed. Thus letAbeanm><nmatrix andB an><pmatrix, andintroduce linear spaces X,YandZwith dimensions n, mandp,respectively. Choose abasis el,...,e,,inthespace X,abasis fl,...,f,,,inthespace Yand abasis gl,...,g,inthespace Z.Using these bases, weassociate alinear operator A:X —>Ywith thematrix Aanda linear operator B:Z ->Xwiht thematrix B(seeSec. 4.23). Then theproduct operator AB:Z —>Ycorresponds totheproduct matrix AB. The range of theoperator ABiscontained intherange oftheoperator A,bythevery deﬁnition ofAB. Since bySec. 4.61 thedimension oftherange ofany operator equals therank ofitsmatrix, weﬁndthat therank oftheproduct of twomatrices does notexceed therank oftheﬁrstfactor. Toprove thatitalso does notexceed therank ofthesecond factor, wegoover totransposed matrices. Using equation (10), p.90,weﬁndthat rank AB=rank (AB)’ =rank B’A’ <rank B’=rank B.I 96 LINEAR FUNCTIONS orAVECTOR ARGUMENT CHAP. 4 4.66. The rank oftheproduct oftwomatrices canactually belessthan therank ofeach factor. Forexample, thematrices 01 10,4= , B= 00 00 both have rank one, buttheir product 00 00AB= hasrank zero. Therefore thefollowing theorem, which gives alower bound rather than anupper bound fortherank oftheproduct oftwomatrices, is ofinterest: THEOREM. LetAbeanm><nmatrix ofrank rAandBann><pmatrix ofrank r,,..Then therank ofthe m><pmatrix ABisnolessthan rA+r,,.—n. Proof First weshow that anyoperator A:X —>Yofrank rcarries every k-dimensional subspace X’CXinto asubspace Y’CYofdimension no lessthan r-(n—k).Choose abasis el,e2,...,eninthespace Xsuch that theﬁrstkbasis vectors lieinthesubspace X’(seeSec.2.43). Thecomponents ofthevectors Ael, Ae2, ...,Aekgenerating thespace Y’occupy theﬁrstk columns ofthematrix oftheoperator A.Byhypothesis, there arerlinearly independent columns inthematrix ofA.Wedivide these columns into two groups, theﬁrstconsisting ofcolumns whose numbers lieintherange 1tok, thesecond consisting ofcolumns whose numbers lieintherange k+1ton. Thesecond group contains nomore than n—kcolumns, andhence theﬁrst group contains nomore than r—(n—k)columns. Thus thesubspace Y’ hasnomore than r—(n—k)linearly independent vectors, asasserted. Now letA:X —>YandB:Z ->Xbelinear operators corresponding tothe matrices AandB.BySec. 4.61, therank ofthematrix oftheoperator AB isjustthedimension oftherange ofAB.Theoperator Bmaps thewhole space Zinto thesubspace T(B) CXofdimension rB.Butasshown above, theoperator Amaps thesubspace T(B) intoasubspace ofdimension noless than rA—(n—r,,»)=rA+rB—n.Thus therange oftheoperator AB, andhence therank ofthematrix ofAB, isnolessthan rA+rB-n.I 4.67. COROLLARY. LetAbeanm><nmatrix andBann><pmatrix, and suppose therank ofoneofthese matrices equals n.Then therank ofAB equals therank oftheother matrix. Proof. Inthiscase, theupper and lower bounds fortherank ofAB, given byTheorems 4.65 and4.66, have thesame value, equal totherank of theother matrix. I SEC. 4.6 THE RANGE AND NULL SPACE OF ALINEAR OPERATOR 4.68. LetAbealinear operator mapping alinear space Xinto alinear space Y.Alinear operator Bmapping YintoXiscalled aleftinverse ofthe operator Aif BA=E istheunit operator inthespace X.The operator Aisthen called aright inverse oftheoperator B.The following theorem gives conditions under which theoperator A(orB)hasaleft(orright) inverse: THEOREM. Theoperator A:X->Y hasaleftinverse ifandonlyifAis amonomorphism. Theoperator B:Y->Xhasaright inverse ifandonly ifB isanepimorphism. Proof. LetAbeamonomorphism with range T(A) CY.Then forevery yeT(A) there isanx6Xsuch thatAx=y,where xisuniquely determined byy since Aisamonomorphism byhypothesis. LetQCYbethesubspace whose direct sum with T(A) isthewhole space Y(seeSec. 2.46). Wenow deﬁne anoperator B:Y->Xbythefollowing rule: ForyeT(A) wesetBy equal tothe(unique) vector xforwhich Ax=y,while otherwise weset By=0 ify€Q, By=ByI ify=y.+yi.y.ET(A).yrEQ- Then itiseasy toseethat theoperator Bislinear andthat BAx =x,for every xeX,sothatBistheleftinverse ofA.However, ifAisnotamono- morphism, there exists anonzero vector xeXsuch that Ax=0.Then for anyB:Y —>Xwehave (BA)x =B(Ax) =B(0) =0,sothat Aindeed fails tohave aleftinverse. Next letB:Y—>X beanepimorphism and letN(B) CYbethenull space ofB, while QCYisthesubspace whose direct sumwith N(B), denoted byN(B) +Q,isthewhole space Y.Since X=B(Y)=B(N(B) +Q)=B(Q). themapping B:Q —>Xisalsoanepimorphism andinfactanisomorphism, since nononzero element y6Qismapped into zero bytheoperator B.We now deﬁne anoperator A:X ->Ybythefollowing rule: Given anyxeX, wesetAxequal tothe(unique) vector y6Qforwhich By=x.Then itis easy toseethat theoperator Aislinear andthat BAx =xforeve_ry xeX, sothat Aistheright inverse ofB.However, ifB:Y->Xisnotanepimor- phism, then BAx qéxforanyoperator A:X ->Yandanyvector xeXsuch that x¢T(B), sothat Bhasnoright inverse. I 4.69. a.Asweknow, theresult ofmultiplying ann><mmatrix Pbyan m><nmatrix Aisasquare n><nmatrix S=PA. 98 LINEAR FUNCTIONS OF AVECTOR ARGUMENT CHAP. 4 IfSistheunitn><nmatrix (seeExample 4.24d), wecallPtheleftinverse ofthematrix A.Similarly, theresult ofmultiplying anm><nmatrix Aby ann><inmatrix Qisasquare I1’!><inmatrix T=AQ, andifTistheunitm><mmatrix, wecallQtheright inverse ofthe matrix A. b.Using theresults ofSec. 4.63, wecannow formulate Theorem 4.68 interms oftherank ofamatrix: THEOREM. Anm><nmatrix Ahasaleftinverse ifand only itsrank equals nandaright inverse andonly itsrank equals m. 4.7.Linear Operators Mapping aSpace K,intoItself 4.71. LetAbealinear operator mapping thespace Xinto itself (this corresponds tosetting Y=XinSec. 4.21). Such anoperator issaid tobe anoperator (acting) inthespace X. Suppose theoperator Aactsinann-dimensional space X=K".Choosing abasis el,...,e,,inthespace X,weusethesame basis inY=Xtoconstruct thematrix oftheoperator A.Then formula (3),p.79becomes Ae,=iZ1a,l’I’e,- (18) (after setting f,=e,-),sothat thecoetﬁcients a,I’I’now form asquare n><n matrix A,called thematrix oftheoperator Ain(orrelative to)thebasis {e}={el,...,e,,}. We will sometimes denote this matrix byAM. The corresponding formula relating thecomponents ofthevectors xandy,where 11 n Y=Ax» Xzzliieii J’=2171191 .7 Iis >1.-=i4.‘."’i.~ (19)i=1 (cf.formula (4), p.80). For aﬁxed basis {e}={el,...,e"}, wegeta one-to-one correspondence between alllinear operators acting inthespace Kn(i.e., mapping K,,into itself) andallsquare n><nmatrices made upof elements oftheunderlying ﬁeld K. 4.72. Examples a.The operator associating thezero vector with every vector ofthe space Xisobviously linear. AsinExample 4.22a, thisoperator iscalled the zero operator. Itisclear that thematrix ofthezero operator relative toany basis consists entirely ofzeros. SEC.4.7 LINEAR OPERATORS MAPPING ASPACE K3INTO ITSELF 99 b.The identity (orunit) operator E,associating thevector xitself with every vector xeX, hasalready been considered inExample 4.22d. Its matrix istheunit(oridentity) matrix oftheform 10---0 ()]...0 E: (cf.Example 4.24d). c.The operator Awhich carries every vector xeXinto Ax,where Aisa ﬁxed number from theﬁeld K,isobviously linear. This operator iscalled thesimilarity operator (with ratio ofsimilitude A).Asinthepreceding example, thesimilarity operator hasthematrix 00---1 inanybasis. d.Wecanspecify avector intheEuclidean plane V2bygiving itspolar coordinates pand <9.The operator Acarrying thevector x=(p,<9)into Ax=(p,<9+<93), where <93isaﬁxed angle, islinear (ascaneasily be veriﬁed bydrawing aﬁgure). This operator iscalled therotation operator through theangle <93. Toconstruct thematrix ofA,wechoose abasis inV2consisting oftwo orthogonal unitvectors elande2.Drawing aﬁgure, weeasily seethat after rotation through theangle <93thevector elgoes into thevector elcos<93+ e2sin<93,while thevector e2goes into -elsin<93+e2cos<93.Hence the matrix oftherotation operator Ahastheform cos<93 -sin <93 sin<93 cos<93 inthebasis el,e2. e.Letel,e2,...,e,, beabasis inann-dimensional space K”, and Suppose thatwith thevector 37‘[\/ls1-\I\XR‘WFTxi I00 LINEAR FUNCTIONS orAVECTOR ARGUMENT CHAP. 4 weassociate thevector "Ii PX=ZEkek k=1 where m<n.Then Pisalinear operator, called theprojection operator onto thesubspace Kmspanned bythevectors el,e2,...,em. Toconstruct thematrix ofP,wenote thatitcarries thevectors el,e2,..., e,,,intothemselves andthevectors e,,,+l, ...,ekintothezero vector. Hence thematrix oftheprojection operator Pinthebasis el,e2,...,ekisjust 1()...()()...() ()1...()()...() (m)00---I0---0. f.Letel,e2,...,e,,beabasis inann-dimensional space Kn,and let Al,A2,...,Akbenﬁxed numbers. Deﬁning anoperator Aforthebasis vectors bytheconditions Ael=Alel, Ae2=A2e2, ..., Aek=Akek, wethen ofcourse uselinearity todeﬁne Aforanyother vector rt. x=Z£1631.- lC=1 bythecondition TI. Ax=2Akikek. k=I Theresulting operator Aissaidtobediagonal relative tothebasis el,e2,.., en;wealso callAadiagonalizable operator. The matrix ofanoperator which isdiagonal relative tothebasis el,e2, ...,ekisofthe form M () ...() 01, 0 QQ 7," inthesame basis. Such amatrix, which canhave nonzero elements only on itsprincipal diagonal, issaid tobediagonal (hence thecorresponding SEC.4.7 LINEAR OPERATORS MAPPING ASPACE K3INTO ITSELF IOI terminology fortheoperator itself). Itshould benoted that thematrix of anoperator which isdiagonal relative tothebasis el,e2,...,ekwill in general notbediagonal inanother basisfl,f2, ...,fk. 4.73. a.Using therules ofSecs. 4.31and4.32toaddlinear operators acting inaspace Xand multiply them bynumbers, weagain getlinear operators acting inX.The rules (7)and (7'), p.82show that thesetofall linear operators acting inaspace X(equipped with theindicated operations ofaddition andmultiplication bynumbers) isagain alinear space over the same ﬁeld K.Moreover, theoperation ofmultiplication described inSec. 4.33 canalways bedeﬁned foroperators acting inaspace X,andtheresult isagain anoperator acting inX.Inparticular, wecandeﬁne thepowers ofa given operator Abytherules A1=A, A’=AA, A“=A’A=(AA)A =A(AA) =A(A’). A"=A"—1A =AA"—1. Wethen have theformula A"'+"=A"A" (m,M=1,2,...), (20) which caneasily beproved byinduction. Next wedeﬁne A°=E, where Eistheidentity operator, andshow that (20)remains valid inthecase where oneoftheindices iszero. Infact, ifBisanyoperator, wehave ﬂiE)x =B(Ex) =Bx=E(Bx), sothat BE=EB=B. Setting B=A”,weobtain A”E=EA”=A", asrequired. b.LetX=Kkbeaﬁnite-dimensional space, and letel,...,ekbean arbitrary basis inX.Then with every linear operator Aacting inthespace X wecan associate thematrix ofAinthebasis el,...,ek. Just like the operators themselves, thecorresponding matrices canbeadded, multiplied and raised topowers inaccordance with therules ofSecs. 4.41-4.43. The dimension ofthelinear space ofallmatrices oforder ncaneasily befound. Infact, letEi,bethematrix whose elements areallzero except forthe I02 LINEAR FUNCTIONS orAvEcT0R ARGUMENT CHAP. 4 element intheithrowandjthcolumn, which, tobeexplicit, wechoose to be1.Then thematrices E,,(i',j=1,...,n)areobviously linearly inde- pendent. Ontheother hand, every matrix oforder nisalinear combination ofthematrices E,-,4Hence thematrices E,,form abasis inthespace ofall matrices oforder n.Since thenumber ofmatrices E,,-isn2,thedimension ofthespace ofallmatrices oforder nisjustn2(seeSec.2.35). Thespace of alllinear operators acting inX=K,,obviously hasthesame dimension n2. 4.74. Examples a.Multiplication bythecomplex number w=oi+iBisalinear trans- formation inthexy-plane, which canbedescribed byarealmatrix oforder two. Itfollows from themultiplication formula (fl+il‘3)(X +iy)=(“XC3)’)+1I(l5x+fly) that thismatrix isoftheform TF1 a ig B oi This ruleestablishes aone-to-one correspondence between complex numbers 6)=oi+iBandrealmatrices rrroforder two, where (asiseasily veriﬁed) thesum (orproduct) oftwonumbers goes into thesum (orproduct) ofthe corresponding matrices. This isdescribed bysaying thatthematrices rrrform anexact representation oftheﬁeld ofcomplex numbers (seeSec. 11.21). b.LetBk(k>0)denote theoperator which “lowers indices byk,”i.e., theoperator carrying each basis vector em(m=1,...,n)into thebasis vector e,,,_k ifm —k>0andinto0ifm —k<0.Obviously B3:E, BkB, :Bk+,, and, inparticular, Bl‘=Bk. Thematrix oftheoperator Blis 0I0---0 00I---0 000---I 1000---01 sEc.4.7 LINEAR oPERAToRs MAPPING AsPAcE K31NTo ITSELF I03 while that oftheoperator Bk(k<n)is (/<-I-1) Q 1Q Q Q Q1 Q Q Q()1(,l_/3) ()...()()...() 4.75. Thedeterminant oftheproduct oftwomatrices. LetA=||a,k|l and B=||b,k|| beanytwo n><nmatrices, and letC=ABbetheir product. Applying Theorem 4.54 totheminor M§:~_:;'f,(AB), which isjustthedeter- minant ofthematrix AB,weget I detAB=detAdetB. (21) Thus wehave proved thefollowing THEOREM. Thedeterminant oftheproduct oftwon><nmatrices equals theproduct ofthedeterminants ofthematrices. There also exist direct proofs ofthistheorem, i.e.,proofs which donot restonaproposition likeTheorem 4.54. Here isonesuch proof. Consider thedeterminant bu but "1 0 0 b2, b2, 0-1 -0 1,”, 1,” Q Q _1 D: 0 ‘‘‘0 an a12 ' arn 0 ‘‘‘0 a21 a22 ' azn () 0 am am ax" oforder 2n.BySec. 1.32, thedeterminant Dequals theproduct ofthedeter- minants ofthematrices an arn an but A= , ,, B= '. ., an! III anri bnl. III bun sothat D=detAdetB. (22) I04 LINEAR FUNCTIONS orAVECTOR ARGUMENT CHAP. 4 Butthere isanother wayofevaluating D.Using theelements —1intheﬁrst nrows andlastncolumns ofD,wecanmake alltheelements inthelastn rows andlastncolumns ofDvanish. This isdone byadding tothe(n+l)st rowofDtheﬁrstrowmultiplied byan,thesecond rowmultiplied byam,..., thenthrow multiplied byam,then adding tothe(n+2)nd row ofDthe ﬁrst row multiplied byan,thesecond row multiplied byan,...,thenth rowmultiplied byah,andsoon,until weﬁnally arrive atthelast(2nth) row. This gives bu bl, -10---0 1;“ b,,, 0-1---0 bill bun 00'''—1D= bnan ‘l‘172131: +'''‘l‘bnlaln '''blnall +'''‘l‘brmarn 0 0 ''' 01 17111121 +bnuzz -i"'''+bnlain '''bu-1121 -i"'''-i"bnnain 0 0'''0 bllanl +banana +'''+bnlann ‘''brnam +‘‘‘+bﬂﬂaﬂﬂ 0 0'''0 andhence, byLaplace’s theorem Sec. (1.81) -1 0 0b11a11+"‘+bn1a1n"' b1na11+""l‘br-narn 0—1 0buau-i" -i"bmazn brnan-i" +bnna2nD=(_1)1+z+~~ +2» 0 0 "1 b11an1+ +bn1ann b1nan1+ ""l‘bnnarm aubu +'''+alnbnl '''allbln ‘l‘'''‘l‘alnbnn = anbn +'''+a2nbn1 '''anbrn +'''+aznbrm =det (AB). (23) anlbll +'''+annbnl ''’anlbln +'''+aﬂﬂbnﬂ Comparing (22)and(23), weget(21), thereby proving thetheorem. Asquare matrix Aissaidtobenonsingular ifdetA7+0andsingular if detA=0.Itfollows from (21)thatifthematrices AandBarenonsingular, then soistheproduct matrix AB,while ifatleast oneofthematrices Aand Bissingular, then soisAB. These conclusions canalso bededuced from Theorem 4.65 andCorollary 4.67. 4.76. The inverse operator. Inkeeping with thedeﬁnition given inSec. 4.68, anoperator Bacting inaspace Xiscalled aleftinverse oftheoperator Aacting inthesame space Xif BA =E, where Eistheidentity operator. Theoperator Aisthen called aright inverse oftheoperator B. sac.4.7 LINEAR OPERATORS MAPPING Asrxcs K”mro ITSELF I05 a.Itispossible foranoperator Atohave many leftinverses and no right inverses atall(seeProblems 25and 26)or,conversely, many right inverses andnoleftinverses atall.However, suppose Ahasboth aleftinverse Pandaright inverse Q,sothat P=PE=P(AQ) =(PA)Q =EQ=Q. Fixing Q,weseethatevery leftinverse coincides with Pandhence isuniquely determined. Injustthesame way, theright inverse Qisuniquely determined under these circumstances. Theuniquely determined operator P=Q,which issimultaneously both aleftandaright inverse oftheoperator A,iscalled theinverse oftheoperator Aandisdenoted byA"1. The operator Aitself, with theinverse Ar‘, issaidtobeinvertible (ornonsingular). b.LetAbeanoperator acting inann-dimensional space X=K”,and letAbethematrix ofA insome ﬁxed basis el,...,en.Then either detA960 ordetA =0.Intheﬁrst case, therank ofthematrix Aequals nand it follows from Theorem 4.69b that Ahasboth aleftand aright inverse. Correspondingly, theoperator Athen hasboth aleftandaright inverse, and hence isinvertible. However, ifdetA=0,then, byTheorem 4.69b again, thematrix Ahasneither aleftnoraright inverse, and hence theoperator A acting inK”hasneither aleftnoraright inverse. 4.77. Thematrix oftheinverse operator. LetAbeaninvertible operator acting inann-dimensional space X,andletB=A“‘beitsinverse. Choosing abasis e1,.. .,en,letA=llafill and B=||b§."’|l bethematrices ofthe operators AandBinthisbasis. Wenow ﬁnd anexplicit formula fortheelements bi.”interms ofthe elements a§"’. Fixing therownumber i,weuseformula (8),p.85towrite down expressions fortheelements oftheithrowofthematrix BA=E: <1»(1) <2»(1) ... <><1»‘bia1+b,» a2+ +b,»”an ~O, b£1)a;i) +bgliaéi) +...+btfmagii Z 1’ b‘(1)a{n) +btfihaéni +...+bttmatnm : Theunknowns bi“, ...,bi?”canbedetermined from thissystem ofequations byusing Cramer’s rule(Sec. 1.73), since detAqé0byhypothesis. Expanding thedeterminant inthenumerator oftheresulting expression forbi”with respect tothejthcolumn, weget bm_Ai“ (24) ' detA’ where Ag.“isthecofactor oftheelement a2"inthematrix A.Inwords, the element bf.”oftheinverse matrix A“1equals theratio ofthecofactor ofthe I06 LINEAR FUNCTIONS orAVECTOR ARGUMENT CHAP. 4 element aj.”oftheoriginal matrix Atothedeterminant ofA.Thus wehave proved thefollowing. THEOREM. Every nonsingular matrix A:Ila:/ll! has aunique inverse matrix B=llbylll such that ABIBA=E. Theelements ofthematrix Baregiven byformula (24). 4.78. LetAr‘betheinverse oftheoperator A,asinSec.4.76a. Then by A"‘ wemean theoperator (A’1)". Itiseasily proved byinduction that formula (20) continues tohold fornegative powers. Powers oftheinverse matrix aredeﬁned injustthesame way, andthen thevalidity oftheformula A"'+"=A'"A" (m,n=1,2,...) fornegative powers ofmatrices isanimmediate consequence ofthevalidity of(20)fornegative powers ofoperators. 4.8.Invariant Subspaces 4.81. Given alinear operator Aacting inalinear space K,wesaythat a subspace K’CKisinvariant with respect to(orunder) AifxeK’implies AxeK’.Inparticular, thetrivial subspaces, i.e.,thewhole space andthe space whose only element isthezero vector, areinvariant with respect toevery linear operator. Naturally, wewillbeinterested only innontrivial invariant subspaces. 4.82. The linear operators given intheexamples ofSec. 4.72 willnow beexamined from thispoint ofview. a-c. Every subspace isinvariant with respect totheoperators ofExamples 4.72a-c (thezero operator, theidentity operator, andthesimilarity operator). d.The rotation operator intheplane (Example 4.72d) hasnonontrivial invariant subspaces, unless theangle ofrotation equals m~rcwhere misan integer (inwhich case, every one-dimensional subspace isinvariant). e.The projection operator (Example 4.72e) hasthefollowing invariant subspaces (among others): Thesubspace K’ofvectors m x:2 gkek k=1 which remain unchanged andthesubspace K”ofvectors 7|. YZ2great . . . k=m+1which arecarried mtozero. ssc.4.3 INVARIANT SUBSPACES I07 f.Every subspace spanned bysome ofthebasis vectors e1,e2, ...,en isinvariant under adiagonal operator (Example 4.72f). 4.83. Suppose anoperator Aacting inann-dimensional space K”hasan invariant m-dimensional subspace Km. Choose abasis e1,...,e,, forK” such that theﬁrstmvectors el,...,emlieinKm. Then Aer=a§"@.+'''+a£i’@.,., Aem Zalmiel +iii+ainmiems andhence thematrix oftheoperator Aisoftheform ail» ... ajmi a;m'I-ll ... apt» aw ax») a£nm+1> ago A: 250 0argj-11> atysll () 0 0 a2m+1I ajni inthegiven basis. Note that alltheelements intheﬁrstmcolumns ofthis matrix vanish ifthey appear inrows m+1through n.Conversely, ifthe matrix ofanoperator Aisoftheform (25), then thesubspace spanned by thevectors el,...,emisinvariant under A. 4.84. Suppose thespace K"can berepresented asadirect sum ofin- variant subspaces E,F,...,H(see Sec. 2.45), and choose abasis forK" such that thevectors el,... ,e,lieinE, f,,...,f, lieinF, hl,...,hulieinH. Then thematrix oftheoperator Ahasthequasi-diagonal form Afei , (26) I08 LINEAR FUNCTIONS orAVECTOR ARGUMENT CHAP. 4 where thesquare matrices AM, Am, ...,Am,along thediagonal aremade upofelements ai”,bill,...,dillinaccordance with theformulas1' T_ I‘)Ae,-~Za,’e,», =1.~. AfrZElbinfi, Ah,=id.<"h.. t1 while alltheelements outside the matrices A(B), Am, ...,Am, vanish. Conversely, ifthematrix ofanoperator Aisoftheform (26)insome basis, then thespace K,,can berepresented asthedirect sum oftheinvariant subspaces spanned bythecorresponding groups ofbasis vectors. 4.9.Eigenvectors andEigenvalues 4.91. Aspecial roleisplayed bytheone-dimensional invariant subspaces ofagiven operator A;they arealsocalled invariant directions (oreigenrays). Every (nonzero) vector belonging toaone-dimensional invariant subspace oftheoperator Aiscalled aneigenvector ofA.Inother words, avector xqé0iscalled aneigenvector oftheoperator AifAcarries xintoacollinear vector, i.e.,if Ax=Ax. The number 7.appearing in(27) iscalled theeigenvalue (orcharacteristic value) oftheoperator A,corresponding totheeigenvector x. 4.92. Wenow reexamine theexamples ofSec.4.72 from thisstandpoint. a-c. InExamples 4.72a~c, every nonzero vector ofthespace isaneigen- vector andthecorresponding eigenvalues 0,1,7.. d.Therotation operator (Example 4.72d) hasnoeigenvectors unless the angle ofrotation equals mirwhere misaninteger. e.Theprojection operator (Example 4.72e) haseigenvectors oftheform m X=Z£1.31;k=1 and 7| J’=2geek’ k=1ri+1 TCf.formula (18), p.98. ssc.4.9 EIGENVECTORS AND EIGENVALUES I09 with corresponding eigenvalues land0.Itcanbeveriﬁed thattheprojection operator hasnoother eigenvectors. f.The diagonal operator (Example 4.72f) byitsvery deﬁnition hasthe eigenvectors el,e2,...,e,,with corresponding eigenvalues 7.1,7.2,...,7.,,. 4.93. Next weprove twosimple properties ofeigenvectors. a.LEMMA. Given anoperator Awith eigenvectors xl,x2,...,xmand corresponding eigenvalues 7.1,7.2,...,7.,,,,suppose 7.,»967.,whenever i#1’. Then theeigenvectors xl,x2,...,x,,,arelinearly independent. Proof. Weprove thisassertion byinduction ontheinteger m.Obviously, thelemma istrueform=l.Assuming thatthelemma istrueforanym—l eigenvectors oftheoperator A,wenow show that itremains true foranym eigenvectors ofA.Infact, assume tothecontrary that x1,x2, ...,x,,, are linearly dependent, sothat there isalinear relation al-x1‘I_ a2-x2‘I_ +amxmzo between theeigenvectors xl,x2,...,xm,with oi,7+0,say.Applying the operator Atothisrelation, weget oi17.1x1 —I—oi27.2x2 —‘,—~~~—I—oim7.mxm =0. Multiplying theﬁrstequation by7.,"andthen subtracting itfrom thesecond equation, weﬁndthat M1()\1 —~7.,,,)x, +oz2(7.2 —~7.m)x2 , oi,,,_1(7.,,,_1 —~7.m)xm_1 =0, which bytheinduction hypothesis implies thatallthecoefﬁcients 110.1 —~km), oz2(7.2 —~hm), ...,0i,,,_1(7.,,,_1 -7.,,,) vanish, inparticular that 01.0. "Am) =0, contrary totheassumption that M1qé0,7.1qé7.,,,.This contradiction shows that theeigenvectors xl,x2,...,x,,,must belinearly independent. I Inparticular, alinear operator Aacting inann-dimensional space cannot have more than neigenvectors with distinct eigenvalues. b.LEMMA. The eigenvectors ofalinear operator Acorresponding toa given eigenvalue 7.span asubspace Km CK. Proof. If Ax!=7.x1, Ax,=7.x.z, then A(oix1 +I-3x2) a.Ax1 +§Ax2 =oi7.x1 +I-}7.x2 =7.(oix1 +j-3x2). | IIO LINEAR FUNCTIONS orAVECTOR ARGUMENT CHAP. 4 Thesubspace Kl“ iscalled theeigenspace (orcharacteristic space) ofthe operator A,corresponding totheeigenvalue 7.. 4.94. Next weshow how tocalculate thecomponents oftheeigenvectors ofanoperator A,where Aisspeciﬁed byitsmatrix insome basis el,e0,...,e,, ofthespace K0.Suppose thevector n xZ2£1.91.k=1 isaneigenvector ofA,sothat Ax=7.x (27) forsome 7..Using (5),p.80,wecanwrite (27)incomponent form as a§"€.+a§”£.+---+a‘."’€..=kit a;"€.+a;”£.+---+a;"’€..=iii, ¢1§."E.+¢1‘..”€2+"'+¢1§."’?m =75.. or (a£"—mi.+a‘.”£.+---+a§"’£..=0, a;"£.+01;”—iii.+--~+a;"’E..=0, ............................. (23) a‘.."£.+a‘..”£.+---+(a§."’—Mi.=0- This homogeneous system ofequations intheunknowns Z1,Z2,...,Z” hasanontrivial solution ifandonlyifitsdeterminant vanishes (seeSec.3.22): iii <2» .. <ia1—7. a1 a1" aéll aéll __A .. agﬂl A(7.)E j j j=0. (29) a(n1i alnli ... ago_X The polynomial A(7.) ofdegree nin7.iscalled thecharacteristic polynomial ofthematrix A.1'Toeach ofitsroots 7.0eKthere corresponds aneigenvector oftheoperator Aobtained bysubstituting 7.0for7.in(28)andthen solving theresulting compatible system forthequantities Z1,2,0,...,in.Moreover, 7.0isobviously theeigenvalue corresponding tothiseigenvector. Inparticular, itfollows that although thematrix oftheoperator Adepends onthechoice ofthebasis el,e0,...,en,theroots ofthecharacteristic polynomial ofthe TCorrespondingly, equation (29)itself iscalled thecharacteristic equation ofA. SEC.4.9 EIGENVECTORS AND EIGENVALUES Ill matrix nolonger depend onthechoice ofbasis. Wewilldiscuss thismatter further inSec. 5.53. 4.95. Wenow study thevarious possibilities which canoccur insolving thecharacteristic equation (29). a.Thecase ofnoroots inthefield K.Ifequation (29) hasnoroots atall intheﬁeld K,then thelinear operator Ahasnoeigenvectors inthespace Kn. For example, asalready noted, therotation operator intheplane V2 corresponding torotation through anangle <90¢m"rc (m=0,il,i2,...) (30) hasnoeigenvectors. This fact, which isgeometrically obvious, iseasily proved algebraically. Indeed, fortherotation operator, equation (29) takes theform cos<90—7. -sin <90 _0 sin<90 cos<90—7. (seeExample 4.72d), which becomes l—27.cos<90+7.’=0 after calculating thedeterminant. Butthisequation hasnorealroots if(30) holds. b.IfK=Cistheﬁeld ofcomplex numbers, then bythefundamental theorem ofalgebra, equation (29) always hasaroot 7.0eK. Thus inthe space C,,every linear operator hasatleast oneeigenvector. c.Thecase ofndistinct roots. Ifallnroots ofequation (29) lieinthe ﬁeld Kandaredistinct, wecanﬁndndistinct eigenvectors oftheoperator A inthespace K"bysolving thesystem (28) for7.=7.1,7.0,...,7.,,inturn. ByLemma 4.93a, theeigenvectors f1,f0,... ,f,,soobtained arelinearly independent. Choosing them asanew basis, wecanconstruct thematrix of theoperator Ainthisbasis. Since Aft ZMfr, Afz Z hzﬁ, Aft= Mfn, thematrix Amhastheform M() ...() 07. 02 . (31) 00 M II2 LINEAR rUNcTioNs orAvEcToR ARGUMENT CHAP. 4 Recalling thedeﬁnition ofadiagonalizable operator (see Example 4.72f), wecanformulate thisresult asfollows: LetAbeanoperator inthespace K", whose matrix (inanybasis) hasacharacteristic polynomial with ndistinct roots intheﬁeld K.Then Aisdiagonalizable. Thematrix ofAinthebasis consisting ofitseigenvectors isdiagonal, with diagonal elements equal tothe eigenvalues ofA. d.Ontheother hand, iftheoperator Ahasadiagonal matrix ofthe form (31)insome basis f1,j0, ...,f,,ofthespace K”with arbitrary, not necessarily distinct numbers 7.1,7.0,...,7.,,along thediagonal, then the vectors f1,f0, ...,f,,areeigenvectors ofAandthenumbers 7.1,7.0,...,7." arethecorresponding eigenvalues. Toseethat Ahasnoeigenvalues other than 7.1,7.0,...,7.0,suppose 7.is aneigenvalue ofAcorresponding totheeigenvector f=hf.» sothat Af=7.f.Then, comparing coeﬁicients off,» intheequations Af= iw.»=B.M.», if=iii.-ft=mt.-, weget 7.8,»=7.0;-3, (i=1,2,... ,n). (32) But atleast one ofthenumbers {-31,{-30,...,8”isnonzero, say81#0. Thus, choosing i=lin(32), weﬁndthat7._7.1,i.e.,7.isalready oneofthe numbers 7.1,7.0,...,7.”. e.Thecaseofmultiple roots. Let7.=7.0bearootofmultiplicity r>1 ofthecharacteristic equation (29). The following question then arises: What isthedimension ofthecorresponding eigenspace KW’, orinother words, how many linearly independent solutions does thesystem (28) have for7.=7.0?This question canbeanswered exactly from aknowledge ofthe rank ofthematrix ofthesystem (28), butwewould likeananswer which involves only themultiplicity roftheroot 7.0. InExamples 4.72a—c and4.72e, itiseasily veriﬁed that thedimension of each eigenspace KW’ isthesame asthemultiplicity of7.0asaroot ofthe characteristic equation ofthegiven operator. However, thisisnottrue in general. Forexample, letAbetheoperator inR0with matrix 7.00 A= , PL 7\0 PROBLEMS II3 where it960isarbitrary. Here thecharacteristic polynomial is(7.0—7.)’ and hasadouble root 7.=7.0.Correspondingly, thesystem (28) takes the form 0'€1‘l“0'€2=0, =0,I J“r--I"3J“to which, towithin anumerical factor, hastheunique solution £1=0, £2Z1- Thus theeigenspace oftheoperator Acorresponding totheeigenvalue 7.0 hasdimension 1,which islessthan themultiplicity oftheroot 7.0. Itcanbeshown that inthegeneral case thedimension oftheeigenspace KW" does notexceed themultiplicity oftheroot7.0(seeChapter 5,Problem 7). Acomplete solution totheproblem ofﬁnding thedimension ofthespace KW’ forthecase K=Cwillbegiven inChapter 6,after showing how to determine the“canonical form” ofthematrix ofthegiven operator. PROBLEMS 1.After deﬁning inanatural wayaddition oflinear forms andmultiplication ofalinear form byarealnumber, construct anewlinear space K*consisting of allthelinear forms deﬁned onsome linear space K.Ifthedimension ofthespace Kisn,what isthedimension ofthespace K*? 2.Which ofthefollowing vector functions deﬁned onthespace V0arelinear operators: a)Ax=x+a(aisaﬁxed nonzero vector); b)Ax=a; c)Ax=(11,x)a;i‘ d)Ax=(a,x)x; e)Ax=(if,E0+E3,ii),where x=(E1,E0,E0); f)Ax=(sin6,1,cosE0,O); g)Ax=(2E»1 _Eat,éz‘l‘éa,Eu)? 3.Consider thefollowing operations inthespace ofallpolynomials int: a)Multiplication byt; b)Multiplicati0n by:2; c)Differentiation. Arethese linear operators? 4.Suppose theoperator Adeﬁned onV0carries thevectors xi.=(0,°,1), xi=(0,1,1), Xi=(1,1,1) THere (a,x)denotes theusual scalar product ofthevectors aandx,i.e.,thenumber equal totheproduct ofthelengths ofthevectors andthecosine oftheangle between them. II4 LINEAR EUNcTioNs orAvEcToR ARGUMENT CHAP. 4 intothevectors )/1=(2.3,5), )/2=(1,0,0), )/a=(0,1, -1)- Form thematrix ofAinthefollowing basw: a)e1=(1,0,0), e0=(0,1, 0),e0=(0,0,1); blX1,x2,xe- 5.Inthree-dimensional space letAdenote theoperator corresponding to rotation through 90°about theaxisOX(taking OYintoOZ), letBdenote the operator corresponding torotation through 90°about theaxisOY(taking OZ intoOX), andletCdenote theoperator corresponding torotation through 90° about OZ(taking OXintoOY).Show that A4-B4=C4=E, ABehBA, AQB2 =B2A2. Istherelation ABAB =A2B2 valid? 6.Inthespace ofallpolynomials int,letAdenote thedifferentiation operator andletBdenote theoperator corresponding tomultiplication bytheindependent variable t,sothat AP(r) -P’(t), BP(r) -rP(t). Istherelation AB-BAvalid? Find theoperator AB—BA. 7.Assuming thatAB-BA,prove theformulas (A+B)?=A’+2AB+B2, (A+B)3=A3+3A2B+3AB’+B3. How must thme formulas bechanged ifAB95BA? 8.Assuming thatAB—BA=E,prove theformula A’"B —BA"‘ =mA'"“‘ (m=l,2,...). 9.Find thedimension ofthelinear space Kjfofalllinear operators mapping an n-dimensional space K0intOanm-dimensional space Km,andconstruct abasis fOrK2". 10.Find theproduct ABofthe matrices AandB,where I123 -1 -2 -4 A=246, BI -1 -2 -4 . 369 1 2 4 11.Raise thefollowing matrices tothenthpower: l1 cos<9-sin <9AI ,BI . 01 sin<9 cos=9 12.Find allmatrices Aoforder twosatisfying thecondition 00A2= .00 PROBLEMS II5 13.Calculate AB-BAwhere 122 411 a) A=212, B=H-420; 123 121 210 3 1-2 b) A=N 112 , B-1} 3-2 4. -1 21 -3 5-1 14.Thesuma11+-''+amofthediagonal elements ofamatrix A=||a,»;,ll iscalled thetrace ofA,denoted bytrA.Prove that tr(A +3) =trA +trB, tr(AB) =tr(BA). 15.Prove thattheformula AB—BA=Eisimpossible foroperators AandB acting onann-dimensional space K0. Comment. Theresult ofProblem 6shows thattheassumption thatthespace K0isﬁnite-dimensional plays anessential rOlehere. 16.Given asquare matrix Coforder twosuch thattrC=0(cf.Problem 14), show thatCcanberepresented intheform C=AB-BA where AandBare(unknown) matrices oforder two. 17.Let X;=Ewe» (/=1,2, ,m) ., _ ... i=1 bemlinearly independent vectors inann-dimensional space, andletAbethe operator deﬁned onthelinear manifold L(x1, x0,...,x,,,)such that m 0 . y,»=Ax,-=Za§€’lx;, (]=l,2,...,m). k=1 Show thatevery minor oforder mofthematrix made upofthecomponents ofy,- (with respect tothebasis e1,e0,...,en)equals theproduct ofdet||a§j"|| withthecorresponding minor ofthematrix made upofthecomponents ofthe vectors x,-. 18.Show that ifthebasis minor ofamatrix ofrank rappears intheupper left-hand corner, thentheratio ofanyminOr Moforder rtotheminor appearing inthesame columns asMbutintheﬁrstrrows depends only onthecolumn indices oftheminor M. 6 LINEAR FUNCTIONS OF AVECTOR ARGUMENT CHAP. 4 19.Show thatifAisamatrix ofrank r,then anyseoond-order determinant of theform M§1.§z.. fr Mi1.i2.....ir£1.12.“-Jr k1.kz..--.kr I k1.kii.----kr kl-k2,...-krMt...-,.....i. Mk1.kz..._.kr consisting ofminors oforder rofthematrix A,vanishes. 20.Show thatevery minor oforder kofthe matrix ABCequals asumofproducts ofcertain minors oforder kofthematrices A,BandC. 21.Find theinverses ofthefollowing matrices: léié 12-312 -%%—%—%,4: ,B=0i 2, c= - 25 O01 %—%%—% -e-atT 22.Prove that<A')**=<A"*>' foranynonsingular matrix A. 23.Find allsolutions oftheequation XA=O,where Aisagiven second-order matrix, Xisanunknown second-order matrix and0isthezero matrix (the matrix allofwhose elements vanish). 24.LetA=Ila)”||beanysquare matrix oforder n,andletA)”bethecofactor oftheelement ai.”inthedeterminant ofA.Thematrix A-||A§"llliscalled the adjugate ofthematrix A.Prove that AA=AA=(detA)E. 25.Inthespace ofallpolynomials inthevariable t,consider theoperators A andBdeﬁned bytherelations A[o0 +a1t+'''+o,1t"] =a1+a0t+"'+a,,t”"1, B[a0+a1t+---+a,,t"] -a0t+a1t2+---+a,,t"+1. Show thatAandBarelinear operators andthat AB=E, BA¢E. Does theoperator Ahave aninverse? 26.Show thattheoperator BofProblem 25hasinﬁnitely many leftinverses. 27.Prove thatifAisanonsingular linear operator acting inann-dimensional linear space, thenevery subspace invariant under Aisalsoinvariant under A“1. 28.Prove that ifthelinear operators Aand Bcommute (i.e., ifAB=BA), then every eigenspace oftheoperator Aisaninvariant subspace ofthe operator B. PROBLEMS II7 29.Prove that ifadirect sum (Sec. 2.45) ofeigenspaces ofanoperator A coincidm with thewhole space Kandifeach eigenspace oftheoperator Ais invariant under anoperator B,then AandBcommute. 30.Letxandybeeigenvectors oftheoperator Acorresponding todistinct eigenvalues. Show thatax+By(ac7E0,I5940)cannot beaneigenvector ofA. 31.Prove thatifevery vector ofthespace Kisaneigenvector oftheoperator A, thenA=7.E(xeK). 32.Prove thatifthelinear operator Acommutm withalllinear operators acting inthegiven space, then A-7.E. 33.Letthelinear operator Ahave theeigenvector e0,with eigenvalue 7.0.Show thate0isalsoaneigenvector oftheoperator A2,witheigenvalue 7.2. 34.Even ifalinear operator Ahasnoeigenvectors, theoperator A2mayhave eigenvectors (e.g., theoperator corrmponding torotation through 90°inthe plane). Show that iftheoperator A2hasaneigenvector with anonnegative eigenvalue 7.=9.2,then theoperator Aalsohasaneigenvector. 35.Find theeigenvalum andeigenvectors oftheoperators given bythefollowing matrices: 2-1—1 -1-2 2 a)0-1 0; b) 0 1 0; 0 2 1 0 0 1 0 0 1-1 -1 0 1-1 0 0 0 0' 0 0 0 12-1 O c) O 1-1 ; d) O 1 3 36.Verify thefollowing facts: a)The relation N(A) I>T(A) isnecessary andsuﬁicient fortheequality A2=0tohold; b)N(A) CN(A2) CN(A3) C---foranyoperator A; c)T(A) I>T(A2) I>T(A3) I>~--foranyoperator A; d)IfT(A'"') I>N(A'"), then T(A)CN<A’"+*-1), T<A"*"“"*> CN(A)- 37.Show thatevery linear operator Aofrank rcanberepresented asthesum ofrlinear operators ofrank one. 38.Find alltheinvariant subspaces ofadiagonal operator withndistinct diagonal elements, andshow thatthere are2"such subspaces. chapter 5 COORDINATE TRANSFORMATIONS Asiswell known, insolving geometric problems bythemethods of analytic geometry avery important roleisplayed bytheproper choice ofa coordinate system. Proper choice ofacoordinate system also plays avery important roleinamuch wider class ofproblems connected with thegeometry ofn-dimensional linear spaces. This chapter isdevoted toastudy ofthe rules governing coordinate transformations inn-dimensional spaces. In particular, theresults obtained herearefundamental fortheclassiﬁcation ofquadratic forms which willbemade inChapter 7. 5.l.Transformation toaNew Basis 5.11. Let {e}={e1,e0,...,e,} beabasis inann-dimensional space K",andlet If} Z{fI~>.f21 ~~'Jfll} beanother basis inthesame space. The vectors ofthesystem {f}are uniquely determined bytheir expansions interms ofthevectors ofthe original basis: ft=i>1"e.+i>§=“e2+-"+i>i."@., f2=i>1"’et+111%+'"+115%., (1) _ II I I7fnZP1”er‘I"Pzniez + IP71"en! II8 sEc. 5.1 TRANsEoRMATioN ToANEw BASIS II9 or,more concisely, f.~=Zi>.""@.- (]=I,2>---,")- (2) ¢=.1 Thecoeﬁicients pi”(i,j=l,2,...,n)in(l)and(2)deﬁne amatrix pp) pp) ...ppm Pzupi,-0!: iv?’iv?’ 111"’ is."ii?’---Pl.“ called thematrix ofthetransformation from thebasis {e}tothebasis {f}. Aswasdone previously insimilar cases (Sec. 4.2ff.),wewrite thecomponents ofthe vectorsf, (with respect tothebasis {e})asthecolumns ofthematrix P. Bythesame token, theformulas (1)together with thematrix Pspecify a corresponding linear operator P,deﬁned bythe relations f,-=Pe,» (i—-l,2,...,n)and called theoperator ofthetransformation from the basis {e}tothebasis {f}. The determinant Dofthematrix Pisnonvanishing, since otherwise the columns ofP, andhence thevectorsf1,f0, ...,fn,would belinearly dependent (Sec. 3.l2a). Amatrix with anonvanishing determinant issaid tobenon- singular (recall Sec. 4.75). Thus thetransformation from onebasis ofthe n-dimensional space Kntoanother basis isalways accomplished byusing a nonsingular matrix. 5.12. Conversely, let{e}I{e1,e0,...,e,,} beagiven basis ofthe space Kn,andletP=|lp§"’|I beanonsingular matrix oforder n.Using the equations (1),construct thesystem ofvectors f1,f0, ...,fn. Itisclear that these vectors arelinearly independent, since thecolumns ofevery non- singular matrix arelinearly independent (Sec. 3.l2a). Consequently, the vectors f1,f0, ...,f,,form anew basis forthespace K0.Thus every non- singular matrix P:|lp§”|l determines via(1)atransformation from onebasis ofthen-dimensional space Kntoanother basis. 5.13. Next wenote aparticular case ofatransformation toanew basis, i.e.,thecasewhere every vector fkisjustthecorresponding vector ekmultiplied byanumber 7.171%0(k:1,2,...,n).Then theequations (1)take theform frZA131, ./pzZ A262» f,,: 7.ne,,, I20 COORDINATE TRANsEoRMATioNs CHAP. 5 andthematrix Phasthediagonal form 711Q...Q P= .0 )\2 ... 0 00 7.,, Inparticular, for7.1=7.0=---:7."=1,weobtain thematrix ofthe identity transformation, namely theunitmatrix E: (theoriginal basis isnotchanged bytheidentity transformation). 5.2. Consecutive Transformations 5.21. LetP:||p§”’|| bethematrix ofthetransformation from thebasis {e} 2{eh e2» ---aen} If} 2{fl-’.f21 ---7f’Vl}9 andletQ=||q§.’°’|| bethematrix ofthetransformation from thebasis {f} tothebasistothebasis {g}I{g1,g2. ---,gi}- Wenow determine thematrix ofthetransformation from thebasis {e} directly tothebasis {g}.By(2),theformula fortransforming from thebasis {e}tothebasis {f}is f,~:2pf»"’e,» (j:-1,2,...,n), (4) i=1 while that fortransforming from thebasis {f}tothebasis {g}is g,:2q‘,"’f, (k=1,2,...,VI). (5) j=1 Substituting (4)into (5),weobtain gtI112"’Zi>$”@.-i=1 i=1 :(p;*"qg’=’)@, (1.=1,2,...,n). (6) '=1 j=1 .~ SEC.5.3 TRANsroRMAT1oN orTI-IECOMPONENTS orAVECTOR I2I Ontheother hand, ifT:||t§."l|| denotes thematrix ofthetransformation from thebasis {e}tothebasis {g},wecanwrite g,.-Z11.‘-’"e.» (/<-1,2,---,n) <1) Comparison of(6)and(7)gives i,<’"’=§p,<*'*q;’" (i,k:1,2,...,n). (8) ;=1 Recalling formula (8), p.85(where thechoice ofindices issomewhat diﬁerent, butnottheir role), weﬁndthat thedesired matrix Tistheproduct PQofthematrices PandQ. 5.22. Consider thefollowing special case ofconsecutive transformations. Since thematrix Pisnonsingular, thesystem ofequations (l)canbesolved forthevectors e1,e0,...,e,,.Theresulting system ofequations e.Iq‘."f.+q§"f2+-""+q‘.."i‘., 32Zqllzlfl +q‘22’f2 +'''+q?if7i1 ........................ (9) e.-q1"‘f.+qt“/2+---+q‘."’f. obviously determines thetransformation from thebasis {f}tothebasis {e}. The consecutive transformation from thebasis {e}tothebasis {f}byusing thematrix Pandthen from thebasis {f}tothebasis {e}byusing thematrix Q:||q§-kl1|isequivalent tothetransformation from thebasis {e}toitself, i.e., totheidentity transformation with unit matrix (3). 5.3.Transformation oftheComponents ofaVector 5.31. Let{e}:{e1,e0,...,en}and{f}:{f1,f0, ...,f,,} betwobases inann-dimensional linear space K".Any"vector x6K,hastheexpansions XZ€1e1"I‘€2e2'I""‘I‘€nenZ7l1f1‘I‘7l2j2‘I""‘I"/inﬂ,» (I0) where Z1,Z0,...,Z,arethecomponents ofthevector xwith respect tothe basis {e}and 111,-q0,...,T1,,areitscomponents with respect tothebasis { Wenow show how tocalculate thecomponents ofthevector xwith respect tothebasis {f}interms ofitscomponents with respect tothebasis {6}- Suppose wearegiven thematrix P=|lp§.’l|| ofthetransformation from thebasis {e}tothebasis {f}. Then thevectors {e}aregiven interms ofthe I22 COORDINATE TRANsEoRMATioNs CHAP. 5 vectors {f}by(9)or,more brieﬂy, by e,»IZq§."’f,. (/<I1.2....,»), (11) k-=1 where thematrix QII|q§j’IIistheinverse ofthematrix P.Substituting (ll) into theexpansion (l0), weget XZiate;Zillitfit Zi1E,~<ki1 qitjlfit) :ki1<ﬁ1qi.j)§,')fi.- j=1 . k=1 ;'= = = = Itfollows bytheuniqueness oftheexpansion ofthevector xwith respect to thebasis {f}that 71»Zzlqi.-Hit (k:1,2,---»'1)» (12),= or,inexpanded form >11:ii’21+q‘.”z.+---+q‘."’z., 712Zq2UE>1 I11952 ‘l‘'''+qllnlznﬂ Y1.Iq‘.."€.+qfii+---+q‘.."’€.- Thus thecomponents ofthevector xwith respect tothebasis {f}arelinear combinations ofthecomponents ofthevector xwith respect tothebasis {e}; thecoeﬂicients ofthese linear combinations form amatrix which isthetranspose ofthematrix ofthetransformation from thebasis {f}tothebasis {e},i.e., thetranspose oftheinverse ofthematrix P.Denoting theinverse ofthematrix PbyP-1andthetranspose ofamatrix byaprime, weﬁndthatthematrix S describing thetransformation from thecomponents Z1,Z0,...,E0tothe components 1,1,110,...,11,,isgiven by SI(P"‘)’. 5.32. Theconverse proposition isalso valid: THEOREM. LetZ1,Z0,...,inbethecomponents ofanarbitrary vector x with respect tothebasis {e}I{e1,e0,...,en}ofthen-dimensional space K0,andletthequantities ~q1,110,...,T1,,bedeﬁned bytheformulas 711ZSitar I512€2 -I‘'''ISlnina 712Z521€1 ‘l‘522€2 ‘l‘'''+S2n€n9 7141 Z3.1151 +Sn2E2 +iii+Swain. where det||s,-,,l| ¢0.Then anewbasis {f}I{f1,f0, ...,fn} canbefound in thespace K"such thatthenumbers 1,1,110,...,11,,arethecomponents ofthe vector xwith respect tothebasis {f}. sEc.5.4 TRANsEoRMAT1oN orTI-IEcoErr1c1ENTs orALINEAR EoRM I23 Proof. Introduce thematrix SIIIS,-kll and thematrix PI(S’)"‘ with elements denoted bypi”. Substituting these elements into theformulas (l), wegetanew basis {f}I{f1,f0, ...,f,,}. Weassert that thisisthedesired basis. Infact, consider thetransformation formulas (12), which givethe components ofthevector xwith respect tothenew basis. Aswehave seen, these formulas can bewritten interms ofthematrix (P"‘)’. But inthe present case, (P-1)’ coincides with S,since (P-1)’I([(S’)"‘l"‘)’ I<s'>'Is. Hence, given anyvector x,thequantities 111,110,...,11,,arejustthecom- ponents ofxwith respect tothebasis {f}. I 5.33. Just asinSec. 5.21, wecanconstruct thematrix corresponding to consecutive transformations ofthecomponents ofavector. LetZ1,Z0,..., Z,bethecomponents ofthevector xwith respect tothebasis {e},andlet thequantities 111,110,...,11,,andT1,T0,...,T"bedeﬁned bytheequations 1t Ylizipiigi (/.2122,---an), i=1 ’ﬂ TkZZqk;’/I; (/(Z112,---1"), j=1 respectively, where thematrices PI||p,~,-l| and QIIlqk,-1| arenonsingular. Then, just asbefore, wecanexpress thequantities T1,T0,...,T"directly interms ofthequantities Z1,Z0,...,inbytheformulas TuZ <E1qlcipi'i) itZEltkizi (kZ1,21---1ll),=,= ,= where thequantities tk,(i,kIl,2,...,n)form amatrix Tequal tothe product QPofthematrices QandP.- 5.4.Transformation oftheCoefficients ofaLinear Form LetL(x) bealinear form deﬁned onaspace Kn.AswesawinSec. 4.1, ifabasis {e}I{e1,e0,...,en}ischosen inK0,then thevalues ofL(x) can becalculated from theformula I-(X)IXltit, k=1 where Z1(kIl,2,...,n)arethecomponents ofthevector xwith respect tothebasis {e},andthecoeﬁicients I1aregiven by l,,IL(e,,) (kIl,2,...,n). I24 COORDINATE TRANSFORMATIONS CHAP. 5 Thecoeﬁicients 1,,obviously depend onthechoice ofthebasis {e}.Wenow derive therule governing thetransformation ofthecoefﬁcients ofalinear form when wegoover toanew basis. Suppose theformulas rtf,I§1i>£»”e.~ (II1,2.....to (13) give thetransformation from thebasis {e}tothenew basis {f}. Wewish to ﬁndthecoeﬁicients ofthelinear form L(x) inthebasis {f}. These coeﬁicients arethenumbers 7.1IL(f,»), which caneasily befound byusing (l3): AtZMfr) Z Pij)L(@¢) Z Finle- iI1 'l=1 Thus thecoeﬂicients ofalinear form transform inthesame wayasthebasis vectors themselves. 5.5.Transformation oftheMatrix ofaLinear Operator 5.51. Given alinear operator Ainann-dimensional space K",letA1,,I ||af-"II bethematrix ofAinthebasis {e}I{e1,e0,...,e,,}, while AmI I|af,”’|I isitsmatrix inthebasis {f}I{f1,ﬁ, ...,f,,}. Moreover, suppose thetransformation formulas from thebasis {e}tothebasis {f}have theform ftIXPI-me,’ (/<I1,Z,---,H), (14) j=1 and letPdenote thematrix ||p§."’||. Wenow ﬁnd therelation between the matrices A1,), A1,,andP. Thematrix A1,,isdeﬁned bythesystem ofequations 1t Ae,IZlaijle,» (jI1,2,...,n), (15) andthematrix A1,,bythesystem ofequations Afm I2ot;,'"’fk (mI1,2,...,n). k=I Inthelastequation, weuse(l4)toreplace thevectors fkbytheir expressions interms ofthevectors e,-.Theresult is Tl Tl Tl TlA/...I211""2tr».I2(211:-'=»=<;."~>)@.. k=1 i=1. i=1 k=1 after changing theindex ofsummation from jtoi.Next weapply theoperator SEC. 5.5 TRANSFORMATION orTI-IEMATR1x orALINEAR OPERATOR I25 Atoboth sides of(l4), changing ktomand using theexpansion forAe, given by(15): TI Tl Afm ZA2Pimlei ZZPi'm)Aei i=1 :l=1- Tl Tl I TI TI VI2,121"2IZ:1= t= = = .~14u. Comparing coefﬁcients ofe,inthelasttwoexpansions, weﬁndthat 7'k)11W 11Xvi11'"IZl1£”i>/".k=1. i=1 O1‘ P/1m ZA(t)P (I6) inmatrix form. This isthedesired relation between thematrices Am, Am and P.Multiplying ontheleftbythematrix P'1, wegetthefollowing expression forthematrix Am: AmIP‘1AmP. 5.52. Itfollows from (l6)andthetheorem onthedeterminant ofaproduct oftwomatrices (Sec. 4.75) that detPdetAmIdetAmdetP, or,since detP I0, detAmIdetAm. Thus thedeterminant ofthematrix ofanoperator does notdepend onthe choice ofabasis inthespace. Therefore wecantalk about thedeterminant ofanoperator, meaning thereby thedeterminant ofthematrix oftheoperator inanybasis. 5.53. Besides thedeterminant, there exist other functions ofthematrix elements ofanoperator which remain unchanged under transformation toa new basis. Toconstruct such functions, consider theoperator AI7.E, where 7.isaparameter. This operator obviously hasthematrices AmI7.E andAm-7.Einthebases {e}and{f}. Bywhat wasjustproved, wehave det(Am -7.E)Idet(Am -7.E) forany7..Both sides ofthisequation arepolynomials ofdegree nin7..Since these polynomials areidentically equal, they have thesame coefﬁcients for anypower of7..Hence these coeﬁicients, which arefunctions ofthematrix elements oftheoperator, areinvariant under changes ofbasis. I26 COORDINATE TRANsEoRMAT1oNs CHAP. 5 Wenow examine thenature ofthese functions. The determinant ofthe matrix Am-7.Ehastheform a§1)__ )\ ail) ___ agnl agl) ag2) gX __ ago) atnll atria) ___ 01111)_AI I(I1)"x" +A11"-1 +---+A,,_1x +A, Itisaneasy consequence ofthedeﬁnition ofadeterminant thatthecoefficient A1of7."-1 equals thesum 0,111+ ag2>+____1_ ail”) ofthediagonal elements, taken with thesign (-l)”"‘:]' The coefficient A0 of7.”"2 isthesum ofalltheprincipal minors oforder 2,taken with thesign (-l)"‘2.I Similarly, thecoefficient A1of7."-"' isthesum ofalltheprincipal minors oforder k,taken with thesign (-l)""". Finally, thecoefficient A" of7.°,i.e.,theconstant term, isobviously equal tojustthedeterminant ofthe operator. The polynomial det(Am —7.E), which, aswehave justseen, is independent ofthechoice ofbasis, iscalled thecharacteristic polynomial of theoperator A. *5.6. Tensors 5.61. Thecomponents ofavector, thecoefficients ofalinear form, theele- ments ofthematrix ofalinear operator, these areallexamples ofageneral class ofgeometric objects called tensors. Before giving thedeﬁnition ofatensor, we ﬁrstrevise and“rationalize” ournotation somewhat. Thebasis vectors ofan n-dimensional space K,willbedenoted, asbefore, bythesymbols e1,e0,..., e,(with subscripts). Thecomponents ofvectors, e.g., xandy,willbedenoted byZ1,Z2,...,Z”and 111,112,...,11"(with superscripts). The coefficients ofalinear form L(x) willbedenoted byl1,I0,...,1,,(with subscripts). The matrix elements ofalinear operator willbedenoted byajf,where thesuper- script designates therownumber and thesubscript designates thecolumn number (incontradistinction tothenotation adopted inSec. 4.23). The convenience ofthisarrangement ofindices isdetermined bythefollowing summation convention: Ifwehave asum ofterms such that thesummation index 1'(say) occurs twice inthegeneral term, once asasuperscript andonce TThesum :11‘)+11$”+-'-+111,")iscalled thetrace oftheoperator A(cf.Problem 14,p.115). __ _ 1Theminor issaid tobeaprincipal minor ifi'1Ij1_ 1',Ij0, ...,1‘.Ij,,. sEc.5.6 TENsoRs I27 asasubscript, then wewillomit tliesummation sign. Forexample, with our convention, theexpansion ofthe vector xwith respect tothe basis {e1,e0,...,e,,}takes theform xIZie, (although thesummation signisomitted, summation over iisimplied). The expression foralinear form L(x) interms ofthecomponents ofthevector x andthecoefficients oftheform becomes to->Ita" (summation over iisimplied). The result ofapplying theoperator Atothe basis vector e,takes theform Ae,Iafe, (summation over jisimplied). The components 11"ofthevector Axare expressed interms ofthecomponents ofthevector xasfollows: 1,.’IH15’ (summation over iisimplied). Wewilldenote quantities pertaining toanew coordinate system bythe same symbols asintheoldcoordinate system butwith primes ontheindices. Thus wedenote new basis vectors bye1,,e0»,...,en»,newcomponents ofa vector xbyZ1’,Z2’,...,5"’,etc.Theelements ofthematrix ofatrans- formation from thebasis e,tothebasis e,-1willbedenoted bypf»,sothat erIPM (17) (summation over iisimplied). The elements ofthematrix oftheinverse transformation willbedenoted byqf,i.e., eiZqilelﬂ (18) (summation over i’isimplied). The matrix istheinverse ofthematrix pi};thiscanbeexpressed bywriting 11,,{OforiIj, (19) pllql Z1foriIj, O1‘ 0fori’Ij’, Iii-l'1l4"Z i (20)1 1fori’Ij’. Tomake thenotation more concise, let31'denote thequantity which depends ontheindices iandj insuch away that itequals 0when theindices are different and lwhen theindices arethesame. Then wecanwrite (19)inthe form i>Z7'1i’.?'= $25 (21) COORDINATE TRANSFORMATIONS CHAP. 5 and(20)intheform P1413'I313- (22) 5.62. Toshow theadvantages ofusing ournewnotation, wederive once again theformulas bywhich thecomponents ofavector, thecoefficients ofa linear form andthematrix elements ofanoperator transform ingoing over toanew basis. Thus suppose wehave avector . .1xIZ‘e1IZ‘e,». Using (18)toreplace e,byq,l'e,»»,weobtain XZEiqi-lei’ Zﬁllet’, which implies 2'"Iqzit‘. <23) since thee,.tform abasis. This isjust thetransformation formula forthe components ofavector. Next suppose wehave alinear form L(x). The numbers l,-,aredeﬁned asusual bytherelations I,»IL(e,»»). Using (l7)tosubstitute theexpression pfte, fore,..,weobtain It’ZL(Pi'@t) ZPi'L(et) ZPi'Ii,sothat Ii’:pi'liv which isthedesired formula. Finally suppose wehave anoperator A.The elements ofitsmatrix in thenewbasis aredeﬁned bytherelations Ae,~Iaﬂiey. Using (17)tosubstitute p§,e,andp§,e, forthequantities e,-Iande,»,weget p,1»Ae, Iaflpjﬂer ButAe,Ia§e1, sothat theresult is pftafe, Ia§.'p’,..e,. Since thee,-arebasis vectors, wehave I1__ 1:ip,..a, Ia,-.p,... Togetof,’ontheright, wemultiply both sides byq§"andsum over theindex j.Using therelation (22), weobtain i>Z-H11-iii?’ I11f5i>§3q’§' I417313- Bythedeﬁnition ofthequantity 319:,thesum overj' reduces tothesingle SEC. 5.6 TENsoRs I29 term corresponding tothevalue j’Ik’. Then SQ:Il (no summation implied) andweget 4'53I112741115} (25) which isthedesired formula. Itisnothard toverify that thethree transformation formulas just derived arethesame asthose derived earlier intheordinary way (seeSecs. 5.3-5.5). Formulas (23)I(25) have much incommon. Intheﬁrst place, these formulas arelinear inthetransformed quantities. Secondly, the coefficients inthese formulas areelements ofthematrix transforming the oldbasis into thenew basis orelements ofthematrix oftheinverse trans- formation or,ﬁnally, elements ofboth matrices. 5.63. Wearenow inaposition togivethedeﬁnition ofatensor. Tensors aredivided intothree classes, covariant, contravariant andmixed. Moreover, every tensor hasadeﬁnite order. Webegin bydeﬁning acovariant tensor, which, tobeexplicit, wetake tohave order three. Suppose there isarule which inevery coordinate system ofann-dimensional space K,allows usto construct n3numbers (components) T111, each ofwhich isspeciﬁed bygiving theindices i,j,kdeﬁnite values from lton.Bydeﬁnition, these numbers T111form acovariant tensor oforder three ifingoing toanew basis, the quantities T,-1,,transform according totheformula Z Acovariant tensor ofanyother order isdeﬁned similarly; atensor oforder mhasn’"components instead ofn3components, andinthetransformation formula there appear mfactors oftheform pf,instead ofthree factors. In particular, thecoefficients ofalinear form, which transform byformula (24), constitute acovariant tensor oforder one. Next wedeﬁne acontravariant tensor oforder three. Suppose wehave arule which inevery coordinate system allows ustoconstruct n3numbers T12", each ofwhich isspeciﬁed bygiving theindices i,j,kdeﬁnite values from lton.Bydeﬁnition, these numbers T”? form acontravariant tensor of order three ifingoing toanewbasis, thequantities T92‘transform according totheformula Ti'i'k' : Acontravariant tensor ofanyother order isdeﬁned similarly. Inparticular, thecomponents ofavector form acontravariant tensor oforder one. Theterms “covariant” and“contravariant,” which have justbeen intro- duced, arevery simply explained. “Covariant” means “transforming inthe same way” asthebasis vectors, i.e.,byusing thecoefficients “Contra- variant” means “transforming intheopposite direction,” i.e.,byusing the coefficients I30 COORDINATE TRANSFORMATIONS C!-[AP_ 5 There isstill thecase ofmixed tensors toconsider. For example, n3 numbers T,l“,.,speciﬁed inevery coordinate system, form amixed tensor of order three, with twocovariant indices andonecontravariant index, ifingoing toanew basis, thequantities T1“,transform according totheformula T131Ii>Z»i>§5»qI'Tl3~- Amixed tensor with lcovariant indices andmcontravariant indices isde- ﬁned similarly. Inparticular, theelements ofthematrix ofalinear operator form amixed tensor oforder two, with onecovariant index andonecontra- variant index. Note theconvenience ofourarrangement ofindices, which has been deliberately chosen toindicate thecharacter ofanytensor ataglance. 5.64. Operations ontensors. Wecandeﬁne theoperation ofaddition for twotensors ofthesame structure, e.g., fortwotensors T1‘,andSf,(with two covariant indices and onecontravariant index). Inthiscase, thesum isa tensor Q1‘,ofthesame structure, deﬁned asfollows: Inevery coordinate system, thecomponent ofQ1‘,with ﬁxed indices i,j,kisthesum ofthe corre- sponding components ofT1‘,and Thefactthatthequantities Q1‘,actually form atensor, andindeed oneofthesame structure asT11‘,andSj.“,_,isimplied bythefollowing equality: Q1-‘$1ITi‘/.~+5.-"7,-IIi>i»i>§3q't§'TlZ~ +i>Zi»i>§5»qi'§'-‘>‘l‘i Ii>Z'i1§'qI'(Tl‘,- +$13»)Ii>Z'i>}q'iY'Ql-‘,-- The operation ofmultiplication isapplicable totensors ofanystructure. For example, letusmultiply atensor T1,byatensor S,f.The result isa tensor Q1”, oforder four. lnany coordinate system itscomponent with ﬁxed indices i,j,k,lisdeﬁned asequal totheproduct ofthecorresponding components ofthefactors T0,»and The tensor character ofQ§,., canbe veriﬁed asfollows: QlﬂwIT.-».-»$i'.-'» I11.-"»i>§3T.~.i>IY»1il'$I~I Pi'Pi'Pii'qi’7i"iSit Ii>Z'»i>}3i>i'§»ql'Ql~n- Next weconsider stillanother operation called contraction. This opera- tion canbeapplied totensors which have atleast onecovariant index and onecontravariant index. Forexample, suppose wehave atensor To contract T1‘,with respect tothesuperscript andtheﬁrst subscript means to form thequantity Ti. inevery coordinate system. Here summation over theindex iisimplied; asaresult, thequantity T,IT1’;depends only ontheindex j.Contraction ofatensor yields another tensor, whose order istirolessthan theorder ofthe original tensor. Weverify thisforthepresent example. Wehave T.-1IT27,-»IPi'Pi'qiiTtki I(i>f»qi§)i>§3TZ‘. I$Zi>i¥Tl‘}-- PROBLEMS |3| Here thesummation over kreduces toonly oneterm, corresponding tothe value kIi.Since SfIl(nosummation implied), weobtain DIMUIMR asrequired. What istheresult ofcontracting amixed tensor Tfoforder’ twowith respect toitstwoindices? The quantity TIT}nolonger haseven asingle index, i.e.,inevery coordinate system itconsists ofjust onenumber. This number isthesame inevery coordinate system, since TIﬁ:¢%UIMﬁIﬁIr Such ascalar quantity, which does notdepend onthecoordinate system, is called aninvariant. Thus, bycontracting tensors, wecanobtain invariants ofthetensors. Forexample, ifwecontract thetensor afcorresponding tothelinear operator A,theinvariant ajlsoobtained isthetrace ofthematrix ofA,i.e., thesum ofitsdiagonal elements. Theinvariance ofthisquantity hasalready been proved inadifferent way inSec. 5.53. Asanother example, thematrix cfoftheproduct oftwooperators with matrices a},andb,l,respectively, is themixed second-order tensor obtained bycontracting thefourth-order tensor aibj.with respect totheindices kandl. PROBLEMS 1.Avector x€K,, hascomponents E1,£0,...,E,with respect toabasis e1,e0,...,e,,.How does oneconstruct anewbasis inK,such thatthecom- ponents ofxwith respect tothisbasis equal l,O,...,O? 2.Abasis e1,e0,...,e,,ischosen inann-dimensional space K0.Show that every subspace K’CK,canbespeciﬁed asthesetofallvectors x6K,,whose components (with respect tothebasis e1,e0,...,en)satisfy asystem ofequations oftheform II§o0eI0 0I1¢VA,n G.>- 3(Contiiiiiation). Show thatevery hyperplane HCK,canbespeciﬁed asthe setofallvectors x6K,whose components (with respect tothebasis e1,e0,..., e,,)satisfy asystem ofequations oftheform W v -Ea,-1;,-Ib, (iIl,2,...,k). :1‘-—-I 4.Letthecomponents ofavector intheplane beZ1,£0withrespect toonebasis, 111,1,0with respect toanother basis, and:1,-:0with respect toathird basis. I32 COORDINATE TRANsroRMATroNs CHAP. 5 Suppose that 7l1Za11E»1 ‘l‘(11252, 1'l2Za21E»1 ‘l‘a22E»2, 71Z121151 ‘l‘121252, T2Z122151 ‘I‘122252, /1Zllaiill, BZllbtill- Express thecomponents 11,10interms ofthecomponents E1,E0. 5.Given alinear form L(x) ,.=é0inthespace K0,ﬁndabasisf1,ﬁ, ...,f,,such thattherelation L(X) Z"I1 holds forevery vector XZiMfr.- k=I 6.Lettheoperator Aacting inann-dimensional space Rhave ak-dimensional invariant subspace R’.Then, temporarily regarding Aasdeﬁned only inthe subspace R’,wecanconstruct thecharacteristic polynomial ofdegree kforA. Show thatthispolynomial isafactor ofthecharacteristic polynomial ofthe operator Aacting inthewhole space R. 7.Let7.I7.0beanr-fold root oftheequation det||Am —7.El|I0.Show thatthedimension moftheeigenspace R121"ofAcorresponding totheroot7.0 does notexceed r. 8.Show that thequantity 81isasecond-order tensor, with onecovariant index andonecontravariant index. 9.Asetofquantities S,-1isdeﬁned inevery coordinate system asthesolution of thesystem ofequationsT“"S.».-Iti. where Til‘isacontravariant tensor oforder twoanddetl|T"‘|| I0.Show that S1,isacovariant tensor oforder two. chapter 6 THE CANONICAL FORM OFTHE MATRIX OFALINEAR OPERATOR Two operators AandBacting inann-dimensional space K"aresaid to beequivalent ifthere exist twobases inK"such thatthematrix oftheoperator Aintheﬁrstbasis coincides with thematrix oftheoperator Binthesecond basis. Clearly, the“linear transformations” inK"corresponding toequivalent operators have identical properties. Buthow canwedecide whether ornot theoperators AandBareequivalent byexamining their matrices inthesame basis? 1nthischapter, starting from agiven linear operator Ainann-dimensional (real orcomplex) space, wewillﬁnd abasis inwhich thematrix Aofthe operator Ahas“canonical form,” i.e.,aform which isthesimplest possible inacertain sense. This canonical form canbeobtained directly from the elements ofthematrix oftheoperator Ainanybasis. Moreover, itturns outthat iftheoperators AandBareequivalent, then their matrices have the same canonical form. Thus anecessary and sufﬁcient condition fortwo operators tobeequivalent isthat their canonical matrices coincide. Webegin ourconsiderations bystudying aspecial class ofoperators (Sec. 6.1). Thegeneral case willbestudied inSec.6.3. 6.l.Canonical Form oftheMatrix ofaNilpotent Operator 6.11. Alinear operator Bacting inann-dimensional space Knissaid to benilpotent ifB’:0(i.e., ifB'x=0forevery xeK,,) forsome positive 133 |34 THE CANONICAL FORM OF THE MATRIX OF ALINEAR OPERATOR CHAP. 6 integer r.Given anilpotent operator Bsuch thatB’=0,wewillassume that B'“‘ ¢0,i.e., that there arevectors xGK" such that B'"‘x ¢0.Bythe height ofavector x6K",wemean thesmallest positive integer mforwhich B’"x =0.Byhypothesis, every vector xeK"isofheight <r,andthere are vectors ofheight equal tor.Given anyk<r,letHkdenote thesetofall vectors ofheight <k. Obviously, Hkisasubspace ofKn.Infact, ifx, yeHk,then Bkx=0,Bky=0andhence B"(<xx +ﬂy):0forarbitrary ax, {-3eK,sothattheheight ofthevector ax+ﬂydoes notexceed k,i.e.,ocx+ ﬂyeHk.Moreover, itisobvious that H,=K"andthatT {0}:Hoc Hlc "'c Hr—lc Hr:Kn- Letmkdenote thedimension ofHk,sothat O=m0<m1<---<m,==n. Next weconstruct abasis inthespace K"asfollows: Aswehave seen, H,_1 does notcoincide with thewhole space K":H,.Therefore wecan ﬁnd vectors fl,...,f1,‘lying inH,and linearly independent over H,_1, where pl:m,—m,_1 (see Sec. 2.44). The vectors Bfl,...,Bf,,_1 liein H,_1 andarelinearly independent over H,_2. infact, ifwehad °‘1Bf1+"'+°‘p,Bfp,:geHr—2 then application oftheoperator B’-2 would give a1B'_y1 +''' °i1|,B'Tlf11, :0, orequivalently alfl +'''+°‘v,f1), GHr~1, which isimpossible, byconstruction. Itfollows that thedimension m,_1 — m,_2 ofthespace H,_1 over H,_2 (again seeSec. 2.44) isequal toorgreater than thedimension m,——m,_1 ofthespace H,over H,_1. Wenow supplement thevectors Bfl,...,Bfpl with vectors fwd, ...,f,,2 inH,_1 tomake the largest system which islinearly independent over H,_2 (p2:m,_1 ~m,_2). Applying theoperator Btoallthese vectors, wegetvectors B¥1, ...,B%,1, Bfp1+1, ...,Bf,“ lying inH,_2 andlinearly independent over H,_3 (this isproved inthesame way asbefore). Itfollows that m,_2 —m,,3 _>m,_1 —m,_2, and wecan construct vectors fp2+1‘ ...,f,,3 inH,_2 which together with thepreceding system form a“full system" ofvectors linearly independent over H,_3. Continuing thisconstruction inthesubspaces H,_3, ...,H0:{O},weﬁnally T{0}denotes thesetwhose only element isthezero vector. sac. 6.1 CANONICAL FORM orTl-IEMATRIX orANILPOTENT OPERATOR I35 getafullsystem ofnlinearly independent vectors. This system canbewritten intheform ofatable fl,...,f,1, Bfb ‘‘'>B.f11l>.fpl+1> ''‘’.fP2 B'*1fl, ...,B'"‘f,,1,B'*%,1+l,. ..,B'*%2, ...,f,,r_1,.l, ...,f,,r, where thevectors intheﬁrstrowareofheight r,those inthesecond roware ofheight r—-l,andsoon,with thevectors inthelastrowbeing ofheight 1 (sothat theoperator Bcarries them allinto thezero vector). 6.12. Every column oftheabove table determines aninvariant subspace oftheoperator B.Theﬁrstplinvariant subspaces allhave dimension r,the next p2—pl invariant subspaces allhave dimension r—l,andsoon,with the last p,—p,_l single-element columns determining one-dimensional invariant subspaces. The whole space K"isthedirect sum ofthese p, invariant subspaces. 6.13. Next wewrite thematrix oftheoperator Binthesubspace deter- mined bythevectors oftheﬁrstcolumn. Forabasis wechoose thevectors B'“1fl, B'“%, ....Bfl,fl,arranged inorder ofincreasing height. With this arrangement, theoperator Bcarries theﬁrstvector ofthebasis intothezero vector, thesecond vector intotheﬁrstvector, etc., andﬁnally therthvector into the(r—l)stvector. Therefore, according toSec. 4.23, thematrix of theoperator Bhasrrows andrcolumns, andisoftheform 010---00 001---00 .. (1) 000---01 000---00 with zeros everywhere except fortheelements (equal tol)along thediagonal just above theprincipal diagonal. The matrix oftheoperator Btakes a similar form intheother invariant subspaces, corresponding totheremaining columns ofthetable, and infact candiffer from thematrix (l)only by having adifferent number ofrows andcolumns. 6.14. Thus thematrix oftheoperator Binthewhole space K"isquasi- diagonal (see Sec. 4.84), with blocks oftheform (1)along theprincipal I36 THECANONICAL roam orTHEMATRIX orALINEAR orsmron CHAP. 6 diagonal: 010---00 001---00 000---01 000---00 010---00 001---00 B: 000---01 000---00 E3 E (Z) Thenumber ofblocks ofsizerequals pl,thenumber ofblocks ofsizer—1 equals p2—pl, ...,thenumber ofblocks ofsize(2)equals p,_l —p,_2, and thenumber ofblocks ofsize(l)equals p,—p,_l. Naturally, ifp,_,+l =p,_, forsome j,then thematrix (2)contains noblocks ofsizej. 6.2.Algebras. TheAlgebra ofPolynomials 6.21. Webegin with some deﬁnitions. Alinear space Kover anumber ﬁeld Kiscalled analgebra (more exactly, analgebra overK)ifthere isdeﬁned ontheelements x,y,...ofKanoperation ofmultiplication, denoted byx-y (orxy),which satisﬁes thefollowing conditions: 1)a(xy) =(ocx)y =x(ay) forevery x,yinKandevery oninK; 2)(xy)z =x(yz) forevery x,y,zinK(theassociative law); 3)(x+y)z:xz+yzforevery x,y,2inK(thedistributive law). SEC.6.2 ALGEBRAS. THEALGEBRA orPOLYNOMIALS 137 Ingeneral, multiplication may notbecommutative, i.e.,wemay have xy¢yx.Ifmultiplication iscommutative, i.e.,if 4)xy-= yxforevery x,yinK, then thealgebra Kissaid tobecommutative. Anelement eeKiscalled aleftunitifex=xforevery x6K,aright unitifxe=xforevery xeK,andatwo-sided unitorsimply aunit(inK)if ex:xe=xforevery xeK. Anelement xeKiscalled aleftinverse oftheelement y6Kifxyisthe unitofthealgebra K;inthiscase, yiscalled aright inverse ofx.Ifanelement 2hasboth aleftandaright inverse, then thetwoinverses areunique andin factcoincide (cf.Sec.4.76a). Theelement 2isthen saidtobeinvertible, and itsinverse isdenoted by2“. Theproduct 2uofaninvertible element 2andaninvertible element uis aninvertible element with inverse u*‘2“‘. Iftheelement uisinvertible, then theequation ux=vhasthesolution x=u“v. This solution isunique, being obtained bymultiplying theequation ux=vontheleftbyu“1. Inthe commutative case, wewrite x=v/uorx=v:u, calling theelement xthe quotient oftheelements vandu. Theordinary rules ofarithmetic arevalid forquotients, i.e., E vluz ~l—ulv2 v . . .-1+ =—i— (ifulandu2areinvertible), "1 "2 "1142 vv vv . . .-1- 2=i (ifulandu2areinvertible), uluz lllllg vv vu . . .—1:i =-1-? (iful,uz,andv2areinvertible). ulug ulvz The proof ofthese facts islefttothereader. Analgebra Kissaid tohave dimension nifKhasdimension nregarded asalinear space. 6.22. Examples a.Given anylinear space K,suppose wesetx'y=0forevery x,yeK. This gives analgebra, called the.trivial algebra. b.Anexample ofanontrivial commutative algebra over aﬁeld Kis given bythesetIIofallpolynomials P()\)=2a,,i" k=0 with coefﬁcients inK,equipped with theusual operations ofaddition and multiplication. This “polynomial algebra” hasaunit, namely thepoly- nomial e()\) with ao=1andallother coefﬁcients equal to0. I38 THECANONICAL FORM orTHEMATRIX orALINEAR OPERATOR CHAP. 6 c.The linear spaceM(K,,) ofallmatrices oforder nwith elements inK, with theusual deﬁnition ofmatrix multiplication, isanexample ofaﬁnite- dimensional noncommutative algebra ofdimension n2(seeSec.4.73b). d.Amore general example ofanoncommutative algebra with aunit is thelinear space ofalllinear operators acting inalinear space K,with the usual definition ofoperator multiplication (seeSec, 4.33). 6.23. a.Asubspace LCKiscalled asubalgebra ofthealgebra Kif xeL,yeLimplies xyeL.Asubspace LCKiscalled aright ideal inK ifxeL, y€K implies x)/EL and aleftideal inKifxeL, yeK implies yxeL.Anideal which isboth aleftandaright ideal iscalled atwo-sided ideal. Inacommutative algebra there isnodistinction between left, right andtwo-sided ideals. There aretwoobvious two-sided ideals inevery algebra K,i.e.,thealgebra Kitself andtheideal {0}consisting ofthezero element alone.T Allother one-sided and two-sided ideals arecalled proper ideals. Every ideal isasubalgebra, buttheconverse isingeneral false. Thus the setofallpolynomials P()\) satisfying thecondition P(0) =P(1) isasubalgebra ofthealgebra IIwhich isnotanideal, while thesetofallpolynomials P()\) satisfying thecondition P(0) :0isaproper ideal ofthealgebra II. b.LetLCKbeasubspace ofthealgebra K,andconsider thefactor space K/L (Sec. 2.48), i.e., thelinear space consisting oftheclasses Xof elements xeKwhich arecomparable relative toL.IfLisatwo-sided ideal inK,then, besides linear operations, wecan introduce anoperation of multiplication fortheclasses XeK/L. Infact, given twoclasses XandY, choose arbitrary elements xeX, y€Y and interpret X-Yastheclass containing theproduct xy.This uniquely deﬁnes X-Y,since ifx’eX, y’eY,then ><’y’—Xy=><’(y'—y)+(><’eX)» andhence x'y'Cxybelongs toLtogether with y’—yandx’—x.More- over, since conditions l)—3), p.136hold inK,theanalogous conditions hold fortheclasses XeK/L. Therefore thefactor space K/L equipped with the above operation ofmultiplication, isalso analgebra, called thefactor algebra ofthealgebra Kwith respect tothetwo-sided ideal L.Ifthealgebra Kiscommutative, then obviously soisthefactor algebra K/L. 6.24. LetK’andK"betwoalgebras over aﬁeld K.Then amorphism co ofthespace K’into thespace K”(Sec. 2.71) iscalled amorphism ofthe algebra K’intothealgebra K”ifbesides satisfying thetwoconditions a)co(x' +y’)=co(x') +co(y') forevery x’,y’eK’, b)m(ax’) =<xco(x’) :<xm(x’) forevery x’6K’andevery a.eK TAsinTheorem 2.l4c, 0-x=0forevery x6K. sec.6.2 ALGEBRAS. TI-IEALGEBRA orPOLYNOMIALS I39 forthemorphism oftwospaces (s'ee p.53), italso satisﬁes thecondition c)co(x’y’) =m(x’)co(y’) forevery x’,y’eK’. Amorphism cowhich isanepimorphism, monomorphism orisomorphism ofthespace K’into thespace K”,asdeﬁned inSec. 2.71, iscalled anepi- morphism, monomorphism orisomorphism ofthealgebra K’intothealgebra K”,provided condition c)issatisﬁed. 6.25. Examples a.LetLbeasubalgebra ofanalgebra K.Then themapping cowhich assigns toevery vector xeLthesame vector xeKisamorphism ofthe algebra Lintothealgebra K,andinfactamonomorphism. AsinExample 2.72a, thismonomorphism issaid toembed LinK. b.LetLbeatwo-sided ideal ofanalgebra K,andletK/L bethecorre- sponding factor algebra (Sec. 6.23b). Then themapping cowhich assigns to every vector xeK theclass XeK/L containing xisamorphism ofthe algebra Kintothealgebra K/L, andinfactanepimorphism. AsinExample 2.72b, thisepimorphism iscalled thecanonical mapping ofK onto K/L. c.Letcobeamonomorphism ofanalgebra K’intoanalgebra K”.Then thesetofallvectors co(x’) eK" isasubalgebra L”CK”,and themono- morphism coisanisomorphism ofthealgebra K’onto thealgebra L". d.Letcobeamorphism ofanalgebra K’into analgebra K”.Then the setL’ofallvectors x’eK’such thatco(x’) =0,which isobviously asubspace ofK’(cf.Sec.2.76b), isatwo-sided ideal ofthealgebra K’.Infact, ifx’eL’, Y’EK’“‘°“ <»<><'y'>=<»<><')<»<y'> =0. sothat x’y’6L’,andsimilarly y’x’eL’,i.e.,L’isatwo-sided ideal ofK’, asasserted. AsinSec. 2.76b, letQbethemonomorphism ofthespace K’/L’ into thespace K”which assigns toeach class X’eK’/L’ the(unique) element 6>(x’), x’EX’. Then Qisamonomorphism ofthealgebra K’/L’ intothealgebra K”.Infact, choosing x’eX’,y’eY’,wehave x’y’eX’Y’ and Q<X'Y'>=‘°(X’)/’)=<»<><'><»<i'> :Q<X'>Q<Y'>- Ifthemorphism coisanepimorphism ofthealgebra K’intothealgebra K”,then themorphism Qisanisomorphism ofthealgebra K’/L’ onto the algebra K”. e.LetAbealinear operator acting inaspace Kover aﬁeld K.Since addition andmultiplication byconstants inKaredeﬁned forlinear operators acting inK,with every polynomial P()\):2a,,>.k i~=l] I40 THECANONICAL roam orTHEMATRIX orALINEAR OPERATOR CHAP. 6 (al,eK)wecanassociate anoperator 1!! _ kP(A) _2a,,A k=0 acting inthesame space KasAitself. Then therule associating P(A) with P(A) hasthethree properties ﬁguring inSec.6.24. Infact, if Po)=P10)+P20)=iaai+ihm=iv.+bk))\ki = k=0 k=0 Pr‘0 then clearly P(A)=§0<a.+1>.)A’“=k§0alA* +§01>,.A*=P.<A>+P.<A>. andsimilarly forproperty b),while if Q(A)=Pl()\)P2()\) =ﬁght§i>,,i’“ =E§a,i>,W*, i=0 k=0 J'=0 k=0 then Q(A) =ZZaibkAj+k =ZalAjZbkAk :P1(A)P2(A), ==0 i=0 k=0 &>0Pr‘ bythedistributive lawforoperators (Sec. 4.34). Note that theoperators Pl(A) andPl(A) always commute with each other, regardless ofthechoice of thepolynomials Pl(A) andPl(A). Theresulting morphism ofthealgebra ITof polynomials (Example 6.22b) intothealgebra B(K) oflinear operators acting inK(Example 6.22d) isingeneral notanepimorphism, ifonly because operators oftheform P(A) commute with each other, while thewhole algebra B(K) isnoncommutativefr f.There exists anisomorphism between thealgebra L(K,,) ofalllinear operators acting inthen-dimensional space K,,andthealgebra M(K,,) ofall matrices oforder nwith elements from theﬁeld K.This isomorphism is established byﬁxing abasis el,...,e,,inthespace K,,andassigning every operator AeL(K,,) itsmatrix inthisbasis. Both algebras L(K,,) andM(K,,) have thesame dimension n2. 6.26. Thesetofallpolynomials oftheform P(7\)Q0(7\), where Q0()\) isa ﬁxed polynomial andP(A) anarbitrary polynomial, isobviously anideal in thecommutative algebra Hofallpolynomials P(A) with coefﬁcients ina ﬁeld K(Example 6.22b). Conversely, wenow show that every ideal I96{0} ofthealgebra IIisofthisstructure, i.e.,isobtained from some polynomial Q0()\) bymultiplication byanarbitrary polynomial P(A). Tothisend, we TExcept inthetrivial case where Kisone-dimensional. sec. 6.2 ALGEBRAS. Tl-IEALGEBRA orPOLYNOMIALS l4l ﬁnd thenonzero polynomial oflowest degree, sayq,intheideal I,and denote itbyQ0()\). Wethen assert that every polynomial Q()\) eIisofthe form P()\)Q0()\), where P(A) EII. Infact, asisfamiliar from elementary algebra, Q0)EP0)Qo0) +R0), (3) where P(A) isthequotient obtained bydividing Q()\) byQ0()\) andR()\) is theremainder, ofdegree lessthan thedivisor Q0()\), i.e.,lessthan thenumber q.Butthepolynomials Q()\) andQ0()\) belong totheideal I,andhence, as isapparent from (3),sodoes theremainder R()\). Since thedegree ofR()\) islessthan qandsince Q0()\) hasthelowest degree, namely q,ofallnonzero polynomials inI,itfollows that R()\) E0,and theitalicized assertion is proved. Thepolynomial Q0()\) issaid togenerate theideal I. 6.27. Thepolynomial Q0()\) isuniquely determined bytheideal Itowithin anumerical factor. Infact, ifthepolynomial Ql()\) hasthesame property asthepolynomial Q0()\), then, asjustshown, Q10) =P1()\)QoO\), Q00“) =P0O\)Q1O\)~ Itfollows that thedegrees ofthepolynomials Ql()\) andQ0()\) coincide and that Pl(A) andP0()\) donotcontain Aandhence arenumbers, asasserted. 6.28. Given polynomials Ql()\), ...,Q,,,()\) notallequal tozero andwith nocommon divisors ofdegree >l,wenow show thatthere exist polynomials Pf()\), ...,Pfn()\) such that Pl0)Q10) +'~~+P',’..0)Q.,.0) E1- (4) Infact, letIbethesetofallpolynomials oftheform Pi0)Qi0) +''~+Pm0)Qm0) with arbitrary Pl(A), ...,P,,,()\) inII.Then Iisobviously anideal inII. BySec.6.26, theideal Iisgenerated bysome polynomial 42.0)=§P;:<i>Q.o>. <5) Inparticular, k=1 Q10) =$10)Q00), ---,Q1,.0) =S7n()\)Q0()\)s where Sl()\), ...,S,,,()\) arecertain polynomials, from which itfollows that Q0()\) isacommon divisor ofthepolynomials Ql().), ...,Qm()\). But, by I42 THE CANONICAL FORM OE THE MATRIX OF ALINEAR OPERATOR CHAP. 6 hypothesis, thedegree ofQ00.) iszero, and hence Q0()\) isaconstant ao, where aof 0since otherwise I={O}. Multiplying (5)byl/aoandwriting Pg()\) :Pg()\)/ao, weget(4),asrequired. 6.3.Canonical Form oftheMatrix ofanArbitrary Operator 6.31. LetAdenote anarbitrary linear operator acting inann-dimen- sional space K,,.Since theoperations ofaddition and multiplication are deﬁned forsuch operators (Secs. 4.3I—4.33), with every polynomial 1!! __ kP(A)_Zall k=0 wecanassociate anoperator P(A)=a,A'= P~Z-0 acting inthesame space K,,(cf.Example 6.25e), where addition andmultipli- cation ofpolynomials corresponds toaddition and multiplication ofthe associated operators inthesense ofSec.4.4. lnfact, if P(A) =P10) +P20) =Zak“ +Zbkhk =Z(ak +bk)7\k, then k=0 k=0 k=0 P(A)=Z011.+bk)Ak=Za,.A" +Zb,.A" =Pl(A)+P2(A)- k=0 k=0 1:=0 Similarly, if Q(A)=Pl().)P2().) =Za,,x"§b,-A" =Z§a,,b,W", ‘ ~="=0 k=0 J=0 >~0 then Q(A) =ZZal.b,A"" "=Za,.A"Zb,A’ =Pl(A)P2(A), k=0i=0 k=0 :i=l1 bythedistributive lawforoperator multiplication (Sec. 4.34). lnparticular, theoperators Pl(A) andPl(A) always commute. Thus themapping co(P()\)) =P(A) isanepimorphism (Sec. 6.24) ofthe algebra IIofallpolynomials with coefﬁcients intheﬁeld Kinto thealgebra IIAofalllinear operators oftheform P(A) acting inthespace K". BySec. 6.25d, thealgebra IIAisisomorphic tothefactor algebra II/Ill, where IAis theideal consisting ofallpolynomials P(A) such that co(P()\)) :P(A) :0. Wenow analyze thestructure ofthisideal. SEC. 6.3 CANONICAL FORM OF TI-IE MATRIX OF AN ARBITRARY OPERATOR I43 6.32. Asnoted inExample 6.25f, thesetofalllinear operators acting in aspace K,,isanalgebra ofdimension n2over theﬁeld K.Hence, given any operator A,itfollows that theﬁrstn2+lterms ofthesequence A°=E,A,A2,...,A’",.. must belinearly dependent. Suppose that 2a,,A" =0 (m<n2). =0K‘ Then, bythecorrespondence between polynomials andoperators established inSec.6.31, thepolynomial Q0)=iaiii k=0 must correspond tothezero operator. Every polynomial Q(A) forwhich the operator Q(A) isthezero operator iscalled anannihilating polynomial of theoperator A.Thus wehave just shown that every operator Ahasan annihilating polynomial ofdegree <n2. 6.33. The setofallannihilating polynomials oftheoperator Aisan ideal inthealgebra II.BySecs. 6.26-6.27 there isapolynomial Q0()\) uniquely determined towithin anumerical factor such that allannihilating polynomials areoftheform P()\)Q0()\) where P(A) isanarbitrary polynomial inII.Inparticular, Q0()\) istheannihilating polynomial oflowest degree among allannihilating polynomials oftheoperator A.Hence Q0()\) iscalled theminimal annihilating polynomial oftheoperator A. 6.34. THEOREM. LetQ(A) beanannihilating polynomial oftheoperator A, andsuppose that Q0)=Q10)Q20), where thefactors Ql()\) andQ2()\) arerelatively prime. Then thespace Kn canberepresented asthedirect sum Kn:T1+T2 oftwosubspaces TlandT2both invariant with respect totheoperator A,’r where Q1(A)x2 =0» Q2(A)x1 :0 forarbitrary xleTl,x26T2,sothat Ql()\) andQ2()\) areannihilating poly- nomials fortheoperator Aacting inthesubspaces T2andTl,respectively. 1'Thus xlET.implies AxlETlandsimilarly x26T2implies Ax,ET,. I44 TI-IE CANONICAL FORM OF TI-IE MATRIX OF ALINEAR OPERATOR CHAP. 6 Proof. BySec.6.28 there exist polynomials Pl(A) andP2()\) such that P1(7\)Q1(7\) -I"P2(7\)Q2(7\) E1, andhence P1(A)Q1(A) +P2(A)Q2(A) EE- LetTl,(k=l,2)denote therange oftheoperator Ql,(A), i.e.,thesetofall vectors oftheform Q,,(A)x, xeK,, (see Sec. 4.61). Then obviously y= Ql,(A)x eTl,implies Ay=Ql,(A)Ax eTl,,sothatthesubspace Tl,isinvariant with respect totheoperator A.Given anyxleTl,there isavector yeK,, such that Q2(A)xl =Q2(A)Qi(A))/ ZQ(A))/ =0, andsimilarly, given anyx26T2,there isavector 2eKl,such that Ql(A)x2 =Q1(A)Q2(A)Z =Q(A)Z =0- Moreover, given anyxeK,,,wehave x=Q1(A)P1(A)x -I‘Q2(A)P2(A)x :x1+x2, where xl,.=Ql,(A)P,,(A)x eTl, (k=l,2). Itfollows that K,,isthesum ofthesubspaces TlandT2.Ifxl,eTlQT2, then Ql(A)xl, =Q2(A)xl, =0,andhence xo=P1(A)Q1(A)x0 ‘l‘P2(A)Q2(A)xn :0- Therefore TlQT2={O},andthesum Kl,=Tl+T2isdirectfr I 6.35. Remark. Byconstruction, the operator Ql(A) annihilates the subspace T2,while theoperator Q2(A) annihilates thesubspace Tl.Wenow show that every vector xannihilated bytheoperator Ql(A) belongs toT2, while every vector xannihilated bytheoperator Q2(A) belongs toTl.Infact, suppose Ql(A)x =0.We have x=xl+x2where xleTl,x2eT2,and hence Ql(A)xl =Ql(A)x —Ql(A)x2 =0since Ql(A)x2 =0.ButQ2(A)xl = 0aswell, since xl6Tl.Itfollows that x1:P1(A)Q1(A)x1 -I‘P2(A)Q2(A)X1 :0, x=x2ET2- Similarly, Q2(A)x =0implies xeTl,andourassertion isproved. 6.36. Representing the polynomials Ql(A) and Q2(A) themselves as products offurther prime factors, wecandecompose thespace Kl,intosmaller subspaces invariant with respect totheoperator Aand annihilated bythe TNaturally, thepossibility isnotexcluded thatoneofthe subspaces TlandT2consists of thezero vector alone. SEC. 6.3 CANONICAL FORM OF TI-IE MATRIX OF AN ARBITRARY OPERATOR I45 appropriate factors ofQl(A) andQ2(A). Suppose theannihilating polynomial Q(A) hasafactorization oftheform Q0)—1110—)‘k)rk <6) k=1 where Al,...,Amareallthe(distinct) roots ofQ(A) andrl,isthemultiplicity of7\l,.For example, such afactorization isalways possible (towithin a numerical factor) intheﬁeld Cofcomplex numbers. Then wehave the following THEOREM. Suppose theoperator Ahasanannihilating polynomial ofthe form (6). Then thespace K"canberepresented asthedirect sum ofmsubspaces Tl,...,Tm,allinvariant withrespect toA,where thesubspace Tl.isannihilated byB§,'=, therl,thpower oftheoperator Bl,=A-Al,E. Proof Apply Theorem 6.34 repeatedly tothefactorization (6)ofQ(A) intomrelatively prime factors oftheform (A~71,)". I 6.37. Byconstruction, theoperator Bl,isnilpotent inthesubspace Tl,. Hence, bySec. 6.14, inevery subspace Tl,(¢{0}) wecanchoose abasis in which thematrix ofBl,takes thecanonical form (2).Inthis basis, the matrix oftheoperator A=Bl,+)\l,Etakes theform i,10~~oo 01,1.--00 000~-1,1 000~~oi, i,,10~~oo 0>.,,1~~oo _(7) 0o0~~i,1 000-~01, E I46 THE CANONICAL FORM OF TI-IE MATRIX OF ALINEAR OPERATOR CI-IAP. 6 Hence thematrix oftheoperator Ainthewhole space Kl,:Tl+~-~+Tm takes theform >.l1 0 QM... 0 00 i, 1,10 Q),,... Q O0---1 00 M -/(A)= (8) El ;\m1 Q O71“-~ O QQ 1 00 ;,m E inthebasis obtained bycombining allthecanonical bases constructed in thespaces Tl,...,Tm.Thus ﬁnally wehave thefollowing THEOREM. Given anyoperator Ainann-dimensional space Kl,with an annihilating polynomial oftheform (6)(inparticular, anyoperator Ainan n-dimensional complex space C,,), there exists abasis, called aJordan basis, inwhich thematrix ofAtakes theform (8),called theJordan canonical form ofAfr Inthecase Kl,:C,,thecomplex numbers Al,...,A,canbearranged in ISynonymously, theJordan normalfvrm ofA. sEc.6.4 ELEMENTARY DIVISORS I47 accordance with anyrule, e.g., inorder ofincreasing absolute valuefr The representation (8)isnotalways possible inthecase ofanoperator Aacting inaspace K,,¢C,,.InSec. 6.6wewillconsider thecanonical form ofthe matrix ofanoperator Aacting inarealspace K,,=R". 6.4.Elementary Divisors I 6.41. Thematrix (8)canbespeciﬁed byatable )\l:nl“, ...,nil’ )\2:nl2’ ...nm , ir2 ("pa >nah) >...>"ifm (9) )\,,,:nl’”’, ...,nix” which foreach diagonal element Al,indicates thesizes nil”, ...,nlflofthe corresponding “elementary Jordan blocks” oftheform Al, 10~~-0 071,,1--~0 "gin . ..... (10) 000~~~1, l000~~~)\l,l appearing inthematrix (8).Wenowshow howtoconstruct thetable (9) andthereby determine theform- ofthematrix J(A) oftheoperator A,from aknowledge ofthematrix Aoftheoperator Ainanybasis ofthespace K,,. 6.42. Asshown inSec.5.53, thecharacteristic polynomial oftheoperator Adoes notdepend onthechoice ofabasis. Forming thispolynomial forthe Jordan basis, weget det(A_IE)=det(J(A)_IE)=110,,_i)"l*’+-""+"i’;.l’, (11) k=1 since every element below theprincipal diagonal in(8)iszero. Thus the numbers Al,(k=I,...,m)aretheroots ofthecharacteristic polynomial, andthenumbers rl,=nil”+-~~+nljilarethemultiplicities ofthese roots. TOrinorder ofincreasing argument 0(varying intheinterval 0<6<211'), inthecase ofidentical absolute values. I48 TI-IECANONICAL FORM orTI-IEMATRIX orALINEAR OPERATOR CHAP. 6 Hence, bycalculating thecharacteristic polynomial (which canbedone by using thematrix A)andﬁnding itsroots, wecandetermine thequantities Xl, andrl,=nil‘)+---+nit’inthetable (9). 6.43. Next (here andinSec. 6.44) weshow how tousethematrix Aof theoperator Aintheoriginal basis tocalculate thenumbers nlklthemselves. Since J(A) andAarematrices ofthesame operator Aindifferent bases, it follows from Sec. 5.51 that J(A) =T“A T, where Tisanonsingular matrix, andhence that J(A)-XE=T—‘(A-XE)T. The minors ofaﬁxed order, sayp,ofthematrix A—XEarecertain poly- nomials inXofdegree 2.;p.LetI,,(A) betheideal inthealgebra TIgenerated byall_these minors, andletI,,(J(A)) have theanalogous meaning. Then the twoideals I,,(A) andI,,(J(A)) coincide. Infact, according toSec. 4.54, every minor oforderp ofthematrix J(A) -—XEisasum ofproducts ofminors of order pofthematrices A—XE,TandT_‘. Buttheelements ofTandT_1 arenumbers. Thus every minor oforder pofthematrix J(A) —XEissimply alinear combination ofminors oforderp ofthematrix A—XE,andhence belongs totheideal I,,(A). Bysymmetry, every minor oforder pofthe matrix A—XE belongs totheideal I,,(J(A)). Itfollows that theideals I,,(A) andI,,(J(A)) coincide, asasserted. Now letD,,(X) bethepolynomial generating this ideal. According to Sec. 6.26, D,,(X) isjust thegreatest common divisor ofthe polynomials generating I,,(A). Thus thegreatest common divisor oftheminors oforderp ofthematrix J(A) —XEisthesame asthegreatest common divisor ofthe minors oforder pofthematrix A-XE,and hence canberegarded as known. Thegreatest common divisor oftheminors oforder pofthematrix J(A) —XEcan becalculated directly asfollows: Instead ofthematrix J(A) —XE,wecan again consider amatrix oftheform S(J(A) —XE)T, where Sand Tareinvertible numerical matrices (not containing X).The operations ofinterchanging rows (orcolumns) and adding anarbitrary multiple ofonerow(orcolumn) toanother lead tomatrices ofjust thiskind (seeExamples 4.44d—4.44g). Wenow assert that theelementary block Xl,—X 10-~ 0 0Xl,—Xl~~~ 0 0 00-~ 1 0 00 Xl.—X SEC.6-4 ELEMENTARY DIVISORS I49 canbereduced totheform 9I1119 <11> 00~-W_w? byoperations oftheindicated type. Infact, toget(12)weﬁrstsubtract the ﬁrst row multiplied byXl,—Xfrom thesecond row, then thesecond row multiplied byXl,—Xfrom thethird row, andsoon.This gives thematrix Xl,—X l0~~~0 —(Xl,—X)2 01 0 (—l)°—2(Xl, —X)°—1 00~~~ 1 (-—-l)°T‘(Xl, —X)“ 00~~- 0 where q=nil". Then from theﬁrstcolumn wesubtract thesecond column multiplied byXl,—X,thethird column multiplied by—(Xl, —X)2,etc., and ﬁnally the(q—l)th column multiplied by(-—l)°_2(Xl, —X)°“‘. This gives thematrix2 0 10---0 0 01---0 0 oo~>1 (AFW—W00-~0 from which thematrix (12)canbeobtained byinterchanging columns.’r Wenow calculate thegreatest common divisor D,,(X) oftheminors of order pofthematrix .I(X) with blocks oftheform (12) along itsprincipal diagonal. Since allnondiagonal elements off(X) vanish, theonly minors of .I(X) which canbenonzero arethose with thesame setofrow andcolumn indices, and such aminor issimply equal totheproduct ofitsdiagonal elements. Among theelements along theprincipal diagonal ofthematrix .I(X), acertain number, sayN,arebinomials oftheform (Xl,-X)"iH, while theother n~Nelements areallequal to1.The number Nisjustthetotal number ofJordan blocks inthematrix J(A), i.e., N= rl+~~~+rm. Clearly D,(X) Elifp<n—N,since some oftheminors of.I(X) oforder p<n—Narecertainly equal to1.Suppose wereplace thematrix .I(X) by TExcept possibly forthesign oftheelement (Xl,~7.)'1, which isirrelevant tothe subsequent determination ofD,,(X).s I50 THECANONICAL roRM orTHEMATRIX orALINEAR OPERATOR CHAP. 6 thediagonal matrix 01~mi" 01_7\)"’l:I (7)2C7\)"i2I J(X)= (Mi‘C7\)":::,I I n—N 1 which obviously has thesame polynomial D,(X) asJ(X). The greatest common divisor oftheminors oforder pofthematrix J(X)areclearly ofthe form m W, 1>..())= 1,11,0,.—))“*. (13) with nonnegative exponents ul,(p). The exponents in(13) areeasily found. Forexample, todetermine ul(p),wenotethatul(p) isthesmallest exponent with which Xl—Xappears inallminors ofJ(X) oforder p.Ifp<n—rl, then there isaminor oforderp which does notcontain Xl—Xatall,sothat ul(p) =0.However, ifp=n—rl+1,then, bearing inmind that the exponents nlll, ...,nil’arearranged indecreasing order, wehave I"'1(P) :"ll,- Moreover, each timepisincreased further by1,theexponent ul(p) increases, ﬁrstbyn§:’_l, then bynj:l_,, andsoon,until ﬁnally wegetul(p) =nil’+~~~+ nlllforp =n.Similarly, 0 ifp<n~—rl,, nit’ ifp=n-rl,—I—l, p"‘(p) 2 nit’+nlﬂl ifp=n—rl,+2, n:::)+...+nikI Note that (kl l*k(") —P-k(" CI)="1, l*k(" —1)'“P-k(" —2)="gm, l*k(" —rk+1)“ P-k(" —rk):"IT,sothat I"k(n-1+1)—I"k(n-1)=~51"(1:1.2.....~-1) (14) (wesetn§.’°l=0ifj>rl,). sEC.6.4 ELEMENTARY DIVISORS l5l 6.44. The ratio Epol) =M‘) D20) iscalled anelementary divisor oftheoperator A.The elementary divisors, likethepolynomials D,,(X) themselves, donotdepend onthechoice ofa basis and hence canbecalculated from thematrix ofAinanybasis. It follows from (I3)that - MH0,.—))“*"’*" ...E-P0‘) :kill :H0%__)\)!J-k1D+1)—I1-kill) H0% _)\)l1-HUI '~‘=1 k=1 _ (p=l,2,...,n—l) orequivalently, En_jO\) :1711'0% _)\)u,-(n~a'+1)—u,-(n—i) Z1’2’lll,n_1), lc=1 Using (14), weget E,_,.())=kf_[1(i,,_i)"§" (j=l,2,...,n—l), where theroots ofE,,_,(X) have multiplicities equal tothesizes ofcertain Jordan blocks inthematrix J(A). Thus bycalculating theelementary divisors ofA,wecanﬁndthenumbers nil",thereby ﬁnally solving theproblem ofconstructing thetable (9). 6.45. Examples a.The“Jordan matrix” 010 011 001 ll 01 1 21 02 21 02 THE CANONICAL FORM OF THE MATRIX OF ALINEAR OPERATOR CHAP. 6 oforder tenhasthree blocks ofsizes 3,2and 1corresponding totheroot Xl=1,andtwoblocks ofsizes 2and2corresponding totheroot X2=2. Hence theelementary divisors are E90) =(1—X)“(2—X)’, E50) I(1—X)’(2—X)’. E70) =1“CIn E60) ="’:E10)=1- b.Suppose agiven matrix A=lla,l,ll oforder tenhas elementary divisors E90) =(3—1)2(4—X)“, E30) =(3—X)’(4—X), E7(X) =4—X, E60) =4—In E1 El1=1 5()~ —() (calculated from theminors ofthematrix A~XE,asinSecs. 6.43-6.44). Then, according toSec.6.44, theJordan matrix J(A) hastwoblocks ofsizes 2and2corresponding totheroot Xl=3,andfour blocks ofsizes 3,1,1and 1corresponding totheroot X2=4.Itfollows that 31 03 31 03 410 041 004 4 4 4J(A)= 6.46. Thus from aknowledge oftheelementary divisors ofanoperator A,wecandetermine allthenumbers ngkland hence thestructure ofthe Jordan canonical form ofA.Inparticular, weseethat theJordan canonical form ofanoperator Aisuniquely determined byA. SEC.6.5 ruRTHER IMPLICATIONS I53 Ontheother hand, since theelementary divisors ofanoperator Aare determined bytheminors ofthematrix A—XEinanybasis, twoequivalent operators AandB,i.e.,twooperators with thesame matrix intwo(distinct) bases, have thesame Jordan canonical form. Conversely, itisobvious that iftwooperators have thesame Jordan canonical form, then theyareequivalent. This completely solves theproblem oftheequivalence oflinear operators (inacomplex space), posed atthebeginning ofthechapter. 6.5. Further Implications 6.51. Ifitisknown thattheoperator Acanbereduced todiagonal form, i.e.,that itsmatrix hastheform Xi Xi ;\2 AZ . 2.2 Am insome basis, then AisjusttheJordan matrix oftheoperator A(allthe Jordan blocks areofsize1).lnparticular, theelementary divisors allhave simple roots. Conversely, ifalltheelementary divisors ofanoperator A have only simple roots, then theJordan matrix J(A) hasblocks ofsize 1 only andhence isdiagonal. 6.52. Given theJordan canonical form ofanoperator A,wecaneasily ﬁnd itsminimal annihilating polynomial. Suppose theoperator Bhasthe matrix 0l0 0 001~~o 000~~-l 000-~~0 I54 THEcANoNioAL roRM orTHEMATR1x orALINEAR OPERATOR CI-IAP. 6 inthebasis el,...,el,,sothat Bel=O,Be2=el,...,Be,=e,,_l. Then B1'x=0 forevery D x=Zcl.el,. k=1 Thus X”isanannihilating polynomial oftheoperator B.The minimal annihilating polynomial isadivisor ofX”(seeSec.6.33), andhence must be oftheform X'",m<p.ButB"_1e, =elqé0,sothatX”isinfacttheminimal annihilating polynomial ofB. Now suppose theoperator Ahasthematrix 10101110 01,1-1-0 000 1 000 X0 inthesamebasisel,...,e,,, sothatA=B+X0E. Asjustshown, (A—X0E)”=B"=0, andhence (X0—X)”isanannihilating polynomial ofA,infacttheminimal annihilating polynomial, bythesame argument asbefore. Next suppose theoperator Ahasthequasi-diagonal matrix 1,1()...() 0X01~~~0 0001--1 0001--1, 9 10101110 01011110 000-111 000-1-1, sEc.0.6 THEREAL JORDAN cANoNicAL EoRM I55 where theblocks along thediagonal have sizes pl>p2>--->p,_Then a polynomial Q(X) annihilating theoperator Amust annihilate each block separately. Clearly thepolynomial (X0—X)“hasthisproperty (cf.Sec.4.52), andinfactistheminimal annihilating polynomial, bythesame argument as before. Finally, inthegeneral case where theoperator AhastheJordan matrix described bythetable (9),thepolynomial Q(X)=(1..—1)"i"’ isclearly anannihilating polynomial ofA,infacttheminimal annihilating polynomial, since none oftheexponents nf”canbelowered, forthereasons given above. Thus thepolynomial Q(X) istheminimal annihilating polynomial ofthe operator A.The degree ofQ(X), equal tonil’+---+n§'"’, isthesum of thesizes ofthelargest Jordan blocks, each corresponding toaroot ofthe characteristic polynomial. Note thatthisnumber cannot exceed theorder of thematrix A,i.e.,thedimension nofthespace inwhich theoperator Aacts. Thecharacteristic polynomial det(A-XE)=1%](1,,-1)"lk’+“‘+"1'£’ 1;==1 oftheoperator A(seeSec.6.42) contains Q(X) asafactor, andhence isalso anannihilating polynomial (aresult known astheHamilton—Cayley theorem). However, thecharacteristic polynomial isingeneral not theminimal annihilating polynomial ofA.Clearly, the characteristic polynomial coincides with theminimal annihilating polynomial ofAifandonly ifeach root ofthecharacteristic polynomial ﬁgures inonly oneJordan" block, of sizeequal tothemultiplicity oftheroot. 6.6.TheReallordan Canonical Form 6.61. LetAbealinear operator acting inarealn-dimensional space R,,. Then ingeneral there isnocanonical basis inwhich thematrix ofAtakes the Jordan form (8),ifonly because thecharacteristic polynomial ofAcanhave imaginary roots. Nevertheless, wecanstillﬁndamodiﬁcation oftheJordan matrix (8)suitable forthecase ofarealspace. LetA=llalklll bethematrix oftheoperator Ainsome basis el,...,e,, ofthespace R,,,andconsider thecomplex n-dimensional space C,consisting ofthevectors x=°I1e1‘I_""I_anem I56 THE CANONICAL FORM OF THE MATRIX OF ALINEAR OPERATOR CHAP. 6 where (X1,...,ix,arearbitrary complex numbers. The matrix Aspeciﬁes a linear operator Ainthespace C,inaccordance with theformula A ‘VI n n Ax:2°‘i¢Aei.- =2”-k(2al'k)e;I)»k=1 k=1 j=1 thesame formula specifying theoperator Aitself forvectors xwith real components ul,. 6.62. First weconsider thecase ofanoperator Awith anannihilating polynomial ofthespecial form P<1)=0*+1*)». where -risapositive number. Fortheoperator Aitmakes sense totalkabout polynomials Q(A) with complex coefﬁcients, inparticular, thepolynomials (A+i-r)" and (A—i-r)”. The polynomial P(X) =(X2+T2)” isalso an annihilating polynomial oftheoperator A.According toTheorem 6.34, the factorization<12+12))’=<1—ir)@<1+tr)” corresponds toadecomposition ofthespace C,into adirect sum oftwo subspaces C1,and Ci,both invariant with respect to inwhich Ahas annihilating polynomials (X—i-r)” and (X+i-r)”, respectively. Moreover, ifthesubspace C},consists ofthevectors xI(X181 +III+amen: with arbitrary complex coefﬁcients xl,...,oc,,,, then thesubspace CZ consists ofthevectors jZ£2181 J7Iii interns where til.isthecomplex conjugate of(Xl,(k=1,...,m).Infact, if (A-i-rE)"x =0, (15) then, taking complex conjugates inboth factors oftheleft-hand side, weget (A+HE)»; =0, (15') andconversely.'l' lnparticular, itfollows that niseven, i.e.,n=2mwhere misthedimension ofeach ofthesubspaces C},andCi. TThe subspaces CI,andCf,areuniquely determined by(15) and (15’), respectively (seeSec.6.35). THEREAL JORDAN CANONICAL FORM I57 SEC. 6.6 h erator Ainthespace C},,asin Now letff betheJordan basis ofteop atrix ofA inthisbasis isoftheform Sec.6.37. According to(7),them it l_0 "1 0 iv 1 OOit. it 10 0 it 1 ll“ 0 0 it Hence theaction ofAonthebasis vectors isdescribed bytheformulas 8/1=0/1. ....Aft=11/1. Af§=f1‘+iTf2‘, 1,Af§=fl+iTf§,A.............q $1.‘.=f.‘._.+111:......AfZ,=fZ,_1 +111.,- The action ofAonthecomplex conjugate vectors inCf,isdescribed by thecomplex conjugates ofthese formulas: AL}=Inf}, _ ..., Ag=:lTfg, _ Ki;=11—11/1. 1.Kf;=i1— 11/2. 31:.--f3.._.—111,--1.$f:._.=f:._.—11/2,. -1‘ Jordan basis fortheoperator Ainthe Thus weseethatthevectors f,.form a taken together form aJordan basis space Ci.Hence allthevectors fortheoperator Ainthewhole space C,,. Wenow construct abasis intherealspace Rn,byreplacing each pair of complex vectors fand byapair ofrealvectors 1 _ fI) (16)1 — .gl=E(f'§+f'§), hl=Z(fiC -- ltfollows from theformulas Ar’;=12:.+1-/1. 3f_l=fZi1— HE<fi=E= 0) THE CANONICAL FORM OF THE MATRIX OF ALINEAR OPERATOR CHAP. 6 that >){E(fl+171):EAg’;=g’i_1—Th’; A1 — —A{5(ii—f’;)}EAh?=hi_.+lg?(gs=hi=0) Thus theaction oftheoperator Aonthevectors gj.‘andhfisdescribed bythe formulas A31= -1111. Ahl‘=Tel‘. Agl‘=gl‘ —Th§, A111‘= hi+Tglf, (17) Agiick Z glii.~—1 _ThI1Ik’ Ahlrik Z hﬁii,--1 TITTglii.-’ Moreover, (16) implies fl‘=gl‘+ihl‘. fi=gl—1'11?- Therefore the(complex) linear manifold spanned byallthevectors gjf,hj.‘is thesame asthelinear manifold spanned byallthevectors fgr‘,fButthe number OfVectors gl,‘,hfisthesame asthenumber ofvectors f, Hence thevectors gf,hfarelinearly independent over theﬁeld C,justlikethevectors ff,fThus, afortiori, thevectors gf,hfarelinearly independent over the ﬁeld R,i.e.,intherealspace R,,. ltfollows from theformulas (17)thatthematrix oftheoperator Ainthe basis g',i,hfisaquasi-diagonal matrix, made upofblocks oftheform OT 1O —'r O O1 0T 10 —'r O O1 0T -1 0. l (13) OT 10 -—~.'O O1 OT —-T0 ofsizes 2nl,...,2n,, respectively. SEC.6.6 THEREAL JORDAN CANONICAL roRM I59 6.63. Wenow consider thegeneral case. LetAbealinear operator ina realn-dimensional space Rn,andletP(X) beanannihilating polynomial of P(X). Then P(X) hasafactorization oftheform P0)=(1..—1)"10—6.)?+ﬁr" (towithin anumerical factor) intherealdomain, where Xl,(k=1,...,n) arethedistinct realroots ofP(X) and cl+i-rl:ul,cl—i-rl:plarethe distinct imaginary roots ofP(X). According tothegeneral theory (Sec. 6.36), thespace R,canberepresented asadirect sum R,=ZEl,+ZFl k=1 l=1 ofsubspaces invariant with respect toA,where (Xl,—X)"isanannihilating polynomial oftheoperator Ainthesubspace El,while (cl—X)?+-cfisan annihilating polynomial ofAinthesubspace Fl.Inthesubspace El,the operator Acanbereduced totheJordan canonical form (7). Asfor thesubspace Fl,letBl =A—cslE. Then (X2+-cf)“ isanannihilating poly- nomial fortheoperator BlinFl,and hence, bySec.6.62, there isabasis inwhich thematrix ofBlisoftheform (18), with -rreplaced by-rl.Inthis same basis thematrix oftheoperator A=Bl+olEisquasi-diagonal, made upofblocks oftheform cl Tl 10 —'rl cl 0 1 csl':l 1O —':l cl O1 61 Ti T“°’ (19) cl Tl 10 —~:l cl 01 cl-rl —'rl cl ofsizes 2nl,...,2n,,respectively. Thus wecanchoose abasis inthespace R,inwhich thematrix oftheoperator Aconsists ofdiagonal blocks ofthe form (10)and(19). This “real Jordan matrix” willbedenoted byJR(A). 6.64. AsinSec. 6.4,thestructure ofthematrix JR(A) canbededuced from theelementary divisors oftheoperator A,which inturncanbecalculated I60 THEcANoNicAL FORM orTHEMATR1x orALINEAR OPERATOR CHAP. 6 from theminors ofthematrix A-—XEintheoriginal basis. Since thepoly- nomials D,,(X) andE,,(X) areobtained from theminors A—XEbyrational operations, thepolynomials E,,(X) have realcoefﬁcients andhence areofthe form E.-.0)=H01—1)"?H10—6.)’+rirl” 0'=1126--l~—1) k=1 l=1 (cf.Sec.6.44). Toevery exponent njklthere corresponds aJordan block of sizen§"’, andtoevery exponent pj.”ablock oftheform (19)ofsize2p§”. 6.65. The above results canbesummarized intheform ofthefollowing THEOREM. Given anyoperator Ainarealn-dimensional space R,,,there exist. abasis inwhich thematrix ofAisquasi-diagonal, made upofblocks of theform (10)and(19), where Xl,(k=1,...,m)aretherealroots andcl3|;i-rl (l=1,...,s)thecomplex roots ofthecharacteristic polynomial ofA.The sizes oftheblocks areuniquely determined bytheelementary divisors ofAin thewayindicated inSec. 6.64. 6.66. COROLLARY. Every linear operator Ainarealn-dimensional space R,hasaninvariant subspace ofdimension 2. Proof. Thebasis vectors g’{andh’;obviously generate atwo-dimensional invariant subspace ofA(see(17)). | The number ofdistinct two-dimensional subspaces ofAcanalways be estimated (from below). lnfact, there areatleast asmany such subspaces asthere aredistinct diagonal blocks (19)ofsize>2inJR(A). *6.7. Spectra, letsandPolynomials Inmany problems ofalgebra andanalysis, theneed arises tocalculate various functions (inparticular, polynomials) ofgiven linear operators acting inaﬁnite-dimensional space. Such functions, which have anumber ofspecial properties, willbeinvestigated inthenext twosections. Anatural arithmetic model forfunctions ofasingle operator isthealgebra ofjets, with which we begin ourdiscussion. 6.71. Byaspectrum, denoted byS,wemean anysetofpoints Xl,...,Xl,, where itisassumed that each point Xl,isassigned a“multiplicity,” i.e.,a positive integer rl,(k=1,...,m),afactindicated bywriting s={>.;1,...,1;;,»}. SEC.6.7 SPECTRA, JETS AND POLYNOMIALS I6I Moreover, weassume thateach point Xl,isassigned asetofrl,numbers from theﬁeld K,denoted by /0.1)=/001.)./'<1..)11 11./‘--"<1..)1 Such asetofnumbers willbecalled ajetf,deﬁned onS. Wenow introduce thefollowing algebraic operations inf(S),thesetof alljetsonagiven spectrum S: _ a.Addition ofjets. Bythesum f+goftwo jetsf= {f(")(Xl,)} and g={g‘1l(Xl,)} wemean thejetdeﬁned bythesetofnumbers (f+ g)”’0)1) =f(j’0k) *1‘g(”0it) (k=1,...,m;j=0,l.,...,rl,-1). b.Multiplication ofajetbyanumber. Bytheproduct otfofajetf= {f”l(Xl,)} andanumber ateKwemean thejetdeﬁned bythesetofnumbers <1»/)~')0..)= 1»/‘"001 These twooperations obviously convert thesetf(S)into alinear space, whose zero element isthejet0whose “components” areallzero. c.Multiplication ofjets. Bytheproduct fgoftwojetsf= {f("l(Xl,)} and g={g(l)(Xl,)} wemean thejetdeﬁned by’r (fg)0k) =f0i¢)g0k)» (fg)'0i.) =f01)g'0).) +f'0).)g01), J (f8)"’(Xi.) =ZC’if"’(X)1)g”_”(X).) l=0 (k=1,...,m;j =0,1,... ,rl,-1), where Clisthebinomial coelﬁcient . ‘IC;:_i___ ll(j—l)! Itiseasily veriﬁed that this operation iscommutative and satisﬁes conditions l)—3) ofSec.6.21. Therefore f(S)isacommutative algebra over theﬁeld K.This algebra hasaunit, i.e.,ajetesuch that ef=ffor every fef(S). lnfact, weneed only choose nj=010)), :e(9 l0if0<j<@ (k=l,...,m). TThese formulas areformally identical withLeibniz’s ruleforrepeated differentiation oftheproduct oftwofunctions /"and g. I62 THECANONICAL FORM orTHEMATR1x orALINEAR OPERATOR CHAP. 6 lnwhat follows, wewillsetupacorrespondence between thealgebra f(S) andthealgebra ofallpolynomials with coeﬂicients intheﬁeld K,forthecase where thepoints Xl,...,X,,,allbelong toK. 6.72. Itwill beassumed that theﬁeld Khasinﬁnitely many distinct elements. Making thisassumption, weﬁrst show how to“reconstruct” the coefﬁcients ofapolynomial from aknowledge ofitsvalues. a.Let P P(X)=Z11,1" k=0 beapolynomial with coefﬁcients intheﬁeld K,whose argument Xalsotakes values inK.Then thecoefﬁcients all,al,...,a,,ofP(X) are uniquely determined bythevalues ofP(X). Infact, letXl,,Xl,...,X,, bedistinct elements ofK,andconsider theequations ao‘l‘a17\0 “I”'''+“M0 :P00)» all—l—alXl +'''+a,,Xf =P(Xl), a0+a1)‘at ‘ITIII+av)‘; :POW)» which can beregarded asasystem ofp+1equations intheunknowns all,al,...,a,,. The system hasanonvanishing determinant (see Example l.55c), and hence, asasserted, hasaunique solution byCramer’s rule (Sec. 1.73). b.Inparticular, itfollows that iftwopolynomials 11 P P0)=Z110". Q0)=Z50k k=0 k=0 coincide forevery value XeK,then al,=bl, (k=0,l,...,p). 6.73. Wewillsubsequently need theconcept ofthederivative ofapoly- nomial P(X), andthenotions ofhigher derivatives andTaylor’s fonnula as well. Inanalysis these concepts areintroduced forthecase ofpolynomials which arefunctions ofareal(orcomplex) argument, buthere wearecon- cerned with polynomials P(X) whose argument Xvaries inanarbitrary ﬁeld K.Wemust therefore introduce thecorresponding deﬁnitions independently, i.e.,without recourse tothenotion ofalimit which maynotexist intheﬁeld K. a.Fixing apoint ueK,wewrite theformula P P l Pb2111.1"=2111.10+0-101*=Z0-0)’: (20)A-=0 11:0 k=0k. SEC.6.7 SPECTRA, JETSANDPOLYNOMIALS I63 where thequantities b)1(1*)i k=0,1,..., k, ( iv) arethepolynomials inliobtained after expanding [li+(X—pL)]kI1'1 powers ofliandX~uandcollecting similar tenns. Thepolynomials bl,(u) arethen given thefollowing names: P . bll(u) =k;)al,u" =-:-P(a), thepolynomial P(u) itself, T bl(u) =k§1kal,u"“ EP’(a), theﬁrst derivative ofP(u), T b2(u) =’gk(k -1)al,u""2 EP”(u), thesecond derivative ofP(u), b,,(u) =p(p—1)'''1'al,EP“”(u), thepthderivative ofP(u). Forapolynomial ofdegree p,wesetP""(l.I) E0ifq>p. lnthenew notation, formula (20)takes theform "1 . ,P0)=ZCP("’(1*)(X —1*)‘. (20)k=0kl known asTaylor’s formula forthepolynomial P(X). b.Inparticular, forthepolynomial P(X) =(X—a)" (aGK), wehave P(a) =P’(a) =----P“""(a) =0, ptmol) =P!’ P“"(X) =0 (q>p). c.More generally, if P0)=0—¢1)"Q0), wehave Q0) 110~at“.P0) 110—a)"*“’1 andhence P(a)=P’(a) =---=P“’“1l(a) =0. (21) d.Conversely, ifitisknown that (21) holds, then P(X)=20iP""(a)(X -a)" =(1-axi3P""’<a)(1 —a)’"-P=0-a)"Q(X)1k=11kl where Q(X) isanewpolynomial. I64 TI-[E CANONICAL FORM OF THE MATRIX OF ALINEAR OPERATOR CHAP. 6 6.74. Itshould benoted that therepresentation ofthepolynomial P(X) intheform ialxEP0)=ib..0)(1 -.11)’: l\‘=II l('=0 where thebl,(u) arepolynomials inli,isnecessarily unique. Iiifact, suppose weﬁxu=alland give Xthedistinct values Xl,,Xl,...,X,,intum. Then T=X—utakes thedistinct values Xl,—ul,,Xl—ul,,...,X,,—ul,,andthe values ofthepolynomial 17 Zbk(I‘I'0)Tk k=0 areknown forthese values ofT,being equal toP(Xl), P(Xl), ...,P(X,,). But then thequantities bl,(ull) areuniquely determined, bySec.6.72a. Since this istrue forarbitrary li=all6K,thepolynomials bl,(u) (k=0,1,...,p) arethemselves uniquely determined. 6.75. a.Given twopolynomials P(X) andQ(X), wenowverify theformulas (P+Q)""(I») =P"°’(I») +Q""(I1)1 (22) <PQ)""(11) =§¢’;P"’(0)Q"'"”(0) (23) i=0 (k=0,1,2,...),where C, /<1 ’:i1(/<—i)1 Infact, bydeﬁnition, (P+(2)0) i(P+Q)""(11)0 —0)’: P0)=ElP""<i1)(1 —10'“,i=0kl Q0) iQ"“’<11)0 —0)’: P0)+Q(X) iIP“"(I»)+Q"“’(11)10 ~0)". sothat (22)follows from theuniqueness theorem ofSec.6.74. Similarly, 1’1 (Io) 1(PQ)0) =20I;-,(PQ) (100T1*). SEC.617 SPECTRA, JETS ANDPOLYNOMIALS I65 while ontheother hand, P0)=12,P"’<11)<1 —0)’.Q(X) iQ‘”01)0 —0)’. ..ab/I“QM-R-P(X)Q(X) =SS%P”’<11)Q"’(1»)0 ~10”’ = L P"’(I*)Q"‘"”(I1)l(X —(Uk-1"</<1')‘ Pr‘0-0 Thus theuniqueness theorem ofSec.6.74 implies 1 ’” 1 _,-E(PQ)""(1*) =20 P”’(1*)Q"‘ ’(1*), which isequivalent to(23). b.Inparticular, formula (23)implies thefollowing important THEOREM. Q’ P"°)(u)=0 (k=0,l,...,m), then (PQ)"‘>(i1I)= 0 (k=0,1,...,m) foranypolynomial Q(X). 6.76. Now suppose wearegiven aspectrum S={1§1,...,1I,;"} (1,e1<) andthecorresponding algebra f(S)ofjetsonS(seeSec.6.71). Then with every polynomial P(X) weassociate thejetP6f(S)which assigns toX,the numbers P011), F011), ---=P(rk_1I()‘k)a where theP"'l(Xl,) arethederivatives ofthepolynomial P(X), asdeﬁned in Sec.6.73. Itfollows from formulas (22)and(23)that theoperations onjets deﬁned inSec.6.71correspond totheusual operations ofaddition andmulti- plication ofpolynomials. Thus themapping P(X) —>P isamorphism (Sec. 6.24) ofthealgebra ofpolynomials IIinto thealgebra ofjetsf(S). Aswe now show, thismorphism isanepimorphism, i.e.,given anyjetf,wecanﬁnd apolynomial P(X) such that P0k) =f0k), F01) =f’0k)1 ---1P("°_1’0k) =f(rk—1’0‘k) (k=1,...,m). Toprove theassertion, itisenough toconsider thecasewhere allthenumbers f"'l(Xl,) vanish except one, corresponding toanygiven value k=kl.Infact, THE CANONICAL FORM OF THE MATRIX OF ALINEAR OPERATOR CHAP. 6 having solved theproblem forthiscase, weneed only construct apolynomial Pl,(X) foreach k=1,...,msatisfying theconditions P1011)=f0.).111.P§.'*""(1i) =f"*""0l.). <24) i>_$j’(1,)= 0(slak;j=0,1,...,r, -1), (25) andthesolution willthen begiven bytheformula P0)=P10) +---+P110)- Thus wemust ﬁndapolynomial Pl,(X) satisfying theconditions (24) and (25). Tothisend, welook forPl,(X) inthefonn P110) =QkO‘)RlcO‘)1 (26) where Ql,(X) isanew polynomial and R110) =I10 —Xl)"1 (27) s¢k BySec.6.73c, wehave R§.”<1.)=0 <s¢k;i=0,1,111.r.—1), andhence, byTheorem 6.75b, i>‘,_3"(1_..)=0 (s¢/<;j=0,1,..., r,.~1) foranypolynomial Ql,(X). Hence thecondition (25) isclearly satisﬁed. We must stillsubject thepolynomial Pl,(X) tothecondition (24). Since R1¢0k) =1101¢_ls)“ ¢0= thecondition iiI f01¢) =P11011) =Qk0k)Rk0k) uniquely determines Ql,(Xl,). Moreover, once Ql,(Xl,) isknown, thecondition f’0ii) =P1101) =Qil0k)Rk0k) ‘I‘Qii0i¢)Ri¢0i¢) uniquely determines Q,;(Xl,). Continuing inthisway, weareable touniquely determine allthenumbers Ql,(Xl,), Q§,(Xl,), ...,Q§,"="1’(Xl,). But once these numbers areknown, wecandetermine thedesired polynomial Ql,(X) by using Taylor’s formula rl,-1(2.0)=2.iQi"0.)<1 ~1..)"1 <28)i=0j! Reasoning backwards, weseethat thepolynomial Pl,(X) deﬁned byfonnulas (26)—(28) satisﬁes thestipulated conditions (24)and(25). 6.77. Next, applying Sec. 6.52d, weﬁnd that thealgebra f(S)ofall jetsdeﬁned onthegiven spectrum Sisisomorphic tothefactor algebra II/I, SEC.617 SPECTRA, JETSAND POLYNOMIALS I67 where Iistheideal inIIconsisting ofallpolynomials forwhich P"'l(Xl,)=O (k=1,...,m;j=O,1,...,rl,—1). Itfollows from Sec. 6.73d that every polynomial P(X) eIisdivisible bythe polynomial T0)=0—1.1)". (29) and from Sec. 6.73c that every polynomial divisible byT(X) belongs toI. Theideal I,likeevery ideal inthealgebra II,isgenerated bythepolynomial inIoflowest degree (see Sec. 6.26), andthispolynomial isjust T(X) itself. Hence thealgebra f(S)isisomorphic tothefactor algebra II/I, where Iis theideal generated bythepolynomial T(X). 6.78. Wenow usetheresult ofSec.6.77tosolve theproblem ofdescribing allinvertible elements (Sec. 6.21) ofthealgebra f(S). Obviously, ajetf forwhich f(Xl,) =0foratleast onevalue ofkcannot beinvertible, since then (f8)0).) =f0).)g0)1) =09'51=@011) forevery jetg.Thus letfbe ajetsuch that /0..)¢0 (/<=1»--1m), andletP(X) bethepolynomial forwhich P(Xl,) =f(Xl,), ...,P"I-"l(Xl,) =f<'_'=C‘)(Xl,) (k=1,...,m) (seeSec. 6.76). This polynomial clearly hasnofactors incommon with the polynomial T(X) deﬁned by(29), and hence, bySec. 6.28, there arepoly- nomials Q(X) andS(X) such that P0)Q0) +T0)-90) E1- (30) Letqbethejetcorresponding tothepolynomial Q(X). Applying theepimor- phism II->f(S)constructed inSec. 6.76 toequation (30), and using the factthat thisepimorphism carries thepolynomial T(X) into 0,weﬁnd that fq=11 i.e.,thejetfef(S)isinvertible. Letubeanyinvertible jet.Then, asweknow from Sec.6.21, theequation ux=v where xisanunknown jetand vanygiven jet,hastheunique solution x==v/u. Wecanﬁndanexplicit expression fortheratio v/ubysuccessively 1168 THE CANONICAL FORM OF THE MATRIX OF ALINEAR OPERATOR CHAP. 6 solving theequations ”0k)x0i~) =U011), "0k)x’0ii) TI‘"'01.-)x0k) =1/01¢), J' b§oCJju(l)()\k)x(;I-—l)()\k) :v(;I)()\k) (k=l,...,m;j=0,l,...,rl,-1). 6.79. a.Aspectrum S={X§1,. ..,Xj,;"} with complex Xl,...,X,,,is saidtobesymmetric ifwhenever Scontains animaginary number Xl,=cl,+ iTl,,italso contains thecomplex conjugate number Xl,=cl,-—iTl,with the same multiplicity rl,.Ajetf={f""(Xl,)} deﬁned onasymmetric spectrum S issaid tobesymmetric ifthenumbers f‘1l(Xl,) andf‘j’(Xl,) arecomplex conjugates (j=0,1,...,rl,—1).IfP(X) isapolynomial with real co- efﬁcients, then thejetdeﬁned onasymmetric spectrum bythenumbers P<~0..) c/<=1.1.1.m;i=o.1..11.r,.-1) issymmetric, since thederivatives P""(X) alsohave realcoefﬁcients andhence PUIOW)=P<~(i..)1 (31) Conversely, given asymmetric jetf={f"')(Xl,)} onasymmetric spectrum S={X§1,. ..,Xj,j"}, wecanalways ﬁnd apolynomial Pl,(X) with real co- efﬁcients such that P,<,”(1l) =f"’(1,,) (/<=1,...,m;j=0,1,...,rl,-1). Infact, bySec. 6.76, wecanconstruct apolynomial P(X) with complex coefﬁcients satisfying thesame conditions. LetP(X) denote thepolynomial whose coefﬁcients arethecomplex conjugates ofthose ofP(X). Then it follows from (31)that %IP‘”(X1) +P""0)1)I =%IP(jIO‘k) +P”’0)1)I =P”’0).) =f"’0)1), i.e.,thepolynomial P00) =%IP0) +130)] with realcoefﬁcients satisﬁes therequired conditions. b.The setofallsymmetric jetsfonasymmetric spectrum Sobviously forms analgebra over theﬁeld ofrealnumbers. According toSec. 6.25d, thisalgebra isisomorphic tothefactor algebra II/I, where IIisthealgebra of allpolynomials with realcoeﬂicients andICIIistheideal consisting ofall SEC. 6-8 OPERATOR FUNCTIONS AND THEIR MATRICES polynomials P(X) 6IIforwhich P<j)(Xl,)=0 (k=l,...,m;j=0,l,...,rl,—l), i.e.,theideal generated bythe(real) polynomial T0)-i111-1..)'*1 1,-=1 *6.8. Operator Functions andTheir Matrices Inthis section weinvestigate functions ofoperators, ﬁnding matrices (and corresponding rules ofoperation) forpolynomials oftheform P(A) andrational functions ofthefonn P(A)/Q(A), where Aisanylinear operator acting inann-dimensional space C,,(orR,,). InSec.6.89 wewillextend the “calculus ofoperators” tothecase ofanalytic functions ofoperators. 6.81. Given anoperator Aacting inann-dimensional space K,,,letIIAbe thealgebra ofalloperators oftheform P(A), where P(X) issome polynomial. Then IIAisisomorphic tothefactor algebra II/IA, where IIisthealgebra ofall polynomials andIAistheideal generated bytheminimal annihilating poly- nomial T(X)oftheoperator A(seeSecs. 6.31-6.33). Suppose itisknown that thepolynomial T(X) hasthefactorization T0)-0-1.)" <32) intheﬁeld K.Then, bySec.6.77, thefactor algebra II/IA isisomorphic to thealgebra f(S)ofalljetsdeﬁned onthespectrum s=sl={1q1,...,1;,-} (called thespectrum oftheoperator A).I-Ience thealgebra HAisitself iso- morphic tothealgebra f(S).The explicit form ofthisisomorphism isthe following: Toevery jetfef(S)there corresponds theclass ofpolynomials P(X) eIIsuch that P"’(1).)=f"’(1)1) (/<=1.111.m;j Z0.1.- __.ri_1). (33) and toeach ofthese polynomials there corresponds thesame uniquely deﬁned polynomial operator P(A), which wedenote byf(A). Below wewillinvestigate theexplicit form ofthematrix oftheoperator P(A) foragiven minimal annihilating polynomial (32), inthecase where the matrix ofAisinJordan canonical form. I70 THEcANoN1cAL I-‘ORM orTHEMATRIx orALINEAR OPERATOR CHAP. 6 6.82. First suppose theoperator Ahasamatrix (oforder n)ofthe special fonn 7,01 () ()7,0 0 .. . (34) ()() 1 0()1, insome basis ofthespace K,,.Then Aisoftheform XOE-1-B,where the operator Bhasthematrix ()]...() 011110 According toExample 4.74b, thematrix ofB1‘is (/<+1) () 1() () () 01 () (35(/<)1 ) where thediagonal consisting entirely ofones hasmoved over ksteps tothe right from theprincipal diagonal. IfP(X) isanarbitrary polynomial ofdegree p,then 111 l l ‘P0)=2-P""11.)0 -'~o)'“.k=0kl byTaylor’s formula (20’). Replacing Xbytheoperator A,wegettheidentity 1’1 "1 FP(A)=2-P""<1.)(A -ME)"-2—P""(1.)B’~111-0kl 11-0kl Then, taking account oftheexpression (35)forthematrix ofB1‘,weﬁndthat SEC.6.8 OPERATOR FUNCTIONS AND THEIR MATRICES I7| P(A) hasthematrix P0.)P0.)§P"0.) (f,;P‘"""(1.) I 1 'Vl'—2 - 0 P( P0,0 ...ii Pt ) 1.) ) M_2), 0.) 0 0 0 i>(1,,) Note thattoconstruct thematrix ofP(A) from thepolynomial P(X), weonly need thenvalues P(Xl,), P’(X2), ...,P‘"'1)(X,l), where nistheorder ofthe matrix ofA. 6.83. Next suppose theoperator Ahasaquasi-diagonal matrix oforder n,made upofblocks oftheform (34), where Xl,takes thevalues Xl,...,X,,, with corresponding block sizes nl,. ..,n,,,.Bytherules foroperating on quasidiagonal matrices (Sec. 4.52), each block ofthematrix oftheoperator P(A) canbecalculated independently. Applying Sec. 6.82, weﬁnd that the matrix ofP(A) isobtained byreplacing each block (34)ofthematrix ofAby theblock (36). Thus toconstruct thematrix ofP(A), wenow need thevalues P~'>(1,,) (k:1,...,m;j=0,l,...,nl,-1). 6.84. LetAbeanyoperator acting inann-dimensional complex space C,,. Then, asonpp.146-147, there exists abasis inwhich thematrix ofAis quasi-diagonal, made upofblocks oftheform 1,,10 0 01,,1 0 nl-"I ---- - (k=1,...,ni:j=l,...,rl,), (37) 0 00 I 000---1,. where thenumbers rl,andnlklarethose ﬁguring intable (9).Corrmpondingly, thespectrum oftheoperator Ais S: SA={1l‘,...,1§,{"}. If f={f"’0).)} (/<=1.-11.m:j=0.1.--1.r).—1) isanyjetdeﬁned onS,then, bySecs. 6.81-6.83, thecorresponding operator /(A) hasaquasi-diagonal matrix, inwhich each block ofthefonn (37) is I72 -msCANONICAL roam orTHEMATRIX orALINEAR OPERATOR CHAP. 6 replaced bytheblock ‘I _1_ /1 ___ 1 ('n(.k)—1)f(M) 1(1).) 2f(M) ("gm_mf (M-) 0/0.)rm) (,,<_%2),f‘"?*’-*’<x.>. <38) 000 /0,.) The isomorphism between thealgebras HAandf(S)hasnow been made perfectly explicit. 6.85. a.Next weconsider functions ofanoperator Awhich hasamatrix oforder 2moftheform or 10 ——'rc 01 or ——'rcs_ <39) G T -—-T G where csand1areelements oftheﬁeld K.Introducing the2><2matrices 10E= ,A=01 wecanwrite thematrix ofAasthefollowing block matrix oforder m: AE0---00 OAE---O0 000---AE 000---0A A00---00 0E0---00 0A0---00 i005---00 2.... ..+... .. 000---A0 000---OE 000---0A 000---00| sac.6.8 OPERATOR ruucrxous AND THEIR MATRICES I73 Therefore itfollows from Sec.6.82 andtheruleformultiplication ofblock matrices (Sec. 4.51) thatthematrix ofP(A) canbewritten intheform ofthe block matrix P(A)P’(A) %P”(A) (7-£—ﬁP‘"'“"(A) 0P(A) P’(A) (7%2_)!P""-*’(A) . (40) 0 0 0 P(A) b.Ifthematrix ofAisquasi-diagonal, made upofblocks oftheform (34) and (39), then, just asinSec. 6.83, wededuce that thematrix ofP(A) is obtained byreplacing each block bythecorresponding block oftheform (36)or(40). c.Inthecasewhere K=R,sothatth'enumbers 0,-randthepolynomial P(A) arereal, wecanﬁnd theexplicit form ofthematrices P""(A) ﬁguring in(40). Infact, introducing thematrix 0l I: , -l 0 weeasily verify thatI2=-—E, sothat thealgebra ofrealmatrices cs-r Re). Im)\ A=0E+'rI== = ()\=c+i'r) ~—'r cs -—Im)\ Rel isisomorphic totheordinary algebra ofcomplex numbers (cf.Example 4.74a). Hence foranypolynomial P(A) with realcoeiﬁcients wehave ReP(A) lmP(A) P(A) IP(6E +T!)= __1m P0) RePO) (X:cs+ir), andcorrespondingly ReP""(7\) ImP<'=>(x)P"‘*(A) =P<">(@E +<1)= .—lmP<’"()\) ReP<'=>(x) 6.86. LetK=RandKn:Rn.Then, given anyoperator Aacting in Kn,theminimal annihilating polynomial T(A) hasrealcoefﬁcients andhence hasasymmetric spectrum SA(seeSec. 6.79a). The algebra IIAofoperators ofthefonn P(A) isisomorphic tothefacior algebra II/IA, where IIisthe algebra ofpolynomials with realcoefﬁcients andIAistheideal generated bythe minimal annihilating polynomial oftheoperator A.According toSec.6.79b, 174 THE CANONICAL FORM OF THE MATRIX OF ALINEAR OPERATOR CHAP. 6 this factor algebra isisomorphic tothealgebra ofsymmetric jets onthe spectrum SA.Ontheother hand, there isabasis inwhich thematrix ofA isquasi-diagonal, with diagonal blocks oftheform (34) and(39). Letfbe any symmetric jetonthespectrum SA.Then itfollows from theabove considerations that thecorresponding matrix f(A)isobtained byreplacing every block (34)byablock (38)andevery block (39)ofsize2mbytheblock matrix /‘(A’A --1 ""*"A >f<> (m1),.‘ <> -f<A> (rn—l2—),,1"'"*”<A> 00 /(A) oforder n1,where thef<’°>(A) are2><2matrices oftheform fa.->(A) : Ref(k)(7\) 1mf(k)(7\) ~—Imf""()\) Ref""()\) 6.87. Given alinear operator Aacting inaspace C,,,suppose Ahasthe Jordan canonical fonn (8)speciﬁed bythetable (9),asonpp.146-147. We now look forallinvertible operators oftheform P(A), where P(X) isapoly- nomial. Itisclear from theform oftheoperator ofthematrix ofP(A) inthe Jordan basis oftheoperator Athat thedeterminant ofthismatrix isjust H[P(7v.)]"‘. 2'1.=" k=1 k=1 (cf.Example l.55b). Therefore theoperator P(A) isinvertible inthealgebra L(C,,) ofalllinear operators acting inthespace C,,ifandonly if P(x,,)¢o (k=1,...,m). (41) Moreover, ifthecondition (41)issatisﬁed, then theinverse operator [P(A)]‘1 already belongs tothealgebra IIA.Infact, inthiscasethejetpcorresponding tothepolynomial P(X) inthealgebra ofjetsf(SA),i.e.,thejetconsisting of thenumbers P”’()\k) (k=1,...,m;j=O,1,...,rk—1), isinvertible inthealgebra f(SA),bySec.6.78. Butthen theoperator P(A) isinvertible inthealgebra IIA,bytheisomorphism between thealgebras f(SA) andIIA. sEc.6.8 OPERATOR FUNCTIONS AND THEIR MATRICES I75 Again using theisomorphism between thealgebras f(SA) and IIA,we seethat ifP(A) isinvertible, then theequation P(A)X(A) =Q(A). where X(A) isanunknown polynomial andQ(X) anygiven polynomial, has theunique solution X(A) =Q(A)/P(A). Letx,pandqbethejetscorre- sponding tothepolynomials X(A), P(A) and Q(A), respectively, sothat in particularpx =q,xq/p Then according toSecs 678and684thematrix oftheoperator X(A) intheJordan basis oftheoperator Aisoibtained by replacing every block oftheform (36) byablock oftheform 0pa.) ip<»>)>-=». - <42) 0 0 ‘$0 ... 1101.) 6.88. The above result canbeinterpreted somewhat diﬁerently. Given a spectrum S={M1, ...,X13}inthecomplex plane, 1et@1,(S) denote thesetof allcomplex rational functions )1/0)=(L).P(X) where P(A) andQ(A) arepolynomials, andP(A) hasnoroots atthepoints of thesetS.Intheset@?,(S) wedeﬁne theoperations ofaddition oftwofunctions, multiplication ofafunction byacomplex number, andmultiplication oftwo functions inaccordance with theusual rules, thereby making @l,(S) into an algebra over theﬁeld C.Moreover, wenote that every function f(7.)e@l,(S) hasderivatives f'()\),f”(7\), ...intheusual sense ofanalysis. Assigning to each function f(7.)€6Ji,(S) thejet f={f”’()\k)} (k=1,...,m;j=0,1,...,rk-1), where f“"(7\,,) denotes theusual jthderivative off(7.),wegetamorphism of thealgebra @l,(S) ofrational functions into thealgebra f(S)ofjetsonthe spectrum S,infactanepimorphism, since bySec.6.76 thejetscorresponding tojustthepolynomials Q(A) already ﬁllthewhole algebra f(S). Now letS=SAbethespectrum ofsome operator Aacting inthespace C".Then thealgebra IIAofoperators P(A) isisomorphic tothealgebra of jetsf(SA), andwecanextend thegiven epimorphism @l,(SA) —>f(SA) to anepimorphism @1,(SA) —>IIA.Inother words, wecan assign toeach Tl-[E CANONICAL FORM OF TI-[E MATRIX OF ALINEAR OPERATOR CHAP. 6 rational function f(7.)€‘Jl(S) alinear operator f(A)eIIAsuch that the correspondence f(7.)—>f(A)isagain anepimorphism, where thematrix of theoperator f(A)isgiven bytherule(42). 6.89. lnstead ofthealgebra ofrational functions, wecanconsider the algebra ofanalytic functions. Thus let6f(S) bethesetofallfunctions f(7.) analytic atthepoints 7.1,...,71",,i.e.,analytic inaneighborhood ofeach ofthepoints 7.1,...,)\m. Then theset 6f(S) equipped with theusual operations ofaddition andmultiplication isagain analgebra over theﬁeld C,infact analgebra containing thealgebra @l,(S). Analytic functions also have derivatives ofallorders (intheusual sense ofanalysis), andusing them, wecanextend theepimorphism@1,(S A)—>IIAconstructed inSec. 6.88 toan epimorphism 5f(SA)—>IIA.Animportant feature ofthis new epimorphism isthat itnow involves many transcendental functions ofanalysis, likee”‘, cost-7.,sint.)\,etc.Iff(A)denotes theoperator corresponding tothefunction f(7.)e€F(SA), then itsmatrix intheJordan basis oftheoperator Aiscal- culated bythesame rule (38) asbefore. Wenote inparticular that the operator formula eu,+mA =et1Aet9A isanimmediate consequence oftheidentity eu,+¢2»>. :el1}.e1g7\' andthefactthat themapping 5f(SA) —>IIAisanepimorphism. Theresults ofSecs. 6.87-6.89, pertaining tolinear operators inacomplex space, canbecarried over tolinear operators inarealspace, byusing the realJordan canonical form andthemethod ofSecs. 6.85—6.86. Weleave the details ofthisextension tothereader, since nonewideas areinvolved. PROBLEMS 1.Thematrix ofanoperator Aisoftheform ).OO---0O 17.O---OO Ol7\---OO O00---7.0 000---l7\ inabasis el,e2,..,en.Inwhat basis does ithave Jordan canonical form? PROBLEMS I77 2.Prove thatthematrix Aandthematrix A’(obtained bytransposing A)are equivalent. 3.Find theJordan canonical form ofthematrix -2-1-1 32 ~4 1-1 32 110-3-2 -4~2-1 5 111-3 0 4.Aretheoperators speciﬁed bythematrices equivalent?110' 41-1 ~OlO,B—~6—l2 002 211 5.Find theelementary divisors ofthefollowing matrices oforder n: A1: 7! 0 /1,: 0 0_1n_ n n— 0 n 3, A4 0 0 ... n11---1 01 ,/‘E20 00...l 2 ...1 1 2 0I O 0 6.Show thatallmatrices oftheform A00¢--a3,, O00--<1°‘"12 a13 0MawO O aln a2n2 n3 n —l 1 0 , .. 1 1 2 O 01... 1 2 ... 2 3 ... 3 . 0 n I78 THECANONICAL FORM orTHEMATRIX orALINEAR OPERATOR CHAP. 6 with arbitrary elements am,aw,...areequivalent iftheelements am,aza,..., a,,_1_,, arenonzero. 7.Find theJordan canonical form ofthematrix Asatisfying theequation P(A) =O,where thepolynomial P(A) hasnomultiple roots. 8.Find theJordan canonical fonn ofthematrix Asatisfying theequation P(A) =O,where thepolynomial P(A) isanarbitrary polynomial. 9.Prove thatiftheannihilating polynomial ofanoperator Aacting inthespace R"isofdegree 2,thenevery vector xliesinaplane orlineinvariant withrmpect toA. 10.Find allmatrices commuting with themXn1matrix a1O---OO OaI OO Am(a): . . . . . . _ OOO a1 O00-"On 11.Find allm><nmatrices Bsatisfying thecondition BA,,(a) 1A,,,(a)B. 12.Find allmatrices commuting with quasi-diagonal matrices oftheform A,,,1(a) 0---0 O Am2(a) '' O O O ...A,,,k(a) 13.Find allmatrices commuting with quasi-diagonal matrices oftheform A,,,1(a1) O --- O . A,,,2(a2) .. O a 0 0---A..,<a..> where thenumbers al,a2,...,akarealldistinct. 14.Find allmatrioes commuting with thegeneral Jordan matrix (8). 15.Under what conditions isevery matrix commuting with agiven matrix A apolynomial inA? chapter 7 BILINEAR AND QUADRATIC FORMS Inthischapter, weshall study linear numerical functions oftwovector arguments. Unlike thetheory oflinear numerical functions ofonevector argument, thetheory oflinear numerical functions oftwovector arguments (such functions arecalled bilinearforms) hasrichgeometric content. Setting thesecond argument equal totheﬁrstintheexpression forabilinear fonn, wegetanimportant new kind offunction ofonevariable, called aquadratic form, which isnolonger linear. The considerations ofSecs. 7.1—7.8 pertain toalinear space Kover an arbitrary number ﬁeld K,while those ofSec.7.9pertain toareallinear space. 7.l.Bilinear Forms 7.11. Anumerical function A(x, y)oftwovector arguments xandyina linear space Kiscalled abilinearform (orabilinearﬁmction) ifitisalinear function ofxforevery ﬁxed value ofyandalinear function ofyforevery ﬁxed value ofx.Inother words, A(x, y)isabilinear form inxandyifand only ifthefollowing relations hold foranyx,yand2: A(x+Z.y)=A(x.y)+A(z.y). A(<v<.y) =<*A(x.y). (1 A(x, y-I-2)=A(x, y)-I-A(x, 2), ) A(x.My)=°@A(x. J’)- I79 I80 BILINEAR AND QUADRATIC roams CHAP. 7 Theﬁrsttwoequations mean thatA(x, y)islinear initsﬁrstargument, and thelasttwoequations that A(x, y)islinear initssecond argument. Using induction andtherelations (1),weeasily obtain thegeneral formula 2:- 3 ,_'l\/1*,_l\/IiA<g1°%xi.g1i51)’1) : °iri'3rA(xi. Yr)» (2) where xl,...,xk,yl,...,y,,,arearbitrary vectors ofthespace Kand ozl, ...,ozk,I-3,,...,I-3",arearbitrary numbers from theﬁeldK. Bilinear forms deﬁned oninﬁnite-dimensional spaces areusually called bilinearfunctionals. 7.12. Examples a.IfL1(x) and L2(x) arelinear forms, then A(x,y) =L1(x)L2(y) is obviously abilinear form inxandy. b.Anexample ofabilinear form inann-dimensional linear space with aﬁxed basis el,e2,...,enisthefunction ;M=.l"1*A(x» Y)=_ aikiiﬂr. where ‘VI 71 X:Z£13.". Y:2WI.~@i- i=1 i-=1 arearbitrary vectors andtheaw(i,k=I,2,...,n)areﬁxed numbers. 7.13. The general representation ofabilinear form inann-dimensional linear space. Suppose wehave abilinear fonn A(x, y)inann-dimensional linear space Kn.Choose anarbitrary basis e1,e2, ...,e,,inKn,andwrite A(e,,e,,)=a,,, (i,k=1,2,...,n). Then foranytwovectors 1L 71 -Y=Z£13.". YZ2911311»i=1 1,-=1 itfollows from (2)that :l‘/1’HI‘/13m A(x. Y): Ez@¢»i‘%@1.) 2,_E="ﬁ1¢A(e1'> ek) Tiﬁat im (3) " kIr‘ i=1k>1 Thus themost general representation ofabilinear form inann-dimensional linear space hasalready been encountered inExample 7.l2b. SEC-7-1 BILINEAR roams I31 Thecoefﬁcients a,»,,form asquare matrix an a12 ''’am A=A<.,= “*‘“*2 "2"=|1a..|| anl an2 ''' arm which wewillcallthematrix ofthebilinear form A(x, y)in(orrelative to) thebasis {e}={el,e2,...,en}. 7.14. Symmetric bilinear forms. Abilinear form iscalled symmetric if A(x.Y)=A()/.16) forarbitrary vectors xandy.Ifthebilinear form A(x, y)issymmetric, then aik=A(ei» ex)=A(ek» er)=am, sothatthematrix A“,ofasymmetric bilinear form inanybasis el,e2,...,e,, ofthespace K,,equals itsown transpose A1,). Itiseasily veriﬁed that the converse isalsotrue, i.e.,ifA(B,=Au,inanybasis el,e2,...,e,,,then the form A(x, y)issymmetric. Infact, wehave ...2:-[\/1:“.9a- A()’a x)=>g_1a.-imtii. =,g_1ai.m-'51. = imi. =A(x» Y)» asrequired. Inparticular, wehave thefollowing result: Ifthematrix ofthe bilinear form A(x, y)calculated inanybasis equals itsown transpose, then thematrix oftheform calculated inany other basis also equals itsown transpose. Amatrix which equals itsown transpose will henceforth be called symmetric. 7.15. Transformation ofthematrix ofabilinear form when thebasis is changed. a.Ofcourse, ifwetransform toanew basis, thematrix ofabilinear form changes according toacertain transformation law. Wenow ﬁnd this law. LetAm=l|a,»,,|| bethematrix ofthebilinear form A(x, y)inthebasis {e};_"{e1s e2’ '''aen}! andletAm=Ilb,-,,|| bethematrix ofthesame form inthebasis Iflifilv“ 'hfn} (i,k=1,2,...,n).Assuming that thetransformation from onebasis to theother isdescribed bytheformula 1.=j;11>§"e. <1=1,2,._..~> I82 BILINEAR AND QUADRATIC roams ci-1A1>, 7 withthetransformation matrix P=||p§."’||, wehave 1».=A</.1.)=A(Z11>‘."e,-,l21 We.)- =Z1>§"’1>l”’A(@.. er)=Z1>‘."1>§”’¢1..- i_1=1 i.l=1 This formula canbewritten intheform :M=.M=‘Q 11..= $”'a..1>§*’. (4) where pl” =pj“isanelement ofthematrix P’which isthetranspose ofP. Equation (4)corresponds tothefollowing relation between matrices (see Sec.4.43): Am=P’/41.1P» (5) b.Since thematrices PandP’arenonsingular, itfollows from Corollary 4.67 that therank ofthematrix Amequals therank ofthematrix Am, i.e., therank ofthematrix ofabilinear form isindependent ofthechoice ofabasis. Hence itmakes sense totalkabout therank ofabilinear form. Abilinear form A(x, y)issaid tobenonsingular ifitsrank equals thedimension nofthe space K". c.LetA(x, y)beanonsingular bilinear form. Then, aswenow show, given anyvector xo¢0,there exists avector yoeK"such thatA(xo, yo)¢0. Suppose tothecontrary thatA(xo, y)=0forevery yeK,,,andconstruct a basis el,e2,...,e,,inthespace K"such that e1=xo.Then thematrix of theform A(x, y)inthisbasis issuch that alm =A(e1a em) =A(x0a em) =0» sothatthewhole ﬁrstrowofthematrix consists ofzeros. Butthen therank ofthematrix islessthan n,contrary tothehypothesis that A(x, y)isnon- singular. This contradiction proves ourassertion. d.Note that aform A(x, y)which isnonsingular inthewhole space K may besingular inasubspace K’CK.Forexample, theform A(x»)/)'= E1‘/11 "E2712 isnonsingular inthespace R2,wherex =(E1,Z2),y=(hl,ha).However, it vanishes identically inthesubspace R;CR2where Z1=Z2(and nl=nz). e.Itfollows from (5)andTheorem 4.75onthedeterminant oftheproduct oftwomatrices that detAm2detAm(detP)2. (6) SEC.7.2 QUADRATIC roams I83 7.2.Quadratic Forms One ofthebasic problems ofplane analytic geometry istoreduce the general equation ofasecond-degree curve tocanonical form bytransforming toanew coordinate system. The equation ofasecond-degree curve with center attheorigin x=0,y=0,hasthefamiliar form Axz+2Bxy +Cy”=D. (7) Acoordinate transformation isdescribed bytheformulas x=aux’ +any’, y=aux’ +any’, where an,am,an,a22arecertain numbers (usually sines andcosines ofthe angle through which theaxes arerotated). Asaresult ofthiscoordinate transformation, (7)takes thesimpler form A’x’2 +B’y’2 =D. Ananalogous problem canbestated foraspace with anynumber ofdimen- sions. The solution ofthisandrelated problems isthefundamental aimof thetheory ofquadratic forms, which wenow present. 7.21. Webegin with thefollowing deﬁnition: Aquadraticform deﬁned onalinear space Kisafunction A(x, x)ofone vector argument xeKobtained bychanging ytoxinanybilinear form A(x,y) deﬁned onK. According to(3),inann-dimensional space K,,with aﬁxed basis {e}= {el,e2,...,en},every quadratic form canbewritten as ...l\/1:[\/I: A(x’ x) Z aikE:i€I:a =1k=1 where Z1,Z2,...,Z"arecomponents ofthevector xwith respect tothe basis {e}.Conversely, every function A(x, x)ofthevector xdeﬁned inthe basis {e}byformula (8)isaquadratic form inx.lnfact, weneed only introduce thebilinear form ilvieiM=B(x, Y)=__¢1.¢.€.-*1», where n1,n2,...,nnarethecomponents ofthevector ywith respect tothe basis {e}.Then thefunction A(x, x)isobviously just thequadratic form B(x, x). I84 BILINEAR AND QUADRATIC FORMS CHAP. 7 7.22. Wecanwrite thedouble sum (8)somewhat diﬁerently bycombining similar terms. Letb,-A=a,-,-andb,»,,=a,»,,+ah,»(i¢k).Then, since aikzizk -I"a1..E1.€. =(an; aki)€i€k =bfkiiika thedouble sum (8)canbewritten as A(x1 x)=2 Zbargain»;;=1iék andhasfewer terms. Itfollows thattwodiﬁerent bilinear forms A(x, Y)=Zarrimk. C(x» Y)=Zcikgink¢.1=1 i'.1=1 canreduce tothesame quadratic form after yisreplaced byx.Allthat is necessary isthata,»,,+ah,»=c,»,,+ck,»forarbitrary iandk. Thus, ingeneral, wecannot reconstruct uniquely thebilinear form generating agiven quadratic form. However, inthecase where itisknown that theoriginal bilinear form issymmetric, itcanbereconstructed. Infact, ifa,»,,=ah,then therelation a,»,,+ah,»=b,,,(i¢k)uniquely determines the coeﬂicients a,»,,,i.e., I an:an=Ebu: (I¢k)» (9) while fori=kwehave an=bu» (9’) sothat thebilinear form itself isuniquely determined. This assertion can beproved without recourse tobases andcomponents. Infact, wehave A(x+1.X+1)=A(x.x)+A(x.1)+A(y.x)+A(y.1) bythedeﬁnition ofabilinear form, and A(x.1)=§1A<x.1)+A<1.x)1=§1A<x+1,x+1)-A(x,x)-A<1,1)1 bytheassumption thatA(x, y)issymmetric. Hence thevalue ofthebilinear form A(x, y)foranypairofvectors x,yisuniquely determined bythevalues ofthecorresponding quadratic form forthevectors x,yandx+y. Ontheother hand, toobtain allpossible quadratic forms, weneed only usesymmetric bilinear forms. Infact, ifA(x, y)isanarbitrary bilinear form, then I A1(xa =E[A(x, + x)l isasymmetric bilinear form, and IA1(x, x)==E[A(x, x)+A(x, x)]=A(x, x), i.e.,thequadratic forms A1(x, x)andA(x, x)coincide. sac.7.3 REDUCTION orAQUADRATIC roam TOCANONICAL roam I85 7.23. These considerations show that inusing bilinear forms tostudy theproperties ofquadratic forms, weneed only consider symmetric bilinear forms, with corresponding symmetric matrices ||a,,,||, an,=ah.Bythe matrix ofthequadratic form A(x, x),wemean thesymmetric matrix A= ||a,,,|| ofthesymmetric bilinear form A(x, y)corresponding toA(x, x).When thebasis ischanged, thematrix Aofthequadratic form A(x, x)transforms justlikethematrix ofthecorresponding symmetric bilinear form A(x, y),i.e., Am=P’/Imp, where Pisthematrix ofthetransformation from thebasis {e}tothebasis {f}.Inparticular, therank ofthematrix ofaquadratic form does notdepend onthechoice ofabasis. Therefore wecantalk about therank ofaquadratic form A(x, x),understanding ittomean therank ofthematrix ofA(x, x)in anybasis ofthespace K".Aquadratic form whose rank equals thedimension nofthespace K,,issaid tobenonsingular. 7.3.Reduction ofaQuadratic Form toCanonical Form 7.31. Suppose wearegiven anarbitrary quadratic form A(x, x)deﬁned onann-dimensional linear space K".Wenow show that there exists abasis {f}={f1,ﬂ, ...,f,,} inK,,such thatgiven anyvector x=Zmf» k=1 thevalue ofthequadratic form A(x,x)isgiven by A(x.x)=mi+mi+'--+mi. (10) where 7.1,72,...,71,,arecertain ﬁxed numbers. Every basis with thisproperty willbecalled acanonical basis ofA(x, x),andtheexpression (I0) willbe called acanonical form ofA(x, x).Inparticular, thenumbers 7.1,72,...,71,, willbecalled canonical coeﬂicients ofA(x, x). Let{el,e2,...,en}beanarbitrary basis ofthespace K,,.If xZZikelpa l\"=1 then, aswehave already seen, A(x, x)canbewritten intheform A(x,x)_§kb..z.z..- (11) I86 BILINEAR AND QUADRATIC roams CHAP. 7 According toSec. 5.22, ourassertion willbeproved ifwecanwrite asystem W1=P11€1 ‘l‘P12€2 -I‘'''+p1n€ns W2=P21€1 ‘l‘P22€2 -I‘'''_i_p2'n€'n9 (12) nu =Pn1€1 ‘l‘Pn2€z +'''+pnnin with anonsingular matrix P= |lp,,,l| such that expressing thevariables hl,*q2,...,nuappearing in(ll) interms ofZ1,Z2,...,inhastheeﬁect oftransforming (ll) into theform (I0). Wewill carry outtheproof by induction onthenumber ofvariables Z,»actually appearing in(II),i.e.,those which have nonzero coeﬂicients, assuming thatevery form containing m——I variables Z1,Z2,...,Z,,,_1, say, canbereduced tothecanonical form (I0) with n=m~I,bymaking atransformation (I2)also with n=m—I. If(ll) actually contains only onevariable Z1,say, i.e., if(ll) hasthe form A(x» x):bniii then theinduction hypothesis issatisﬁed foranychoice ofpm¢0.Consider aform (ll) which actually contains mvariables Z1,Z2,...,Em.First we assume that oneofthenumbers bu,b22,...,bmm, saybmm, isnonzero, and wegroup together alltheterms in(II)which contain thevariable Em.This group ofterms canbewritten intheform blmglgm +b2m€2€m +i'.+bm—1,m€m—1€m +bmmzjn mm zbmm 1 zbmm 2 zbmm m—1 m 1a a where A1(x, x)denotes aquadratic form which depends only onthevariables Z1,Z2,...,Z,,,_1. Now consider thecoordinate transformation "71=£1» T2Z £2, Tm—1 €’m—1a _ blm bim bm—l,m TmMzbmm £1+zbmm £2+ +zbmm €m_1 +gm- Thematrix ofthistransformation isnonsingular (itsdeterminant isactually 1).Inthenewcoordinate system, A(x, x)clearly hastheform sac.7.3 REDUCTION orAQUADRATIC roam TOCANONICAL roam I87 where thequadratic fonn B(x, x)depends only onthevariables ‘:1,-r2,..., 'rm_1. Bytheinduction hypothesis, there exists anewtransformation ‘ﬁi=P1171 +P1272 ‘l‘'''‘l‘P1.m-1"7m-11 W2=P2111 -I"P2212 -l-‘‘‘-I"P2.m-1‘Fm-11 (127) ‘Wm-1 =Pm-1,171 +Pm-1.2172 +'''-l‘Pm-1,m-17m-1, with anonsingular matrix P=“Pall, which carries B(x, x)intothecanonical form B(x1 x)=7\m1 +Mn: ‘l‘'''‘l‘)‘m—1n€n—1" Ifwesupplement thesystem ofequations (l2') with theadditional equation *qm=-rm,weobtain anonsingular transformation ofthevariables -rl,-:2,,_,, -rmintothevariables nun2,...,nm,which carries A(x, x)intothecanonical fonn A(x, x) =B(x, x) +bmmlrzm =17\1'ﬁi‘l“ +.-'+)\m—1nfn—1 +bmmnfw According toSec. 5.33, thedirect transformation from thevariables {E}to thevariables {n}isaccomplished byusing thematrix equal totheproduct ofthematrix ofthetransformation from {-r}to{n}andthematrix ofthe transformation from {E}to{'r}.'l' Since both ofthese matrices arenonsingular, theproduct ofthematrices isalso nonsingular. Wemust stillconsider thecase ofaquadratic form A(x, x)inmvariables E1,E2,...,Emwhich hasallthenumbers bu,b22,...,bmm equal tozero. Consider oneoftheterms b,»,,E,~E,, with anonzero coeﬂicient, sayb12¢0. Then carry outthefollowing coordinate transformation, where forconve- nience wewrite thetransformation from thenewvariables totheoldvariables: E1=ii+El. E2=ii—Eé. £32iii’ E...=E2.- Thedeterminant ofthematrix ofthetransformation (I4)equals -2,andhence thistransformation isagain nonsingular. Theterm b12E1E2 istransformed into b12£1€2 =1712512 _17125112» sothat twosquared terms with nonzero coefﬁcients areproduced simultane- ously inthenew form. (Clearly these terms cannot cancel anyoftheother 1“{E}isshorthand fortheset{EuE2,...,Em}, {1}}fortheset(11,,11,,...,11",}, etc. I88 BILINEAR AND QUADRATIC roams CHAP. 7 terms, since alltheother terms contain avariable E;with i>2.)Wecan now apply ourinductive method tothequadratic form (ll) written inthe newvariables Eg. Thus, ﬁnally, wehave proved ourtheorem foranyinteger m=I,2,.... Inparticular, thecase m2nsuﬂices toprove thetheorem foranarbitrary quadratic form inann-dimensional space. The idea ofourproof, i.e.,consecutive splitting oﬁofcomplete squares, canbeused asapractical method forreducing agiven quadratic form to canonical form. However, inSec.7.5wewilldescribe another method, which permits ustoobtain directly both thecanonical form andthevectors ofthe canonical basis. 7.32. Example. Toreduce thequadratic form A(x» x)= -I"6€1£2 +5&2_4€1€a _lzieia +4&3_4€2£4 _sgaiq _ tocanonical form, weﬁrst complete thesquare inthegroup ofterms con- taining E1,writing W12&1 -I‘3&2 “2&3- Then theform istransformed into A(x.X)=ni~4€§-452E. ~35351 -Ei- Next wecomplete thesquare inthegroup ofterms containing E2,writing W2=2&2 +£4- This reduces theform to A(x, x)=Wi"Wii"8€3€4- There arenosquares ofthevariables E2andEA.Hence wewrite E3:W37W4. £4:W3-I‘W4, sothat E2EA 2ng~1r,§.Thus theform A(x, x)isreduced tothecanonical form A(x.X)=ni—ni—8n§+8ni bythetransformation W1=£1-I‘3&2 "2&3, W2=2&2 +£4» W322&3+2&4» W4="T253 -I‘2&4- Itisapparent from theconstruction that thistransformation isnonsingular, afactwhich iseasily veriﬁed directly. SEC.7.3 REDUCTION orAQUADRATIC roam TOCANONICAL roam I89 7.33. a.Neither thecanonical basis northecanonical form ofaquadratic form isuniquely determined. Forexample, anypermutation ofthevectors ofacanonical basis gives another canonical basis. InSec.7.5itwillbeshown, among other things, that with afewrare exceptions acanonical basis fora given quadratic form canbeconstructed bychoosing anarbitrary vector of thespace astheﬁrst vector ofthebasis. Moreover, ifA(x, x)iswritten in thecanonical form A(x’ x) 2KIWI +WZWZ +'''+)\nW3n where nun2,...,nmarethecomponents ofthevector x,then thetrans- formation W1=@1171, W2=°‘2"72. W11 1 aﬂlrﬂ (where 0&1,012,...,amareﬁxed numbers alldiﬁerent from zero and‘rl,:2, ...,-rmarenewcomponents) carries A(x, x)into thenewform A(x,x)=(1.1%?+().1€)1€+---+<).15)1;i, which isalsocanonical buthasdilferent coeﬂicients. Hence there stillremains theproblem ofdescribing allthecanonical forms towhich agiven quadratic form canbereduced. This problem canbemade more precise ifwerestrict thedeﬁnition ofacanonical form (asforexample willbedone inSec. 7.93 forthecase ofarealspace) orifwerestrict theclass ofadmissible coordinate transfonnations (asforexample willbedone inSec. 10.1forthecase ofa Euclidean space). b.Itshould benoted that inthepreceding example thenumber ofnon- zero coeﬂicients remains unchanged when wetransform from thevariables {n}tothevariables {-r}.Ingeneral, thenumber ofnonzero canonical coeﬂicients isobviously therank ofthematrix ofthequadratic form inthe corresponding canonical basis. Since therank ofthematrix ofaquadratic form does notdepend onthechoice ofabasis (Sec. 7.23), thenumber of nonzero canonical coeﬂicients ofaquadratic form does notdepend onthechoice ofacanonical basis. Moreover, this number obviously coincides with the rank ofthequadratic form (Sec. 7.23). Thus from aknowledge ofaquadratic form A(x, x)inanybasis {e},wecanpredict thenumber ofnonzero canonical coeﬂicients ofA(x, x)inanycanonical basis, namely therank ofA(x, x). Inparticular, thecanonical coeﬁicients ofanonsingular quadratic form are allnonzero inanycanonical basis. I90 BILINEAR AND QUADRATIC roams CHAP. 7 7.4.TheCanonical Basis ofaBilinear Form 7.41. a.Thevector x1issaidtobeconjugate tothevector ylwithrespect tothebilinear form A(x, y)if A(x1> Y1)=0- Inthiscase, ylisalsosaid tobeconjugate tox1. b.Let||a,,,|l bethematrix oftheform A(x, y)inanybasis el,e2,...,en. Then, if x1=2E.-er. Y1=ZWi=e;.-, i=1 k=1 thecondition forx1andy,tobeconjugate canbewritten intheform Tl A(x1» Y1)=Zai»€iWrt =0- i.lt‘=1 c.Ifthevectors xl,x2,...,x,,areallconjugate tothevector yl,then every vector ofthelinear manifold L(x1, x2,...,x,,)spanned byxl,x2,..., x,,isalsoconjugate toyl.Infact, itfollows from theproperties ofabilinear form that A(°‘1-x1 +052-x2 +'''‘I’95;;-xiv Y1) =°‘1A(x1» Y1)-I"°i2A(x2» Y1)'l‘'''‘l‘°%A(x1t» Y1)=0- Avector ylconjugate toevery vector ofasubspace K’CKissaid tobe conjugate tothesubspace K’. d.ThesetK”ofallvectors yl6Kconjugate tothesubspace K’isobviously asubspace ofthespace K.This subspace K”issaid tobeconjugate toK’. 7.42. Abasis el,e2,...,emofthen-dimensional space K,,iscalled a canonical basis ofthebilinear form A(x, y)ifthebasis vectors areconjugate toeach other, i.e.,if A(e,, e,,):0for i¢k. For example, inthespace V2letthebilinear form A(x, y)bethescalar product ofthevectors xandy.Then tosaythatxandyareconjugate with respect toA(x, y)means that xandyareorthogonal. Inthis case, any orthogonal basis ofthespace V2isacanonical basis. 7.43. The matrix ofabilinear form relative toacanonical basis is diagonal, since a,-2.=A(e,~, e2.):0for i¢k. sac. 7.4 TI-IEcANoN1cA1. BASIS orABILINEAR roam I9I Since adiagonal matrix coincides with itsown transpose, abilinear form which hasacanonical basis must besymmetric. (We recall from Sec. 7.14 that whether ornotthematrix ofabilinear form issymmetric does not depend onthechoice ofabasis.) Conversely, wenow prove that every symmetric bilinear form A(x, y)hasacanonical basis. Tosecthis, consider thequadratic form A(x, x)corrmponding tothegiven bilinear form A(x, y). Weknow that there exists abasis el,e2,...,eminthespace Kminwhich A(x, x)canbewritten inthecanonical form A(x,x)=1.-at Itfollows from formulas (9)and(9'),p.I84thatthecorresponding symmetric bilinear form A(x, y)takes thecanonical form ,_l\/l=Z’gr53’ /57-‘5.’) A(x.1)=S inthisbasis, where Tl Y=;1Wi‘ei‘. andhence itsmatrix isdiagonal. Butthisjust means thatthebasis el,e2,..., emiscanonical fortheform A(x, y),andourassertion isproved. 7.44. Inanalytic geometry itisshown that thelocus ofthemidpoints of thechords ofasecond-degree curve which areparallel toagiven vector isa straight line. Wenow prove this theorem. Asecond-degree curve inthe x1x2-plane hasanequation oftheform 2 2a11x1 -I"2a12x1x2 +a22x2 +b1x1 +b2x2 -I"4'=0 or A(x, x)+L(x) +c=0, where A(x» x)=anxi +2a12x1x2 +(122752 isaquadratic form and L(x) =blxl +b2x2 isalinear form inthevector x=(xux2).Letxbethevector giving the position ofthemidpoint ofachord parallel toaﬁxed vector e.This means that theequations A(x-I-te,x+te)-I-L(x+ te)+ c=0, (16) A(x —te,x——te)-I-L(x——te)+ c=0 I92 BILINEAR ANDQUADRATIC roams CHAP. 7 aresatisﬁed forsome t¢0.LetA(x, y)bethesymmetric bilinear form corresponding tothequadratic form A(x, x).Then wecanwrite (I6)as A(x, x)+2tA(x, e)+t2A(e, e)+L(x) +tL(e) +c=0, A(x, x)—2tA(x, e)+t2A(e, e)+L(x) -tL(e) +c=0. Subtracting thesecond equation from theﬁrstanddividing by2t,weget 2A(x, e)+L(e) =0. (I7) This equation islinear inxandhence determines astraight lineinthex1x2- plane, thereby proving thetheorem. Letx’beanother point ofthesame line, sothat 2A(x’, e)+L(e) =0. (I8) Then subtracting (I8)from (I7), weget A(x -x’,e)=0, i.e., thevector eand thevector x—x’determining thedirection ofthe straight line inquestion areconjugate with respect tothebilinear form A(x, y),inthesense ofSec. 7.41. 7.45. Letel,...,embeacanonical basis oftheform A(x, y)inak- dimensional subspace K’CK,and let1-:1,...,1-:,,bethecorresponding canonical coeﬂicients. Expressing thenumbers A(x,e,.) interms ofthe components ofavector xeK’,weget 2. I: A(x» er)=A(Z1€;@;» er)=21E.;A(@,', er)=€iA(eia er)=Sign 1 I sothat thenumbers A(x, e,)areuniquely determined bythecomponents of thevector x.Iftheform A(x, y)isnonsingular inthesubspace K’,then the numbers c,»areallnonzero. Inthiscase, theconverse isalso true, i.e.,the values A(x, e,»)oftheform A(x, y)uniquely determine thecomponents of thevector x. 7.5.Construction ofaCanonical Basis byJacobi’s Method 7.51. The construction ofacanonical basis given inSec. 7.31 hasthe drawback that thecomponents ofthevectors ofacanonical basis andthe corresponding canonical coeﬂicients 7.,cannot bedetermined directly from a knowledge oftheelements ofthematrix Amofthesymmetric bilinear form A(x, y)inagiven basis {f}={f1,ﬂ, ...,f,,}. .lacobi’s method, which will now bepresented, dom allow ustodojust this. However, wemust now impose thefollowing supplementary condition onthematrix Am: The SEC.7.5 CON$TRUCTlON orACANONICAL aAs1s avJAcoal’s METHOD I93 descending principal minors ofAmoforder uptoandincluding n-I,i.e., theprincipal minors oftheform a11 a12 81=a11» 82= 1 1 (121 a22 a11 a12 'a1.»-1 (121 a22 'a 871-1 :__ _ 2.n—1 ’ an—1.1 an—1.2 ''' an—1.n—1 must allbenonvanishing. 7.52. Thevectors el,e2,...,emareconstructed bytheformulas e1=f1» ) -. 92=°‘i1f1 +12» e3=°1i2)f1 +@2272 ‘l‘,/3» ................. @;,+1 =<7-iki/'1 +12"-ifa -I"flgkifs +'''-I"°!j;’"f;, "l‘f1t-+1, en Za1"_”.f1 +7';’L_1).f2 +ai1n_1).f3 +---+ailiqnfn-—1 +.fna where thecoefﬁcients oil?"(i=I,2,...,k;k=1, 2,...,n-1) arestill tobedetermined. First ofall,wenote that thetransformation from the vectors fuﬂ, ...,f,,tothevectors el,e2,...,ekisaccomplished byusing thematrix 10 0 00 11;" 1 0 00 9 apt-1) mgr-1) aé1¢—1) 041:1) 1 whosedeterminant isunity. Hence fork =I,2,...,nthe vectorsfuﬂ, ..., f,.canbeexpressed aslinear combinations ofe,,e2,...,e,,,sothatthelinear manifold L(f,,ﬂ, ...,f,,)coincides with thelinear manifold L(e1, e2,...,eh). Wenow subject thecoeﬂicients oz?"(i=I,2,...,k)tothecondition that thevector emu beconjugate tothesubspace L(e,, e2,...,em).A necessary andsufﬁcient condition forthisisthat therelations A(e2.+1,f,) =0,A(e,,+,,ﬂ) =0,...,A(e,,+,,f,,) =0 (21) 194 BILINEAR AND QUADRATIC FORMS CHAP. 7 besatisﬁed. Infact, itfollows from (21)that thevector e,,+1isconjugate to thelinear manifold spanned bythevectorsfuﬂ, ...,f,,,which, aswehave just proved, coincides with thelinear manifold spanned bythevectors e,,e2,...,eh.Conversely, ifthevector e,,+1 isconjugate tothesubspace L(e1, e2,...,em),itisconjugate toevery vector inthesubspace, inparticular, tothevectorsfuﬂ, ...,f,,,sothattheconditions (21)aresatisﬁed. Substituting theexpression (20)fore,,+1into(21)andusing thedeﬁnition ofabilinear form, weobtain thefollowing system ofequations inthe quantities at?"(i=I,2,...,k): A(@1=+1»f1) =°°ik)A(f1.f1) "I"°<2’”A(f2»f1) "'1''''"I"<7-i=’”A(f1=»f1) "I"A(f1=+1»f1) =0» A(ert+1»f2) =°‘ik)A(f1»f2) +°i2k)A(f2»f2) +'"‘l‘°‘irk)A(f1t»ﬁ) ‘l‘A(f1i+1»/2) =01 A(e1t+1» fit)=°‘ik)A(f1.f») ‘l‘°‘2k)A(f2»f1=)‘l‘ '"‘l‘°‘itMA(f1r1f1t) ‘l‘A(fk-1-1.f1t) =0- (22) Byhypothesis, thisnonhomogeneous system ofequations with coefﬁcients A(f,~,f,»)=a,-,- (i,j=l,2,...,k) hasanonvanishing determinant, andhence canbesolved uniquely. There- fore wecandetermine thequantities oz?”andthereby construct thedesired vector emu. Todetermine allthecoeﬂicients oz?”andallthevectors e,,,we must solve theappropriate system (22) forevery k.Thus, inall,wemust solve n—Isystems oflinear equations. LetE1,E2,...,Emdenote thecomponents ofthevector xand n1,n2, ...,nmthecomponents ofthevector ywith respect tothebasis el,e2,..., emjustconstructed. Then thebilinear form A(x, y)becomes A(x,1)=1.-2.-1. <23) inthisbasis. 7.53. Tocalculate thecoefﬁcients 71,,weargue asfollows: Consider the bilinear form A(x, y)only inthesubspace Lm=L(e1, e2,...,em)where m<n.Theform A(x, y)clearly hasthematrix "11 a12 am (121 a22 @2111 aml am2 ''' amm SEC. 7.5 CONSTRUCTION orAcANoN1cA1. BASIS BYJAcoB1’s METHOD I95 inthebasisfuﬂ, ...,fmofthesubspace Lmandthematrix A1Q...0 0A2...0 00 Am inthebasis el,e2,...,em.Aswehaveseen, thematrix ofthetransformation (20)from thebasisf1,j2, ...,fmtothebasis el,e2,...,emhasdeterminant 1.Hence byequation (6),p.182wemust have an a12 '''am 7\1 0 '‘‘0 (121 a22 ''112m 0 7\2 '''0 det =det , am, am2 --amm 00 71m or,inthenotation (19), 8m=7.17\2---71m (m=1,2,...,n) (8,,=detAm). Itfollows immediately that s s 2:, 7\1=81=a111 72:?’ 7\3=§3, ---» 7\n=§—- (24) 1 2 11-1 Using (24), wecanﬁnd thecoefﬁcients ofthebilinear form A(x, y)ina canonical basis without calculating thebasis itself. 7.54. Consider once again thekthequation inthesystem (20), which we write intheform fk+1 ="°‘ik)f1 _'''"°‘i=k)f» *1‘e1i+1 =31.-1"er.-+1» where gmliesinthesubspace L(fu ...,fm) and emu isconjugate tothis subspace. The coeﬂicients 0111'", ...,affi areuniquely determined bythe system (22)subject tothecondition thatdet||A(f,,f,)l| ¢0or,equivalently, thattheform A(x, y)benonsingular inthesubspace L(f1, ...,fm). Since the vector f,,+1 isarbitrary inthisconstruction, then, writing f=ﬁt+1> g:git’ h:ek+1’ L(.f1’ '''afk) =K’CK» wearrive atthefollowing THEOREM. Suppose thebilinear form A(x, y)1'snonsingular inasubspace K’CK,andsuppose thevector fdoes notbelong toK’.Then there exists a I96 BILINEAR AND QUADRATIC FORMS CHAP. 7 unique expansion f=g+'1» (25) where geK’andhisconjugate tothespace K’. 7.55. LetK”denote thesubspace conjugate tothesubspace K’with respect totheform A(x, y).Then theexistence anduniqueness oftheexpan- sion (25)shows that thewhole space Kisthedirect sum ofthesubspaces K’ andK”(see Sec. 2.45). Thus, given asubspace K’CKinwhich abilinear form A(x, y)deﬁned onthewhole space Kisnonsingular, Kcanbewritten asthedirect sum K=K’+K”, where K”isconjugate toK’with respect totheform A(x, y). 7.6. Adjoint Linear Operators 7.61. Let(x,y)denote aﬁxed nonsingular symmetric bilinear form in thespace KmLetAandBbelinear operators acting inKn,and usethe formulasA<x.1)=<Ax.1), Bo.1)=(x.B1) todeﬁne functions A(x, y)andB(x, y)oftwo vector arguments xandy. Then A(x,y) and B(x,y) arebilinear forms. Infact, itfollows from the deﬁnition ofalinear operator (Sec. 4.21) and thedeﬁnition ofabilinear form (Sec. 7.11) that A(x1 -1"x2»Y) =(A(X1 -1"X2)» Y)=(A-Y1 -1"A752» Y) =(Axvy) +(Axz.1)=A(x1.1)+A(x.»1). Aw.1)=(A(¢x). 1)=WAX.1)=¢(Ax.1)=2A(x,1). which shows that A(x, y)islinear initsﬁrst argument. Similarly, the linearity ofA(x, y)initssecond argument isaconsequence ofthelinearity of (x,y)iny.Then A(x, y)isabilinear form, andsimilarly soisB(x, y). Next lete,,...,embeacanonical basis oftheform (x,y), sothat (@1132) =0ifjik. (em, em)=am6K, am¢0. Wenow compare thematrix oftheoperator Awith that oftheform A(x, y) inthisbasis. The matrix |la}f’|| oftheoperator Aisdeﬁned bytheformula A6.=Za§.”@. (1=1,..A.11), k=1 sec.7.6 ADJOINT LINEAR OPERATORS I97 where here (incontradistinction tothenotation adopted inSec. 4.23) the superscript indicates therownumber andthesubscript thecolumn number. Thematrix ||a,,,|| oftheform A(x, y),where theﬁrstsubscript indicates the row number andthesecond thecolumn number, isdeﬁned bytheformula a,-m=A(e,-, em)=(Ae,-, em)=<g1a§,”e,,, em) =a(,§,’(em, em)=1-zmaij’. (26) Hence themthcolumn oithematrix ||a,m|l isobtained (forevery m=1,..., n)bymultiplying themth column ofthematrix |la§j,"|| bythecanonical coefﬁcient 1-:moftheform (x,y).Similarly, forthematrix ||b§,”||oftheoperator B(inthesame basis el,...,em)andthematrix I15,-1,11 Oftheform B(x, y), weget b,-m=B(e,, em)=(ej,Bem) =(em21b§,"”e,,) =bl-””(e,-, e,)=1-:,b(/’”, (27)2: i.e.,thejth rowofthematrix ||b,m|| isobtained (foreveryj =1,...,n)by multiplying thejthcolumn ofthematrix oftheoperator Bbythecorrespond- ingcanonical coeﬂicient 1-:,. 7.62. Conversely, given two bilinear forms A(x, y)and B(x, y)inthe space Km,weassert thatthere exist unique linear operators AandBsuch that A(x,y) =(ALY). B(x»Y)=(X.By) (23) Toshow this, wespecify AandBinthesame basis el,...,embythematrices with elements ,,,1 m1. ah)=—A(e:i> em)» ):_Bieiv em)’ Em El respectively. Wethen usethese operators toconstruct theforms A1(x, y)= (Ax, y)and B1(x, y):(x,By). Itfollows from Sec. 7.61 that thematrix oftheform A1(x, y)coincides with thematrix oftheform A(x, y)inthe basis e,,...,e,,, while thematrix oftheform B,(x,y) coincides with the matrix oftheform B(x, y).Butthen (Any) =A1(x.1)=A(x.1). (X.By)=B1(x.y) =B(x,y) forarbitrary x,yeK,,(recall Sec.7.13), sothattheoperators AandBsatisfy (28). Toprove theuniqueness, weneed only verify that ifanoperator A satisﬁes thecondition (Ax, y)20forarbitrary x,yeKm (29) then AxI0forevery xeK,,, sothat AistheZero operator. Suppose Axoqé0forsome xoeK,,. Then, since thefonn (x,y)isnonsingular, it follows from Sec.7.15cthatthere isavector yoeKnsuch that (Axo, yo)¢0. I98 BILINEAR AND QUADRATIC FORMS CHAP. 7 This contradicts (29) and establishes therequired uniqueness ofA.The uniqueness ofBisproved similarly. 7.63. Wenow prove thefollowing important THEOREM. Let(x,y)beanonsingular symmetric bilinear form inthespace Km Then, given any linear operator Aacting inKn,there exists aunique linear operator A’acting inK,,such that (AX.1)=(x.A’y) forarbitrary x,yeKmThematrix oftheoperator A’inanycanonical basis oftheform (x,y)isobtained from thematrix ofAbytransposition, followed bymultiplication ofthemthrowbythecanonical coeﬂicient 1-:manddivision ofthejth column bythecanonical coeﬂicient i-:,-(j,m=1,...,n). Proof. Weusethegiven operator Atoconstruct theform A(x, y)= (Ax, y),andthen wedeﬁne theoperator A’bytheformula (Ax.Y)EA(x.Y)=(X»AU’)- Theexistence anduniqueness ofA’follow from Sec. 7.62. Inanycanonical basis oftheform (x,y),thematrix ||a§j’|| oftheoperator A,thematrix Ila,-m|| oftheform A(x, y)and thematrix ||a,Tl""|| oftheoperator A’are related byformulas (26) and(27): . a. a.ax‘) Z Jm’ aglm) Z am_ am 51' Itfollows that a. 5 . a;-(ml =i‘=1am’. | (30) 2, 2,- The operator A’iscalled theadjoint (orconjugate) oftheoperator A with respect totheform (x,y). 7.64. The operation leading from anoperator Atoitsadjoint A’has thefollowing properties: 1)(A’)’ =Aforevery operator A; 2)(A-1-B)’=A’-1-B’forevery pair ofoperators AandB; 3)(7A)’ =71A’forevery operator Aandevery number 7eK; 4)(AB)’ =B’A’ forevery pair ofoperators AandB. Toprove property 1),weusetheformula (xv (A’)’Y) :(A’x>Y) =(xvAY) implied bythedeﬁnition of(A’)’, together with theuniqueness oftheoperator SEC, 7.7 ISOMORPHISM OF SPACES EQUIPPED WITI-I ABILINEAR FORM deﬁned byabilinear form (Sec. 7.62). Theremaining propertim areproved similarly. Thus (X.(A+3)’)/)=((A+B)x»)/) =(AL)/) +(316,)/) =(x.A'1)+(x.B'1)=(x.(A'+B')1) implies property 2). (X.(7A)’y) =(Mr.1)=MAX»1)=Mr.A’y)=(X.W1) implies property 3),and (X.(AB)’y) =(ABX.1)=(Bx.A’y)=(X.B’A’y) implies property 4). 7.65. Wepoint outanother connection between theoperators AandA’. Suppose thesubspace K’CKmisinvariant under theoperator A.According toSec. 4.81, thismeans that theoperator Acarries every vector x6K’into another vector ofthesame subspace K’.LetK”bethesubspace conjugate toK’(Sec. 7.55). Then K”isinvariant under theadjoint operator A’.In fact, suppose yeK”,sothat (y,x)=0forevery xeK’.Then (A’y, x)= (y,Ax)=0,since xeK’implies Ax6K’.Butthismeans that thevector A’yisconjugate toevery vector xeK’andhence belongs toK”,asrequired. 7.7.Isomorphism ofSpaces Equipped with aBilinear Form 7.71. Deﬁnition. LetK’and K”betwo linear spaces over the same number ﬁeld K.Suppose K’isequipped with anonsingular symmetric bilinear form A(x’,y’), while K”isequipped with anonsingular symmetric bilinear form A(x”, y"). Then K’andK”aresaid tobeA-isomorphic if 1)They areisomorphic regarded aslinear spaces over theﬁeld K(see Sec. 2.71), i.e., there exists aone-to-one mapping (morphism) cox’=x" preserving linear operations; 2)The values oftheforms A(x’, y’)and A(x”, y”)coincide forall corresponding pairs ofelements x’,y’andx”=wx’, y”=wy’, i.e., A(x‘.1’)=A(x”.1")- 7.72. T1-lE0REM. Given twoﬁnite-dimensional linear spaces K’and K”, suppose K’isequipped with anonsingular symmetric bilinear form A(x’, y’), while K”isequipped with anonsingular symmetric bilinear form A(x”, y"). Then K’andK”areA-isomorphic andonly if a)They have thesame dimension n; b)There exists acanonical basis forA(x’, y’)inK’andacanonical basis forA(x", y”)inK”relative towhich thetwoforms have thesame setofcanonical coeﬂicients 1-:1,...,em 200 BILINEAR AND QUADRATIC FORMS CHAP. 7 Proof. Suppose K’andK”areA-isomorphic. Then they areisomorphic aslinear spaces andhence have thesame dimension, sayn(seeSec.2.73d). Ife1,...,ei,isacanonical basis fortheform A(x’, y’)inthespace K’,then II 0ifi2j, A(@='» er)2 ...2,»1ft2j. Lete’{,...,e',’,bethevectors inK”corresponding tothevectors e1,...,e,’, inK’under thegiven A-isomorphism. Byhypothmis, I I 'II II 0 iiii,A(o)e,, o)e,.) 2A(e,», e,)2 ___ 2,1f12j. Thus ea’,...,ef,isacanonical basis forA(x”, y")inthespace K”.Moreover, A(x”, y")hasthesame canonical coefﬁcients 1-:1,...,1-:,,inthebasis e’{,..., exasA(x’, y’)hasinthebasis e1,...,e;,. Conversely, suppose K’and K”have thesame dimension n,and let e1,...,ej,eK’and el’,...,exeK”becanonical bases with thesame canonical coefﬁcients 1-:1,...,a,,,sothat I I II II 0 i¢ j’A(er, er)=A(@r» er)= .. .2,»1f121. Given anyvector Tl x’=ZEx!»_ i=1 1nK’,let 71 x”2o)(x’) 22E,~e§’ i=1 (with thesame components E1,...,E,,)bethecorresponding vector inK”. This correspondence deﬁnes anisomorphism o)ofthespaces K’andK”(see Sec.2.73d). Moreover, if 1’=mi. 1"=w(y’)=me?-’. then H A(x’, Y’)=Z13i§iWr =A(x”) Y”), sothat theisomorphism o)isanA-isomorphism. I 7.73. Given ann-dimensional space K,,equipped with anonsingular symmetric bilinear form A(x, y),consider anA-isomorphism ofK,,,i.e.,an invertible linear mapping y2Qxwhich does notchange theform A(x,y) inthesense that A(Qx, Qy) 2A(x,y). (31) sec.7.7 ISOMORPHISM orSPACES EQUIPPED WITH ABILINEAR FORM 201 Wewillhenceforth denote A(x, y)simply by(x,y).IfQ’istheadjoint of theoperator Qwith respect totheform (x,y),then (Qx.Qy)=(Q'Qx, )1) (32) Itfollows from (31) and(32)that Q’Q2E, (33) andhence that Q’istheinverse oftheoperator Q(since Qisnonsingular, soisQ’). Conversely, (33) implies (32) andthen (31), sothat thecondition (33) completely speciﬁes theclass ofoperators which donotchange theform (x,y).Thme operators aresaidtobeinvariant withrespect totheform (x,y). 7.74. IfQisinvariant, then soistheinverse operator Q_1 2Q’,since (Q’x.Q'y)=(QQ’x. 1)=(X.1) forevery xandy.Theproduct oftwoinvariant operators QandTisalso aninvariant operator, since (QTX1 QTY) Z(Txi TY) 2(xiY) forevery xandy. 7.75. Letel,...,e,,beacanonical basis oftheform (x,y),with canonical coefﬁcients 1-:1,...,emThen, applying aninvariant operator Qtothevectors el,...,e,,,wegetthevectors fl2Qe1,.. .,f,,2Qe,,, (34) where.f.2k, (fiifk) Z(Qei> Qek) =(911eh)=[Si ifiikl Thus fl,...,f,,isalso acanonical basis oftheform (x,y),with thesame canonical coefﬁcients :1,...,1-:,,. Conversely, iffl, ...,fmisacanonical basis oftheform (x,y)with the same canonical coeﬂicients 1-:1,...,1-:,,asthebasis el,...,e,,, then the operator Qdeﬁned by(34)isinvariant. Infact, 2,»1fj2k, e- e2 -, 2e-,e 2 (Q..Q1.) (/1f») (11.)10ifjik, andhence (31)holds foranypairofbasis vectors. Butthen. bythelinearity, (31) holds forarbitrary vectors x,yeK,,,asrequired. Thus aninvariant operator Qischaracterized bythefactthat itcarries every canonical basis ofthespace K,,(with respect totheform (x,y))into another canonical basis with thesame canonical coeﬂicients. 202 BILINEAR AND QUADRATIC FORMS CHAP. 7 7.76. Wenow ﬁnd conditions characterizing thematrix ofaninvariant operator Qinacanonical basis oftheform (x,y).Letel,...,e,,besuch abasis, and lets1,...,anbethecorresponding canonical coeﬂicients. Moreover, letQ=||q§"’|l bethematrix ofQinthebasis el,...,en.Then, according toSec.7.63, thematrix oftheadjoint operator hastheform I I I E IQ=Hql-U)“, Q5"=—’1121 51' Interms ofmatrix elements, wecanwrite equation (33) as " i_ W,__ 1ifj=k,zqlqtnqin =2E:_qg_1>q’(c1> __=8210 ={ 'I i=1 i=12. 0If,ask. Inother words, l.. 7! 1 Z21q2~q:.~=‘f’k’ 65> 0ifjatk. Equation (35)isequivalent to(33), andcanalsoserve asthedeﬁnition ofan invariant operator Q. Thus aninvariant matrix, i.e.,thematrix ofaninvariant operator inany canonical basis oftheform (x,y),ischaracterized bythefactthat thesum ofthesquares oftheelements ofitsjthcolumn taken with coefﬁcients 2;‘,...,2;‘equals thenumber sf(j=l,...,n), while thesum ofthe products ofthecorresponding elements oftwodiﬁerent columns also taken with the coefﬁcients sf,...,2;‘equals zero. Since (33) also implies QQ' ==E,wealso have therelations-.>-NF) W W (' ( Ek (' (Y (' Zqknqinki =2 _q1¢”q1=m =813)’i=1 ;,=1sm O1‘ Zixqki qr; " __ (35)k=1 if1¢m. This gives another characterization ofaninvariant matrix, namely thesum ofthesquares oftheelements ofitsjthrowtaken with coefﬁcients 21,...,an equals thenumber :-:,-(j=1,...,n),while thesum oftheproducts ofthe corresponding elements oftwodiﬁerent rows alsotaken with thecoeﬂicients 21,...,anequals zero.n‘ (H ("H _-{E14 lf j2m, I 0 *7.8. Multilinear Forms 7.81. Byanalogy with bilinear forms wecanconsider linear functions of alarger number ofvectors (three, four ormore). Allsuch functions are called multilinearforms. SEC7-8 MULTILINEAR roams 203 Deﬁnition. Afunction A(x1, ...,x,,)ofkvector arguments xl,...,x,, varying inalinear space Kiscalled amultilinear (more exactly, ak-linear) form ifitislinear ineach argument x,-(j=l,...,k)forﬁxed values ofthe remaining arguments xl,...,x,-_1, x,-+1, ...,x,,.Amultilinear form A(x1, ...,xk)iscalled symmetric ifitdoes notchange when anytwoofits arguments areinterchanged, andantisymmetric ifitchanges sign when any twoofitsarguments areinterchanged. Anexample ofanantisymmetric ‘multilinear form inthree vectors x,y andz(atrilinear form) ofthespace V3isthemixed triple product ofx,yand z.’[Anexample ofanantisymmetric multilinear form innvectors x1=(all? a12» ~~~aam), x2=(a21v 1122, ---»112"), xn=(am, anz,...,am) ofann-dimensional linear space K,,I isthedeterminant an a12 '''am A<x..x..---.x.>= “*1“*2 "2" (36) anl an2 ''i arm Asomewhat more general example istheproduct ofthedeterminant (36) with aﬁxed number AeK. 7.82. Wenow show that every antisymmetric multilinear form A(x1, x2,...,x,,) innvectors xl,x2,...,xnofann-dimensional linear space K"with aﬁxed basis el,e2,...,enequals thedeterminant (36) multiplied bysome constant AeK. LetAdenote thequantity A(e1, e2,...,e,,).Then wecaneasily calculate thequantity A(e,l, e,-2,...,ein)where i1,1'2,...,inarearbitrary integers from lton.Iftwo ofthese numbers areequal, then A(e,l, eiz,...,ein) vanishes, since ontheone hand itdoes notchange when thearguments corresponding tothese numbers areinterchanged, while ontheother hand it must change sign because oftheantisymmetry property. Ifallthenumbers 1'1,1'2,...,inarediﬁerent, then bymaking thesame number ofinterchanges TI.e.,(x,y><z)where (,)denotes thesmlar product and ><thevector product. :13)’ X1:(all! am»: --1am) we mean X:duel +‘H292 +‘''+ainem where er,eh ...,e,,isaﬁxed basis ll'lK,,,andsoon. 204 BILINEAR AND QUADRATIC roams C]-[AP_ 7 ofadjacent arguments asthere areinversions inthesequence ofindices i1,i2,...,i,,,wecancause thearguments tobearranged innormal orderT; lettherequired number ofinterchanges beN.Then wehave A(e,-1, eiz,...,e,~n)=(——1)N)\. Now let W x,-=Za,»,»e,» (i=1,2,...,n) j=1 beanarbitrary system ofnvectors ofthespace K,,, and consider the multilinear form Tl 7! 7| A(x1, x2,...,xn)=A<2amen, Za2,»2e,-2, ...,Za,,,~'_e,»,_) 111 ,, -= t,=1 1=1 7| = 2 a1i'1a2i'g '''am',,A(ei,, eip---,ea“)t,.t,.....i,,=1 Tl =7\ 2 ("1)Na1t,a2¢, '''am,- i1.iz.....1',,=1 Since ineach term ofthelastsum, Ndenotes thenumber ofinversions inthe arrangement ofthesecond subscripts oftheelements a,-,-when theﬁrst subscripts areinnormal order, itfollows that each term isoneoftheterms inthedeterminant (36)with theappropriate sign. Hence thesum ofallthe terms isjustthedeterminant (36), andourassertion isproved. Inparticular, thisshows that themixed triple product ofthree vectors x,yand2ofthespace V3inanybasis canbewritten asthethird~order determinant made upofthecomponents ofx, yand2,taken with acoeﬂicient equal tothetriple product ofthebasis vectors. 7.9.Bilinear andQuadratic Forms inaRealSpace 7.91. Every real number hasadeﬁnite sign (+or~—), and hence the theory ofbilinear andquadratic forms inarealspace canbecarried some- what further than inaspace over anarbitrary ﬁeld K.According tothe general theory ofSec.7.3l,aquadratic form A(x, x)canbereduced insome basis tothecanonical form A(x.x)=mi+mi+---+mi. where thenumber ofnonzero coefﬁcients )\,,A2,...,kn,equal totherank oftheform A(x, x)(Sec. 7.33b), does notchange when thecanonical basis ischanged. These coeﬂ‘icients areeither positive ornegative. Itturns out TCf.theproof ofTheorem 4.54. sec.7.9 BILINEAR AND QUADRATIC roams 1NAREAL SPACE 205 that changing thecanonical basis also hasnoeﬁect onthetotal number of positive coeﬂicients andthetotal number ofnegative coeﬂicients: THEOREM (Law ofinertia for quadratic forms). Ifaquadratic form A(x, x)inarealspace iswritten incanonical form, thetotal number ofpositive coeﬂicients andthetotal number ofnegative coeﬂicients areinvariants ofthe form, i.e.,donotdepend onthechoice ofthecanonical basis. Proof. Suppose A(x, x)hastheform A(x» x):2aikiiik i.lc=1 inthebasis {e}={el,e2,...,en},where Z1,Z2,...,E"arethecomponents ofthevector xwith respect to{e}.Suppose A(x, x)hastwocanonical bases {f}=-{f1»fz» ---»f..}=1nd{g} ={g1.g2, ---,g..}-I-HW1»nz.---.n..d¢n<>t¢ thecomponents ofxwith respect tothebasis {f}, and let‘rl,-:2,...,1,, denote thecomponents ofxwith respect tothebasis {g}. Letthecorre- sponding transformation formulas be W1=bnii +buzz -l"'''-l"burg", W2=b21€1 +bzziz +'''+1721.5», ﬁn=bmii -l"bnziz -l"'''+bnnzn(37) and "71=cnii +@1252 +'''+(min, T2=c21€1 +Fzziz -l"'''+czﬂgﬂ’ Tn =cnlgl +cn2€2 +'ii+cnngrn(37') where thematrices ||b,,,|| andIlcikll arenonsingular. Inthebasis {f}, A(x, x) hastheform A(x» x):“17li -l"'''-l"ﬁlm: “‘at-+1'ﬁi¢+1 “''''“‘“m7lht> (38) while inthebasis {g}ithastheform A(x» X)=B17: +'''+B12121 '_Ba1+1TfH-1 'T'''—B075» where thenumbers otl,...,am,B1,...,B,areassumed tobepositive. Wewish toshow that k==p,m=q.Equating theright-hand sides of(38) and(39), andtransposing negative terms toopposite sides oftheequation, weobtain 2“mi +'''+1mi+ Bp+1"7f»+1 +'''+Bf‘: =°‘x+1W:+1 +'''+amnin +Bfri +'''+Bp"7iw- (40) 206 BILINEAR AND QUADRATIC roams CHAP. 7 Now suppose k<p,andconsider thevectors xwhich satisfy theconditions =0, =,...,.=, W1 W2 0 W1. 0 0 (41) ‘r,,+1=0,...,-ra~:0,‘ra+1=O,...,-rn==. There areclearly lessthan nofthese conditions, since k<p. Using (37) and (37') toexpress nl,. ..,nk,1%,, ...,1,,interms ofthevariables £1,Z2,...,Zn,weobtain ahomogeneous system oflinear equations inthe unknowns Z1,Z2,...,Zn.Thenumber ofequations islessthan thenumber ofunknowns, and therefore this homogeneous system has anontrivial solution x=(Z1,Z2,...,Zn). Ontheother hand, because of(40), every vector xsatisfying theconditions (4l) alsosatisﬁes theconditions ‘r1=‘r2='~'=‘r,,=O. However, since detl|c,,,|l ¢0,anyvector xforwhich 11:12=-~-=19:-|;'p+1=-~-z-|;'n=() must bethezero vector, with allitscomponents Z,,Z2,...,Enequal to zero. Thus theassumption that k<pleads toacontradiction. Because of thecomplete symmetry oftherole played bythenumbers kandpinthis problem, theassertion p<kalso leads toacontradiction. Itfollows that k=p.Moreover, examining theconditions -r,=0,'r2=O,...,-r,,=O, ‘q,,+1=0,...,‘qm=0,'ra+,=O,...,'r,,=O, wecanusethesame argument toshow that m<qisimpossible andhence, bysymmetry, thatq<m.Thus weﬁnally ﬁndthat k=p,m=q.I 7.92. The total number ofterms appearing inthecanonical form ofa quadratic form A(x, x),i.e.,itsrank (seeSec. 7.33b), isalsocalled itsindex ofinertia. The total number ofpositive terms iscalled thepositive index of inertia, andthetotal number ofnegative terms iscalled thenegative index ofinertia. Ifthepositive index ofinertia equals thedimension ofthespace, theform issaid tobepositive definite. lnother words, aquadratic form A(x, x)ispositive deﬁnite ifandonly ifallnofitscanonical coefﬁcients are positive. Itfollows that apositive deﬁnite quadratic form takes apositive value atevery point ofthespace except theorigin ofcoordinates. Conversely, ifaquadratic form deﬁned onann-dimensional realspace takes positive values everywhere except attheorigin, then itsrank isnand itspositive index ofinertia isalso n,i.e.,theform ispositive deﬁnite. In fact, foraform ofrank lessthan norwith lessthan npositive canonical coeﬁicients, itiseasy toﬁndpoints inthespace other than theorigin where sac.7.9 BILINEAR AND QUADRATIC roams 1NAREAL SPACE 207 theform takes either thevalue 0ornegative values. For example, the quadratic form A(x.X)=‘ii+£2 ofrank 2inathree-dimensional space takes thevalue 0foranynonzero vector with components Z1=0,Z2qé0,Z3=0.Forthese vectors theform A(x.X)=ii—Z3+ii ofrank 3inathree-dimensional space takm negative values. Clearly, these examples illustrate thefullgenerality ofthesituation. 7.93. The law ofinertia just proved forquadratic forms generalizes immediately tothecase ofsymmetric bilinear forms, i.e.,thetotal number ofpositive coeﬂicients and thetotal number ofnegative coeﬂicients inthe canonical form (22) ofasymmetric bilinear form A(x, y)isindependent of thechoice ofacanonical basis. Thus thepositive and negative indicw of inertia arewell-deﬁned concepts forasymmetric bilinear form. Thevalues ofthepositive and negative indices ofinertia ofthebilinear form A(x,y) andhence ofthequadratic form A(x, x)canbedetermined from thesigns ofthedescending principal minors ofthematrix oftheform inanybasis (provided only thattheminors arenonzero) byusing theformulas (24), p.195. ltshould benoted thatgiven anyquadratic form A(x, x)inarealspace Rn,acanonical basis canalways befound such thatthecorresponding canonical coeﬂicients canonly take thevalues i1.Infact, having reduced A(x, x)tothe form A(x.X)=Mi+'''+mi—Pm-Li—"''—tmi+. where thenumbers 7.1,...,AD,ul,...,uaareallpositive, wemake another coordinate transformation T1: 711: --->T11=\/Ta: nan T114-1 = ‘%+1, ---s70+-a =\/Q 7111-}-qs thereby reducing A(x, x)totheform A(xvx)=TI+"‘ +T12»HT?1+1_"'_T121+u‘ This shows that inarealspace thenumbers pandqaretheonly invariantsT ofthequadratic form A(x, x)andthecorresponding symmetric bilinear form A(x.y)- THEOREM. Twoﬁnite-dimensional real spaces R’andR”,equipped with nonsingular symmetric bilinear forms A(x’, y’)andA(x”, y”),respectively, are A-isomorphic ifandonly they have thesame dimension andtheindices of TApart from anyfunction ofp andq(like therank r=p+q),which isobviously aninvariant ofA(x, x)andA(x, y). 208 BILINEAR ANDQUADRATIC poms CHAP. 7 inertia p’,q’oftheform A(x’, y’)coincide with thecorresponding indices of inertia p”,q”oftheform A(x”, y"). Proof An immediate consequence oftheabove considerations and Theorem 7.72. | 7.94. LetA(x, y)beasymmetric bilinear form inarealspace R,,.Then, asinSec.7.15b,A(x, y)issaidtobenonsingular ifitsrank equals thedimension ofthespace, i.e.,ifallthecoefﬁcients X1,X2,...,7."inthecanonical form A(x? =Mimi +7\2€2‘ﬁ2 +''i+7\n€1t'% (see Sec. 7.43) arenonzero. Suppose that inaddition allthecoefﬁcients 7.1,7.2,...,7.,arepositive, sothatthecorresponding quadratic form A(x, x) ispositive deﬁnite (see Sec. 7.92). Then thebilinear form A(x, y)issaid to bepositive deﬁnite. Thus, according toSec. 7.92, A(x, y)ispositive deﬁnite ifand only ifthecorresponding quadratic form A(x,x) takes apositive value forevery nonzero vector x.- Byitsvery deﬁnition, apositive deﬁnite form A(x,y) inaspace R,is nonsingular. But, because ofthefact that A(x, x)>0,apositive deﬁnite form A(x, y)remains positive deﬁnite inanysubspace R’CRn.Hence a positive deﬁnite bilinear form, unlike thegeneral bilinear form (see Sec. 7.l5d), remains nonsingular inanysubspace R’CRn. Thus, given anyk linearly independent vectors fl,...,ﬁc,thedeterminant A(fvf1) A(.fI=.flc) D: . . . A(f1~f1) A(f1~ﬁ.) must benonzero. Wewillseeinamoment that Dmust infactbepositive. 7.95. Animportant example ofasymmetric positive deﬁnite bilinear form inthespace V3isgiven bythescalar product (x,y)ofthevectors xand y.Infact, itfollows atonce from thedeﬁnition ofthescalar product that (X.y)=(y.X), (x,x) =|x|2>0for xqé0. Theﬁrst ofthese relations shows that thebilinear form (x,y)issymmetric, while thesecond shows thatthecorresponding quadratic form takm apositive value forevery vector xqé0.Thus thebilinear form (x,y)ispositive deﬁnite. Positive deﬁnite bilinear forms will play aparticularly important role below. Infact, byusing such forms wewillbeable tointroduce theconcepts ofthelength ofavector and theangle between two vectors inageneral linear space (Chap. 8). sEc.7.9 BILINEAR AND QUADRATIC roams INAREAL SPACE 209 7.96. The problem now arises ofhow tousethematrix ofasymmetric bilinear form A(x, y)todetermine whether ornotA(x, y)ispositive deﬁnite. Theanswer tothisproblem isgiven bythefollowing THEOREM. Anecessary andsuﬂicient condition forthesymmetric matrix A=||a,,,|| todeﬁne apositive deﬁnite bilinear form A(x, y)isthat the descending principal minors an a12 a13 an a12 1111, ,(121 a22 (123 7---.ddiiaikii (42) 1121 a22 1131 1132 aaa ofthematrix |la,~,,|| allbepositive. Proof. Iftheprincipal minors (42) ofthematrix Aareallpositive, then bytheformulas (24), p.195, allthecanonical coefﬁcients 2,,oftheform A(x, y)arealso positive insome basis, i.e.,A(x, y)ispositive deﬁnite. Conversely, suppose theform A(x, y)ispositive deﬁnite. Then the descending principal minors (42) ofthematrix Ila,-kll arepositive. Infact, theprincipal minor an a12 '" alm a a aM: 21 22 2m aml am2 ''' amm corresponds tothematrix ||a,»,,|| (i,k=l,2,...,m)ofthebilinear form A(x, y)inthesubspace L,"spanned bytheﬁrstmbasis vectors. Since A(x, y) ispositive deﬁnite inthesubspace L,"(A(x, x)>0forxqé0),there exists a canonical basis inL,"inwhich A(x, y)canbewritten incanonical form with positive coeﬁicients. Inparticular, thedeterminant ofA(x, y)inthisbasis is positive, being equal totheproduct ofthecanonical coefﬁcients. Bearing in mind therelation between determinants ofabilinear form indiﬁerent bases (equation (6),p.182), weseethatthedeterminant ofA(x, y)intheoriginal basis ofthesubspace Lmisalsopositive. Butthedeterminant ofA(x, y)inthe original basis ofL,,,isjusttheminor M.Itfollows that M>0.I Remark. Inthesecond part oftheproof, wecould have taken Mtobe any principal minor instead ofadmcending principal minor, without changing theargument inanyessential way. Thus every principal minor of thematrix ofapositive deﬁnite bilinear form ispositive. 7.97. Forapositive deﬁnite form A(x, y)there always exists acanonical basis el,...,e,,inwhich allthecanonical coefﬁcients equal +1(see Sec. 7.93). Hence two n-dimensional real spaces R;and R:equipped with 2l0 BILINEAR AND QUADRATIC roams CI-IAP. 7 positive deﬁnite forms A(x’, y’)andA(x”, y”),respectively, areA-isomorphic, byTheorem 7.72. 7.98. Thesolution ofthefollowing problem isoften needed inapplications oflinear algebra toanalysis (i.e., inthetheory ofconditional extrema): Given thematrix A=llaikll ofasymmetric bilinear form A(x, y),determine whether theform ispositive deﬁnite inthesubspace speciﬁed bythesystem ofkindependent linear equations 2b,»,-2,,-=0 (i=l,2,...,k;k<n). i=1 Itturns outthat anecessary andsufﬁcient condition forthistobethecase isthat thedescending principal minors oforders 2k+l,2k+2,..., k+nofthematrix 00~~0b,, 00~~0b,, 00~~0b,,,I712 b22 bk2bi. b2n bknA=(_1)* bu Z721 bu an a12 aln biz I722 b;;2 a21 a22 a2" ,b1n b2n bkn anl an2 arm bepositive, under theassumption that therank ofthematrix lib,-,-ll equals k and that thedeterminant made upoftheﬁrst kcolumns ofIlb,-,.|| isnon- vanishing.T PROBLEMS 1.Dotheelements ofthe matrix ofabilinear form constitute atensor (Sec._5.61), andifso,ofwhat type? 2.Reduce thequadratic form 515.2+E353+5.351 tocanonical form. 3.Letp bethepositive index ofinertia ofaquadratic form A(x, x)(deﬁned on thespace Rn),andletqbeitsnegative index ofinertia. Moreover, letA1,7.2,..., 7.1,beanyp positive numbers andul,v.2,...,v.3anyq negative numbers. Show thatthere exists abasis inwhich theform A(x, x)takes theform A(x, x)=7.11% +---+)\D‘l’.‘% +u1r;l,+, +~~~+u3¢;1,+a. TSeethenote byR.Y.Shostak, Uspekhi Mat. Nauk, vol.9,no.2(1954), pp.199-206. PROBLEMS 2|I 4.Show thatthematrix ofaquadratic form ofrank ralways hasatleast one nonvanishing principal minor oforder r. 5.Reduce thebilinear form A(X,_Y) :5.1111 +Elﬂg +zignl 'l‘2Zg7)2 +2é27i3 +2€37l2 +5E_,3'Y)3 tocanonical form. 6.Apply Jacobi’s method toreduce-the bilinear form ,1“>-5>- A(x,)/) = *£192 “£291 +E-17l3 +£391 +2E»2"la +2E»3l'l2 +éalla ++E»2l'l2 tocanonical form. 7.State theconditions under which asymmetric matrix llamll deﬁnes anegative deﬁnite bilinear form. 8.Given asymmetric matrix A=llaiill with theproperties an "12an>O, >O,...,detllaikll >O, a21 a22 show thatam,>O. 9.Prove that anantisymmetric multilinear form inn+1vectors ofann- dimensional space K,,vanishm identically. 10.LetA(x1, ...,x,,_1) beanantisymmetric multilinear form inn~1vectors ofann-dimensional space. Prove thatA(x1, ...,x,,_1) canbewritten inany basisasadeterminant whose ﬁrstn—1rowsconsist ofthecomponents ofthe vector arguments andwhose last(nth) rowisﬁxed. 11.Prove thatevery antisymmetric bilinear form A(x,y)i0canbereduced tothecanonical form A(x, y)=cl-:2 —0211 +03¢, —@413 +'''-1-o2k_1'r2k —o2k¢2k_1. 12.PrOve thatarealquadratic form Tl‘/l=.-1: A(x, X)= 1:65:51.- isnonnegative forallxeR,, ifandonly ifallprincipal minors ofthematrix A=Ila,-kll arenonnegative. Comment. Thedescending principal minors 81and82vanish forthematrix O O 0-1 butthecorresponding form fails tobenonnegative. Thus theconditions 81>O,82>0arenotsufficient fornOnnegativity oftheform. 2l2 BILINEAR AND QUADRATIC FORMS Cl-IAP, 7 13.LetA(x,y)beanonsingular symmetric bilinear form inann-dimensional space K,,,andletK’CK,,beasubspace ofdimension r.Prove thatthespace K”<1Kconjugate toK’with respect toA(x,y)isofdimension n~r. 14.Consider thesymmetric bilinear form (x.y) =Elm—E2122 inthespace R2.Find theoperator which istheadjoint with respect tothisform oftherotation operator with matrix COS onSinon A= . -sin <2cos<2 15.Let(x,y)beanonsingular quadratic form inthespace K.,,.Forthesystem §a,.,,z,,. =b,~(j=1,2,...,n) (43) 1;:>- ofnlinear equations innunknowns, prove Fredholm’s theorem which asserts thatthesystem (43)hasasolution forprecisely those vectors b=(bl,...,b,,) which areconjugate toallthesolutions ofthehomogeneous system ‘VtZ%m=Q (W k=1 where llajkll isthematrix conjugate tolla,-kll with respect totheform (x,y). From thisdeduce thatthenumber ofindependent linear conditions onthevector bwhich arenecessary andsufficient forthesystem (43)tohave asolution equals thedimension ofthespace ofsolutions ofthehomogeneous system iajkii-=0 (j=1,2,...,n). k=>- Comment. Forageneral system Z%%=M U=LL~Um¢M, ow k=1 thetwoquantities inquestion nolonger coincide, andtheir difference, equal tom~n,iscalled theindex ofthesystem (43'). 16.Prove thatevery nonnegative bilinear form ofrank rinthespace R,canbe represented asasumofrnonnegative bilinear forms ofrank 1. 17.Prove thatevery bilinear form ofrank linthespace K,isoftheform A(x.y)=f(x)g(y). where f(x)andg(y) arelinear forms. 18.Prove thatif and arenonnegative bilinear forms inthespace R,,,thentheform isalsononnegative.A(x,y) =Zaikéjnk I,k=1 B(x,y)=lbikéink 5-5 C(X.y) =..envi-“taikbikEma.PROBLEMS 2I3 chapter 8 EUCLIDEAN SPACES 8.l.Introduction The explanation ofalarge variety ofgeometric facts rests toagreat extent onthepossibility ofmaking measurements, basically measurements of thelengths ofstraight linesegments andtheangles between them. Sofar, wearenotinaposition tomake such measurements inageneral linear space; ofcourse, thishastheeﬁect ofnarrowing thescope ofourinvestigations. Anatural way toextend these “metric” methods tothecase ofgeneral linear spaces istobegin with thedeﬁnition ofthescalar product oftwo vectors which isadopted inanalytic geometry (and which issuitable asof now only forordinary vectors, i.e.,elements ofthespace V3introduced in Sec.2.l5a). This deﬁnition reads asfollows: Thescalar product oftwovectors istheproduct ofthelengths ofthevectors andthecosine oftheangle between them. Thus thedeﬁnition already rests onthepossibility ofmeasuring the lengths ofvectors andtheangles between them. Ontheother hand, ifwe know thescalar product foranarbitrary pair ofvectors, wecandeduce the lengths ofvectors and theangles between them. lnfact, thesquare of thelength ofavectorequals thescalar product ofthevector with itself ,while the cosine oftheangle between twovectors isjusttheratio oftheir scalar product totheproduct oftheir lengths. Therefore thepossibility ofmeasuring lengths andangles (and with it,thewhole ﬁeld ofgeometry associated with measurements, so-called “metric geometry”), isalready implicit inthe concept ofthescalar product. Inthecase ofageneral linear space, the 2l4 SEC.8.2 DEE1N1rioN orAEucLrDEAN SPACE 215 simplest approach istointroduce theconcept ofthescalar product oftwo vectors, and then usethescalar product (once itisavailable) todeﬁne lengths ofvectors andangles between them. Wenow look forproperties oftheordinary scalar product which canbe used toconstruct asimilar quantity inageneral linear space. Forthetime being, werestrict ourselves tothecase ofrealspaces. Asalready noted inSec. 7.95, inthespace V3thescalar product (x,y)is asymmetric positive deﬁnite bilinear form inthevectors xand y.Quite generally, wecandeﬁne such aform inanyreallinear space. Thus weare ledtoconsider aﬁxed butarbitrary symmetric positive deﬁnite bilinear form A(x, y)deﬁned onagiven reallinear space, which wecallthe“scalar product” ofthevectors xandy.Wethen usethescalar product todeﬁne thelength ofevery vector and theangle between every pair ofvectors bythesame formulas asthose used inthespace V3.Ofcourse, only further study will show how successful this deﬁnition is;however, in thecourse ofthisandsubsequent chapters, itwillbecome apparent thatwith thisdeﬁnition wecaninfactextend themethods ofmetric geometry togeneral linear spaces, thereby greatly enhancing our technique forinvestigating various mathematical objects encountered inalgebra andanalysis. Atthis point, itisimportant tonote that theinitial positive deﬁnite bilinear form canbechosen inavariety ofdiﬁerent ways inthegiven linear space. The length ofavector xcalculated byusing onesuch form willbe diﬁerent from thelength ofthesame vector calculated byusing another form; asimilar remark applies totheangle between twovectors. Thus thelengths of vectors andtheangles between them arenotuniquely deﬁned. However, this lack ofuniqueness should notdisturb us,forthere iscertainly nothing very surprising about thefact that diﬁerent numbers will beassigned asthe length ofthesame linesegment ifwemeasure thesegment indiﬁerent units. Infact, wecansaythatthechoice oftheoriginal symmetric positive deﬁnite bilinear form isanalogous tothechoice ofa“unit” formeasuring lengths of vectors andangles between them. Areallinear space equipped with a“unit” symmetric positive deﬁnite bilinear form will henceforth becalled aEuclidean space, while alinear space without a“unit” form willbecalled anaﬂine space. Thecaseofcomplex linear spaces willbeconsidered inChapter 9. 8.2.Deﬁnition ofaEuclidean Space 8.21. Areallinear space RissaidtobeEuclidean ifthere isaruleassigning toevery pair ofvectors x,y6Rarealnumber called thescalar product of thevectors xandy,denoted by(x,y),such that a)(x,y)=(y,x)forevery x,yeR(thecommutative law); b)(x,y+2)=(x,y)+(x,2)forevery x,y,26R(thedistributive law); 2l6 EucLrDEAN SPACES Q]-1AP_ 3 c)(Ax,y)=7\(x,y)forevery x,y6Randevery realnumber 7.; d)(x,x)>0forevery x1+0and(x,x)=0forx=0. Taken together, these axioms imply that thescalar product ofthevectors x andyisabilinear form (axioms b)and c)),which issymmetric (axiom a)) and positive deﬁnite (axiom d)). Conversely, any bilinear form which is symmetric andpositive definite canbechosen asthescalar product. Since thescalar product ofthevectors xandyisabilinear form, equation (2)ofSec. 7.1holds, andinthepresent case becomes PE‘ § ,.l\/1’-“are(§mgwJ=_=Mam. (n where xl,...,x,,,yl,...,y,,,arearbitrary vectors oftheEuclidean space R,andM1,...,otk,T-31,...,T-3,"arearbitrary realnumbers. 8.22. Examples a.Inthespace V3offree vectors (Sec. 2.l5a), thescalar product is deﬁned asinthebeginning ofSec. 8.1,andaxioms a)—d) express thefamiliar properties ofthescalar product, proved invector algebra. b.Inthen-dimensional space R,(Sec. 2.l5b) wedeﬁne thescalar product ofthevectors x=(Z1,Z3,...,Zn)andy =(nl,n3,...,nn)bytheformula (xs = glnl +52% +'''+gnnn‘ This deﬁnition generalizes thefamiliar expression forthescalar product of three-dimensional vectors interms ofthecomponents ofthevectors with respect toanorthogonal coordinate system. The reader caneasily verify that axioms a)—d) aresatisﬁed inthiscase. Wenote that formula (2)isnottheonly way ofintroducing ascalar product inR,,.Adescription ofallpossible ways ofintroducing ascalar product (i.e., asymmetric positive deﬁnite bilinear form) inthespace R, hasessentially already been given inSec. 7.96. c.Inthespace R(a, b)ofcontinuous real functions ontheinterval a<t<b(Sec. 2.l5c), wedeﬁne thescalar product ofthefunctions x= x(t)andy=y(t)bytheformula Dmn=jnwmo m Axioms a)—d) arethen immediate consequences ofthebasic properties of theintegral. Henceforth thespace R(a, b),with thescalar product deﬁned by(3),willbedenoted byR2(a, b). 8.3.Basic Metric Concepts Equipped with thescalar product, wenowproceed todeﬁne thebasic met- ricconcepts, i.e.,thelength ofavector andtheangle between twovectors. sEc.8.3 BASIC METRIC CONCEPTS 2|7 8.31. Thelength ofavector. Bythelength (ornorm) ofavector xina Euclidean space Rwemean thequantity ]x]=-I-\/(x, x). (4) Examples a.Inthespace V3ourdeﬁnition reduces totheusual deﬁnition ofthe length ofavector. b.Inthespace R,thelength ofthevector x=(Z1,Z3,...,Zn)isgiven by |x|=+~/€i+£€+---+£i. c.Inthespace R3(a, b),thelength ofthevector x(t)turns outtobe ]x]=-l—\/(x, x)=+]jbx2(t) dt. This quantity issometimes written ]]x(t)]] andisbest called thenorm ofthe function x(t) (inorder toavoid misleading connotations connected with thephrase “length ofafunction”). 8.32. Itfollows from axiom d)that every vector xofaEuclidean space Rhasalength; thislength ispositive ifx960andzero ifx=0(i.e., ifxis thezero vector). Theformula IMI=~/(ix.xx)=~/tux.x)=In~/(x.x)=inIx] (5) shows that thelength ofavector multiplied byanumerical factor 7.equals the absolute value of7.times thelength ofx. Avector xoflength 1issaid tobeaunitvector. Every nonzero vector x canbenormalized, i.e.,multiplied byanumber 7.such that theresult isa unit vector. Infact, solving theequation ]7.x]=lfor7.,weseethat 7.need only besuch that 1 W71x1‘ AsetFCRissaidtobebounded ifthelengths ofallthevectors xeF arebounded byaﬁxed constant. Thesetofallvectors x6Rsuch that ]x]<l isabounded setcalled theunitball, while thesetofallx6Rsuch that ]x]=l isabounded setcalled theunitsphere. 8.33. Theangle between twovectors. Bytheangle between twovectors x andywemean theangle (lying between 0and180degrees) whose cosine is theratio (X,y) IX]I)/I 2l8 EucL1DEAN SPACES CHAP. 8 Forordinary vectors (inthespace V3)ourdeﬁnition agrees with theusual way ofwriting theangle between twovectors interms ofthescalar product. Toapply thisdeﬁnition inageneral Euclidean space, wemust ﬁrst prove that theratio hasanabsolute value nogreater than unity foranyvectors x andy.Toprove this, consider thevector 7.x-y,where 7.isarealnumber. Byaxiom d),wehave (7.x—y,7.x—y)>0 (6) forany7..Using (1),wecanwrite thisinequality intheform 7*(X,X)—27(X.y)+(y,y)>0- (7) Theleft-hand sideoftheinequality isaquadratic trinomial in7.with positive coeﬁicients, which cannot have distinct realroots, since then itwould not have thesame signforall7..Therefore thediscriminant (x,y)“—(x,x)(y, y) ofthetrinomial cannot bepositive, i.e., (><.y)* <(X.><)(y.y)- Taking thesquare root, weobtain l(X,)/)l <IX]I)/I. (3) asrequired. Theinequality (8)iscalled theSchwarz inequality.']‘ 8.34. Wenow examine when theinequality (8)reduces toaninequality. Suppose thevectors xandyarecollinear, sothaty=7.x,7.6R,say. Then obviously l(X.)/)l =l(X.XX)!=I7](X.X) =I7]IXI“=IX]I)/I, and(8)reduces toanequality. Conversely, iftheinequality (8)reduces toanequality forsome pair of vectors xandy,then xandyarecollinear. Infact, if l(X.)/)l =IX]I)/I. then thediscriminant of(7)vanishes andhence (7)hasaunique realroot 7.0 (ofmultiplicity two). Therefore 7302X)—27..(><.y)+tr.y)=(Mr—y.Mr—y)=0. whence itfollows byaxiom d)that 7.0x—y:0ory=7.0x, i.e.,thevectors xandyarecollinear. Thus theabsolute value ofthescalar product oftwo vectors equals theproduct oftheir lengths andonlyifthevectors arecollinear. Examples a.Inthespace V3theSchwarz inequality isanobvious consequence of thedeﬁnition ofthescalar product astheproduct ofthelengths oftwovectors andthecosine oftheangle between them. TSometimes alsoassociated withthenames ofCauchy andBuniakovsky. sEc.8.3 BAs1c METRIC CONCEPTS 2l9 b.Inthespace R,theSchwarz inequality takes theform <.]iii,/iii. <9) = 1:1 j=1 and isvalid foranypair ofvectors x=(Z1.Z2,...,Zn)andy=(nl,n2, ...,'q,,), orequivalently, foranytwosetsofreal numbers Z1,Z2,...,in and“n1,n2,...,-q,,. v c.Inthespace R2(a, b),theSchwarz inequality takes theform~._l\/1=ivy5... '> b b|><(r)y(t) dtl<.\/I.X2(l)dt\/j y2(t)dt. (10) 8.35. Orthogonality. Two vectors xandyaresaid tobeorthogonal if (x,y)=0.Thus thenotion oforthogonality ofthevectors xandyisthe same asthenotion ofxandybeing conjugate (Sec. 7.4la) with respect to thebilinear form (x,y).Ifxqé0andyqé0,then, bythegeneral deﬁnition oftheangle between twovectors, (x,y)=0means that xandymake an angle of90°with each other. The zero vector isorthogonal toevery vector xeR. Examples a.Inthespace Rntheorthogonality condition forthevectors x= (Z1,E2,. ..,Zn)andy=(nl,“n2,...,'q,,)takes theform ifm +€2'ﬁ2 +''' Em. :0' Forexample, thevectors e1=(l,0,...,0), e2=(0,l,...,0), e,,=(0,0,...,l) areorthogonal (inpairs). b.Inthespace R2(a, b)theorthogonality condition forthevectors x:x(t)andy=y(t)takes theform I! jx(t)y(t) dt=0. Cl The reader caneasily verify, bycalculating theappropriate integrals, that inthespace R2(—"rt, ~rc)anytwovectors ofthe“trigonometric system” l,cost,sint,cos2t,sin2t,...,cosnt,sinnt,..-. areorthogonal. 220 EucL1DEAN SPACES CI-IAP. 8 8.36. Wenowderive some simple propositions associated with theconcept oforthogonality. a.LEMMA. Ifthenonzero vectors xl,x2,...,x,,areorthogonal, then they arelinearly independent. Proof Suppose thevectors arelinearly dependent. Then arelation of theform ot1x1+ot2x2-I----+ot,,x,,=0 holds, where M1950,say. Taking thescalar product ofthisequation with xl,weobtain ot1(.X1, x1)=0,since byhypothesis thevectors xl,x2,...,x,, areorthogonal. Itfollows that (xl,x1)=0and hence that x1isthezero vector, contrary tohypothesis. I Theresult ofthislemma isoften used inthefollowing form: Ifasumof orthogonal vectors is2ero, then each term inthesumiszero. b.LEMMA. Ifthevectors y1,y2, ...,y,,areorthogonal tothevector x, thenanylinear combination 011)/1+Ugyg+-~-+ot,,y,,isalsoorthogonal tox. Proof Weneed merely note that (°‘1)’1 +°‘2)/2 +'‘'+akykf -X) =°‘1()’1i -X)+°12()/2’ X)+'''+°%()/ki -X)=0- l The setofalllinear combinations (x,y, -I-ot2y2 +---+otkyk forms a subspace L=L(y,, y2,...,yk),namely thelinear manifold spanned bythe vectors yl,y2,...,y,,(Sec. 2.51). Therefore ifxisorthogonal tothevectors y1,y2, ...,y,,, itisorthogonal toevery vector ofthesubspace L.Inthis case, wesaythat thevector xisorthogonal tothesubspace L.Ingeneral, if FCRisanysetofvectors inaEuclidean space R,wesaythat thevector x isorthogonal tothesetFifxisorthogonal toevery vector inF.According toLemma 8.36b, thesetGofallvectors xorthogonal toasetFisitself a subspace ofthespace R.Themost common situation isthecase where Fis asubspace. Then thesubspace Giscalled theorthogonal complement ofthe subspace F. 8.37. ThePythagorean theorem anditsgeneralization. Letthevectors x andybeorthogonal. Then, byanalogy with elementary geometry, wecan callthevector x+ythehypotenuse oftheright triangle determined bythe vectors xandy.Taking thescalar product ofx+ywith itself ,andusing the orthogonality ofthevectors xandy,weobtain IX+yl*= (X+)/.X+)/)= (x,x)+2(X.)/) +(Jay) =(X.X)+(JG)/)=IX!“+lyl*- SEC.83 BASIC METRIC CONCEPTS 22l This proves thePythagorean theorem inageneral Euclidean space, i.e.,the square ofthehypotenuse equals thesum ofthesquares ofthesides. Itiseasy togeneralize this theorem tothecase ofany number ofsummands. ln fact, letthevectors xl,x2,...,x,,beorthogonal andlet 2:-x1—l—x2—l—---+x,,. Thenwehave ]Z]2=(-X1‘]‘-x2+"'+-X1.’-x1']"x2"]C"'+-xx) =]X1]2 "I"]X2]2 "I"'''"I"]X1.]2- (11) 8.38. Thetriangle inequalities. Ifxandyarearbitrary vectors, then by analogy with elementary geometry, itisnatural tocallx-I-ythethird side ofthetriangle determined bythevectors xandy.Using theSchwarz inequality, weget IX+)/I*= (X+)/,X +)/)=(X.X) +Z(X.)/) +0'.)/) <.IXI2+2IX]I)/I+I)/I2=(IXI+I)/])2. >IXI*~2IX]I)/I+I)/I2=(IXI—I)/I)*. O1‘ IX+y]<Ix]+I)/I. (12) IX+)/I>I]XI-—I)/I] (13) Theinequalities (12)and(13)arecalled thetriangle inequalities. Geometric- ally, they mean that thelength ofanysideofatriangle isnogreater than the sumofthelengths ofthetwoother sides andnolessthan theabsolute value of thediﬂerence ofthelengths ofthetwoother sides. 8.39. Wecould nowsuccessively carry over allthetheorems ofelementary geometry toanyEuclidean space. Butthere isnoneed todoso.Instead we introduce theconcept ofaEuclidean isomorphism between two Euclidean spaces, i.e., two Euclidean spaces R’and R”aresaid tobeEuclidean- isomorphic ifthey areisomorphic regarded asreallinear spaces (seeSec.2.71) andifinaddition (X’.y’)=(X”.y”) whenever thevectors x”,y”6R”correspond tothevectors x’,y’eR’.Then itisobvious that every geometric theorem (bywhich wemean anytheorem based ontheconcepts ofalinear space andascalar product) proved fora space R’isalsovalid foranyspace R”which isEuclidean—isomorphic toR’. According toSec.7.97, anytwoEuclidean spaces with thesame dimension n areEuclidean—isomorphic. Hence any geometric theorem valid inann- dimensional Euclidean space R;isalso valid inany other n-dimensional Euclidean space R1.lnparticular, thetheorems ofelementary geometry, 222 EucL1DEAN SPACES CI-IAP, 8 i.e., thegeometric theorems inthespace R3,remain valid inanythree- dimensional subspace ofany Euclidean space. Inthissense, thetheorems ofelementary geometry areallvalid inanyEuclidean space. 8.4.Orthogonal Bases 8.41. THEOREM. Inanyn-dimensional Euclidean space Rnthere exists a basis consisting ofnnonzero orthogonal vectors. Proof There exists acanonical basis el,e2,...,e,,forthebilinear form (x,y),just asforanyother symmetric bilinear form inann-dimensional space (seeSec. 7.43). Thecondition (eh eh") I0 i satisﬁed bythevectors ofthecanonical basis isinthiscasejustthecondition fororthogonality ofthevectors e,»ande,,.Thus thecanonical basis el,e2,..., e,,consists ofn(pairwise) orthogonal vectors. | InSec. 8.6wewillconsider apractical method forconstructing such an orthogonal basis. 8.42. Itisoften convenient tonormalize thevectors ofanorthogonal basis bydividing each ofthem byitslength. Theresulting orthogonal basis inR"issaid tobeorthonormal. Letel,e2,...,e,,beanarbitrary orthonormal basis inann-dimensional Euclidean space R,,.Then every vector xeR"canberepresented intheform -X= £161 +€2e2 ‘l7'''+E36... where Z1,Z2,...,Z,"arethecomponents ofthevector xwith respect tothe basis el,e3,...,e,,.Wewillalso callthese components Fourier coeﬂicients ofthevector xwith respect totheorthonormal system el,e2,...,e,,. Taking thescalar product of(l4)with e,-,weﬁndthat Z,=(x,e,-) (i=l,2,...,n). (15) Lety=:rue,+-/j3e3+---+-/j,,e,, beany other vector ofthespace R,,. Then itfollows from (l)that (x,y) :£1271 +£2702 -I‘"'+En"/Iw (16) Thus inanorthonormal basis thescalar product oftwovectors equals thesum oftheproducts ofthecomponents (Fourier coeﬂicients) ofthevectors. In particular, setting y=:x,weobtain IXIZ=(X.X)=ii+ii+---+ii (17) SE0 3-5 PERPENDICULARS Z23 8.5. Perpendiculars 8.51. LetR’beaﬁnite-dimensional subspace ofaEuclidean space R, andletfbeavector which isingeneral notanelement ofR’.Wenow pose theproblem ofreprmentingfin theform f=,g+h, (18) where thevector gbelongs tothesubspace R’andthevector hisorthogonal toR’.The vector gappearing intheexpansion (18) iscalled theprojection offonto thesubspace R’,andthevector hiscalled theperpendicular dropped from theendoffonto thesubspace R’.This terminology calls tomind certain familiar geometric associations, butitisnotintended todomore than just suggest these associations.T Thesolution ofthisproblem hasineﬁect already been given inSec. 7.54 foranysymmetric bilinear form which isnonsingular inthesubspace R’. Since thepositive deﬁnite form (x,y)isnonsingular inevery subspace R’CR(Sec. 7.94), theexistence anduniqueness ofasolution ofourproblem follows from Sec.7.54. Moreover, asshown inSec.7.55, theexistence ofthe expansion (18)shows thatthewhole space Risthedirect sum ofthesubspace R’anditsorthogonal complement R”.Adirect sum whose terms areorthog- onal iscalled anorthogonal direct sum. Thus wehave expanded thespace Rasanorthogonal direct sum ofthesubspaces R’andR”.IfRandR’have dimensions nandk,respectively, then thedimension ofR”equals n~k, since thedimension ofthedirect sum isthesum ofthedimensions ofits terms (Sec. 2.47). Wenote that theproblem isalso solved inthecase where fliesinthe subspace R’,since then f=f+0. This solution isobviously unique. Infact, if f=g—]-/1 (geR’,heR”), then h=f— geR’ which implies h=0,g:f 8.52. Applying thePythagorean theorem (Sec. 8.37) totheexpansion (18), weobtain Ifl“=Igl“+]hI*. (19) which implies theformula 0<Ihl<If]. (20) TSince theconcept ofthe“end ofavector” plays norole inouraxiomatics, itis inappropriate tolook foranylogical content inthisterminology. 224 EucLrDEAN SPACES CHAP. 8 expressing thegeometric fact that thelength ofaperpendicular does not exceed thelength ofthelinesegment from which itisdropped. Consider the cases where oneoftheinequalities in(20) becomes anequality. The ﬁrst equality sign holds if]h]=0;thismeans thatf: g+0,i.e.,fis anelement ofthesubspace R’.The second equality sign holds if]h]=]f|;according to(19), this means that g:0orf= 0-I-h,i.e.,fis orthogonal tothe subspace R’.Thus ]h]=0means thatfbelongs toR’,while ]h]=IfImeans thatfisorthogonal toR’.lnanyother conﬁguration off,the(inherently positive) length ofhisIessthan that off Now letel,€2,...,e,,beanorthonormal basis inthesubspace R’,and let R‘ g=Zlaje,-.,= Then, bySec. 8.42, It IgI2=Z11?- J'=1 Substituting thisvalue of]g]2into (19), weget k.If!“=|h1“+Ea?-;I=1 Inparticular, forany (ﬁnite) orthonormal system el,ez,...,e,,and any vectorf, wehave theinequality <Iflz,,_il\/]’~'ta....,= known asBessel’s inequality. The geometric meaning ofthis inequality is clear: Thesquare ofthelength ofthevector fisnolessthan thesum ofthe squares ofitsprojections onto anykmutually orthogonal directions. 8.53. Intheapplications, wesometimes need anexplicit solution ofthe problem ofdropping aperpendicular onto asubspace R’,given some basis {b}=={b1,b2, ...,b,,} inR’(ingeneral, notanorthonormal basis). To solve thisproblem, weﬁrst expand therequired vector g(the “foot ofthe perpendicular”) with respect tothebasis {b},i.e.,wewrite 8: P1171 P2172 Cl‘'''Cl‘I51-be Wethen impose onthevector h:f—gthecondition that itbeorthogonal toallthevectors bl,b2,...,bk,thereby obtaining thesystem ofequations (hib1):(f‘_g» bl)T: bl)‘Ci'31(b1» bl)‘Ci'32(b2. bl)‘C'''‘Cl5k(bk»b1) 20» (h»b2) :(f‘C 8»I72): I72)‘Ci51(b1» I72)Ci'32(b2» I72)‘C'''‘Cl5k(bkvb2) :0» (hibk):(f‘C g.bl")I: bk)Ci51(b1> bi-)‘Ci'32(b2, bk)‘C'''‘Ci'3;¢(b;,-,b;;) C;0» SEC3-5 PERPENDICULARS 225 with determinant (bi,bi) (b2, bi) '''(bk, bi) (bl!b2) (b2, b2) '''(bk, b2) 1)Z . . . . _ (bl, bk) (b2,bk) '''(bk! bk) ButDisnonzero, being thedeterminant ofthematrix ofthepositive deﬁnite form (x,y)inthebasis bl,b2,...,bk(see Sec. 7.96). Hence wecansolve thesystem byCramer’s rule, obtaining thefollowing expression forthe coeﬁicients I-3,(j=1, 2,... ,k): (blibl) (b2,b1) '''(bd—-Dbl) bi) (b1'+1,b1) '''(bkvbl) (b1,b2) (b2,b2) '''(bi-1,172) b2) (bi+1,b-3) '''(bI.~,b2) $125 ' (bl,bk) (b2,bi.-) '''(bi-1» bk) (fibk) (b;I+1»bi.-) '''(blvbk) 8.54. Theproblem ofdropping aperpendicular canbeposed notonly for asubspace, butalsoforahyperplane, inwhich casetheproblem isformulated asfollows: Suppose thatinaEuclidean space R,wearegiven avector fand ahyperplane R”,generated byparallel displacement ofasubspace R’.We wish toshow that there exists aunique expansion f=g+h, (21) where thevector gbelongs tothehyperplane R”andthevector hisorthogonal tothesubspace R’.T Thegeometric meaning oftheexpansion (21)isillustrated inFigure l(a). Note that theterms intheexpansion (21) areingeneral no longer orthogonal. Theproblem isnow easily reduced totheproblem ofSec. 8.51. Infact, ifweﬁxanyvector inthehyperplane R”andsubtract itfrom both sides of (21), weobtain theproblem ofrepresenting thevectorf -—f, asasum oftwo vectors g-—f0 andh,ofwhich theﬁrst belongs tothesubspace R’andthe second isorthogonal toR’(seeFigure l(b)). Bytheresult ofSec. 8.51, such arepresentation exists. Therefore therepresentation (21) also exists. It TSaying thatgbelongs tothehyperplane R‘means geometrically thattheendpoint of glieinthehyperplane R‘,while itsinitial point is,asusual, attheorigin ofcoordinates. One must notimagine thatthewhole vector gliesinthehyperplane R”! 226 EucLiDEAN SPACES CHAP. 8 I7 R‘ I/f /¢ I / ' 9 Z’ FIGURE 1 remains only toprove theuniqueness oftherepresentation (21). Ifthere were twosuch representations f=g1+h1:g2+h2, then wewould have 0=(81‘C82)Jr(hl‘Ch2), where g1-—g3belongs tothesubspace R’andhl-—h2isorthogonal toR’. Itfollows thatg1-—g3=hl-—hz=0,asrequired. 8.6.TheOrthogonalization Theorem 8.61. Thefollowing theorem isoffundamental importance inconstructing orthogonal systems inaEuclidean space: THEOREM (Orthogonalization theorem). Let xl,x2,...,xk,...bea ﬁnite orinﬁnite sequence ofvectors inaEuclidean space RandletL=9 k L(x1, x2,...,xk)bethelinear manifold spanned bytheﬁrst kofthese vectors. Then there exists asystem ofvectors yl,yz,...,yk,...such that 1)The linear manifold L;=L(y1,y2, ...,y,,) spanned bythevectors yl,yz,...,ykcoincides with thelinear manifold Lkforevery positive integer k;2)The . ...vector y,,+1 ISorthogonal toLkforevery positive integer k. Proof Wewill prove thetheorem byinduction, i.e., assuming that k vectors y1,y3, ...,y,,have been constructed which satisfy theconditions sEc.8.6 THEORTHOGONALIZATION THEOREM Z27 ofthetheorem, wewillconstruct avector yk,1such that thevectors yl,y2, ...,y,,,y,,+1 also satisfy theconditions ofthetheorem. First letyl=x1. Then thecondition L1: L1isobviously satisﬁed. The subspace Lkis ﬁnite-dimensional, andhence bySecl 8.51 there exists anexpansion xk-H =gk+hkv (22) where g,,isanelement ofLkandhkisorthogonal toLk.Setting y,,+1 =h,,, wenow verify that theconditions ofthetheorem aresatisﬁed forthischoice ofy,,+1. Bytheinduction hypothesis, thesubspace L,,contains thevectors y1,y2, ...,y,,,and hence thelarger subspace L,,+1 also contains these vectors. Moreover, itfollows from (22) that L3,, contains thevector h,,= y,,+1. Therefore thesubspace LH1 contains allthevectors yl,y2,...,yk,y,,+1, andhence also contains thelinear manifold L;+1spanned bythese vectors. Conversely, thesubspace L22, contains thevectors xl,x2,...,xk,and moreover by(22), LLH contains thevector x,,+1 aswell. Itfollows that L;+1contains thewhole subspace LH1. Therefore L;+1=Lk+1,andtheﬁrst assertion ofthetheorem isproved. The second assertion isanobvious consequence oftheconstruction ofthevector y,,+1 =h,,.This completes the induction, thereby proving thetheorem. I 8.62. lnthepresent case, theinequality (20)takes theform 0<lYk+1l <lxk+1l- (23) Asshown inSec. 8.52, theequality |y,,+1| :0means that thevector x,,+1 belongs tothesubspace L,,,and istherefore alinear combination ofthe vectors xl,x2,...,xk.The opposite equality |y,,+1| _=|x,,+1| means that thevector x,,+1 isorthogonal to_the subspace L,,,andhence isorthogonal to each ofthevectors xl,x2,...,xk. 8.63. Remark. Every system ofvectors 21,z2,...,22.,...satisfying the conditions oftheorthogonalization theorem coincides towithin numerical factors with thesystem y1,y2, ...,yk,...constructed intheproof ofthe theorem. Infact, thevector 22+, must belong tothesubspace LH1, and at thesame time 22+, must beorthogonal tothesubspace L,,.The ﬁrstofthese conditions implies theexistence ofanexpansion Zk+1 =ciyi Ci‘92)/2 Ci‘'''Ci‘ckyk Ci‘ck+1.yk+1 :)7»Ci‘¢’k+1)’k+1» where y,1:clyl+c2y2 -I—---+c,,yk 6L,,andc,,,1y,,+1 isorthogonal toLk. Thesecond condition implies thaty,,=0andhence that Z1.-1:61.41)’).--1. asrequired. 228 EucL1DEAN SPACES CHAP. 8 *8.64. Legendre polynomials. Suppose weapply theorthogonalization theorem tothesystem offunctions x0(t)=l,x1(t) =t,...,x,,(t)=t",... intheEuclidean space R2(—l ,l).Then thesubspace Lk=L(l, t,...,tl‘) isobviously thesetofallpolynomials intofdegree n<k.The func- tions x0(t), x1(t), ...,x,,(t) are linearly independent (see Sec. 2.22d), and hence thefunctions y0(t), y1(t), ...obtained bytheorthogonalization method areallnonzero, bySec.8.62. Byitsvery construction, y,,(t) must bea polynomial intofdegree k.Inparticular, direct calculation bytheorthog- onalization method gives J/0(1)=1.J/1(1)=1.)/2(1)=1“—ii‘-.J/3(1)=1”—%t.---- These polynomials were introduced in1785 bytheFrench mathematician Legendre, inconnection with certain problems ofpotential theory. The general formula fortheLegendre polynomials wasfound byRodrigues in l8l4, who showed that thepolynomial y,,(t) isgiven by p..(r>=i;[(12-1>"1(ti=0.1.2.---> (24)dt towithin anumerical factor. Wenow prove thisformula, using theremark ofSec.8.63,i.e.,wewillshow thatthepolynomial p,,(t) satisﬁes theconditions oftheorthogonalization theorem, whence itwillfollows from theremark inquestion thatp,,(t) must equal c,,y,,(t) forevery n,asrequired. a.The linear manifold spanned bythevectors p0(t), p1(t), ...,p,,(t) coincides with thesetofallpolynomials ofdegree nogreater than n.Infact, itisobvious from (24)that thepolynomial p,,(t) isclearly apolynomial int ofdegree k.Inparticular, P0(7) =boo, I710) =amCl‘911', ”*‘f1.C.f‘%".T.?%:ff.?*?" <1» /71.-(7) =ak0Cl‘aklt Cl‘'''Cl‘akktkr p'n(t) =a'n0 +a'n1t_l_ iii+anktk +'II+aunt"? where theleading coeﬂicients a,,,,,an, ...,am,arenonzero. Thus allthe polynomials p0(t),p1(t),...,p,,(t) areelements ofthe linear manifold sEc.8.6 THEORTI-IOGONALIZATION THEOREM 229 spanned bythefunctions l,t,...,t",which isobviously justthesetL,,of allpolynomials intofdegree nogreater than n.Conversely, thefunctions l,t,...,t"can beexpressed aslinear combinations ofp0(t),p1(t), ..., p,,(t), since thematrix ofthelinear relations (25) hasthenonvanishing determinant aman ---a,,,,. Hence thelinear manifold L(p0(t),p1(t), ...,p,,(t)) coincides with thelinear manifold L(l, t,...,t")andtherefore coincides with thesetL,,,asrequired. b.Thevector p,,(t) isorthogonal tothesubspace L,,_1. Itissuﬂicient to verify that thepolynomial p,,(t) isorthogonal inthesense ofthespace R2(—l ,l)tothefunctions 1,t,...,t"C1. Toshow this, weusetheformula forintegration byparts, familiar from elementary calculus, which inthe case ofpolynomials involves derivatives ofthetype considered inSec.6.73c from apurely algebraic point ofview. Inparticular, thederivatives ofthe polynomial (F2*1)" =(I—l)"(t+1)" oforders 0,1,...,n—lvanish fort=i1.T Thus, calculating thescalar product oft"andp,,(t) fork<nandintegrating byparts, weobtain k __ +1k2___ n(rt)(I.P..(t)) ~j I[(1 1)Idf-1 ___ k 2_ 'n(11-1) +1 +1 k-1 2 'n(11-1)—tl(t 1)] -kI[(1—1)] dt. -1 —1 where theﬁrst term ontheright vanishes. Integrating thesecond term by parts again, and continuing thisprocess until theexponent oftbecomes zero, weget (#2p..(t>>=—kt’"“i(r’ —1>"1‘"-2’ +1+/<(/<—1>l+1"CW-1>"1‘"‘*’dr-1 -1 =___= I-bl [O2 _1)n](n—k)dt 1 +1 :ikj[(t2 _1)n](n—k—1) Z0’ -1 i.e.,p,,(t) isorthogonal toL,,_1, asasserted. Thus, ﬁnally, wehave proved that forevery nthepolynomial y,,(t) is thesame asthepolynomial p,,(t) =[(t2—1)"]]"I, except foranumerical factor. TCf.formula (21), p.163. 230 EucL1DEAN SPACES CHAP. 8 Wenow calculate p,,(l), byapplying theformula forn-fold diﬁerentiation ofaproduct tothefunction (t2—1)"=(t+l)"(t -—1)". Theresult is 2.0)=10+1>"(r—1>"1‘"’ =(1+1)"l(t-1>"1‘"*+cite+1)”]’l(r—1>"1‘"-"+--- =(t +l)"n! +C{’n(t +l)”“n(n —1)---2(t—1)+---, where C};=n!/k!(n —k)!. Thesubstitution t:lmakes alltheterms ofthis sum vanish from thesecond term on,andweget pn(l) =2"n!. For numerical purposes, itisconvenient tomake thevalues ofour orthogonal functions equal lfort=1. Toachieve this, weneed only multiply p,,(t) bythefactor l/2"n!. Infact, itisactually these normalized polynomials which arecalled theLegendre polynomials, i.e., theLegendre polynomial ofdegree n,denoted byP,,(t), isgiven bytheformula 1 2 1)P,,i=—— t~1" ". ()2,"!l( )I 8.7.TheGram Determinant 8.71. ByaGram determinant ismeant adeterminant oftheform (X1, X1) (X1, X2) '''(X1, Xk) )=(X2, X1) (X2, X2) '''(X2, Xk) G(-X17 X27 'I'7xk (Xk, X1) (Xk) X2) '''(Xk, Xk) where xl,x2,...,xkarearbitrary vectors ofaEuclidean space R.lnSec. 7.96wesawthatthisdeterminant ispositive inthecaseoflinearly independent vectors xl,x2,...,xk.Tocalculate thevalue ofG(x1, x2,...,xk),weapply theorthogonalization process tothevectors xl,x2,...,xk.Thus letyl=x1 andsuppose thevector Y2:°‘1Y1 Cl‘X2 isorthogonal toyl.Replacing thevector x1byyleverywhere inthedeter- minant G(x1, x2,...,xk),wemultiply theﬁrstcolumn ofG(x1, x3,...,xk) byax,(associating oi,with thesecond factors ofthescalar products) andadd ittothesecond column. Then wemultiply theﬁrstrow ofthedeterminant byax,(associating ax,with theﬁrstfactors ofthescalar products) andaddit sEc. 8-7 THEGRAM DETERMINANT 23] tothesecond row. Asaresult, thevector y2appears atevery place inthe determinant where x3appeared formerly. Next let Y3=I51)/1 Cl‘I52)/2 Cl‘Xs beorthogonal toylandy2.Multiply theﬁrstcolumn byI51andthesecond column byI52,andaddthem tothethird column. Then carry outthesame operations ontherows. Asaresult, x3isreplaced byy3everywhere in G(x1, x2,...,xk).Wecancontinue thisprocess until wearrive atthelast column (and row). Since these operations donotchange thevalue ofthe determinant, weﬁnally obtain (YDY1) 0 0 0 (J/2.)/2) "-0 G(x1, x2,...,xk)= ' ' ' (26) 0 0 '''(ykr yk) =(Y1,)’1)()’2» Y2)'''()/toYk)- Moreover, bytheresult ofSec.8.62, wehave theinequality 0<G(X1» X2,---7Xk)<(X1, X1)(X2» X2)'''(Xk» Xk)- (27) Next weexamine theconditions under which thequantity G(x1, x2,...,xk) can take thevalues 0or(x1,x1)(x3, x2)---(xk,xk). Itfollows from the form (26)oftheGram determinant that itvanishes ifandonly ifoneofthe vectors yl,y3,...,y,,vanishm. But according toSec. 8.62, this implies that thevectors xl,x2,...,xkarelinearly dependent. Moreover, according to(26) andSec. 8.62, thesecond equality sign holds intheinequality (27) only inthecase where thevectors xl,x3,...,xkarealready orthogonal. Thus wehave proved thefollowing THEOREM. TheGram determinant ofthevectors xl,x2,...,xkvanishes if thevectors arelinearly dependent andispositive ifthey arelinearly independ- ent.Itequals theproduct ofthesquares ofthelengths ofthevectors xl, x3,...,xkifthey areorthogonal andislessthan thisquantity otherwise. 8.72. Thevolume ofak-dimensional hyperparallelepiped. Asiswellknown from elementary geometry, thearea ofaparallelogram equals theproduct of abase andthecorresponding altitude. Iftheparallelogram isdetermined bytwovectors x1andx3,then forthebase wecantake thelength ofthe vector x1andforthealtitude wecantake thelength oftheperpendicular 232 EucL1DEAN SPACES CHAP. 8 dropped from theendofthevector x2onto thelinecontaining thevector x1. Similarly, thevolume oftheparallelepiped determined bythevectors xl,x2 and x3equals theproduct ofthearea ofabase and thecorresponding altitude; forthearea ofthebase wechoose thearea oftheparallelogram determined bythevectors x1andx2,andforthealtitude wetake thelength oftheperpendicular dropped from theendofthevector x3onto theplane ofthevectors x1andx2. These considerations make thefollowing avery natural inductive deﬁ- nition ofthevolume ofak-dimensional hyperparallelepiped inaEuclidean space: Given asystem ofvectors xl,x2,...,x3inaEuclidean space R,let h,denote theperpendicular dropped from theendofthevector x,+1onto the subspace L(x1,x2,...,x,) (j=l,2,...,k-—l), andintroduce thefollowing notation: V1=]x1| (aone-dimensional volume, i.e., thelength ofthe vector x1), V2=V1|h1| (atwo-dimensional volume, i.e., thearea ofthe parallelogram determined bythevectors xl,x2), V3=V2|h2] (athree-dimensional volume, i.e.,thevolume ofthe parallelepiped determined bythevectors xl,x2,x3), V3=V3_1 ]h3_1| (ak-dimensional volume, i.e., thevolume ofthe hyperparalleliped determined bythevectors xl,x2, ...,x3). Obviously thevolume V3canbewritten intheform VkEV[x17 X2,---7Xkl:lX1llhll'''lhk-1l- Using equation (26), wecanexpress thequantity V3interms ofthevectors xl,x2,...,x3asfollows: (X1,X1) (X1,X2) '''(X1, Xk) 2 (X2, X1) (X2,X2) '''(X2, Xk)V3= . (xki X1) (xkt X2) '''(xki Xk) Thus theG_ram determinant ofthekvectors xl,x2,...,x3equals thesquare ofthevolume ofthek-dimensional hyperparallelepiped determined bythese vectors. 8.73.Let 5]" (j=l,2,...,k;i=l,2,...,n) SEC.8-7 THEGRAM DETERMINANT 233 bethecomponents ofthevector x,-with respect toanorthonormal basis el,e2,...,e,,.Expressing thescalar products interms ofthecomponents ofthevectors involved, weobtain thefollowing formula forV3: _§§1>g§1> _j_..._j_Eqligqlt ...alright +...+£21152.) V2 anal) _j_ ...+ Eifbzilll ...Eiﬁzgkl + ...+ Eifzlnk) k Eiktgqlt +..._j_gljtgglt ...Eqktgikt +...+Eqktzqi Wenow useanargument similar tothat used inSec.4.54. Every column ofthedeterminant just written isthesum ofn“elementary columns” with elements oftheform E§"IE§.°‘I, where theindices atand iareﬁxed ineach elementary column, while jranges from 1tok.Therefore thewhole deter- minant'equals thesum ofn"“elementary determinants” consisting only of elementary columns. Ineach elementary column thefactor Eff‘)isconstant andhence canbefactored outoftheelementary determinant. Asaresult, each elementary determinant takes theform Elf’E]-1‘ E51’ ~¢2) --(,2) ___ <2) £21723) ... ‘:11 E": £17. 7 2:1"ti-1"---at-1:’ where i1",i2,...,i3arenumbers from 1ton.Ifsome ofthese numbers are thesame, then thecorresponding elementary determinant obviously vanishes. Thus weneed only consider thecase where i1,i2,...,i3arealldiﬁerent. In theentire sum wegroup together those terms oftheform (28) which have thesame indices i1,i2,...,i3butarranged indiﬁerent orders. Let M2[j1vj22 '--vjkl denote thesum ofallsuch terms, where j1,j2, ...,j3aretheindices i1,i2, ...,i3rearranged inincreasing order. Anargument similar tothat used in Sec.4.54 then leads tothefollowing result: Inthen><kmatrix ||.£§"’|| (i=l,2,...,n;j=l,2,...,k), thequantity M2[j1,j2, ...,j3]isthesquare oftheminor oforder kformed from thecolumns ofthismatrix with indices j1,j2, ...,j3.The sum ofall theterms (28)equals thesum ofthesquares ofalltheminors oforder kof thematrix ||E§"I||. Thus thesquare ofthevolume ofthek-dimensional hyperparallelepiped determined bythevectors xl,x2,...,x3equals thesum ofthesquares ofalltheminors oforder kinthematrix consisting ofthe components ofthevectors xl,x2,...,x3with respect toanyorthonormal basis el,e2,...,e,,. 234 EucL1DEAN SPACES CHAP. 8 8.74. Inthecase k:n,thematrix ||EIFIII hasonly oneminor oforder k, equal tothedeterminant ofthematrix. Hence thevolume ofthen-dimensional hyperparallelepiped determi:1ed bythevectors xl,x2,...,x,,equals the absolute value ofthedeterminant formed from thecomponents ofthevectors xl,x3,...,x,,with respect toanyorthonormal basis. 8.75. Hadamard’s inequality. Using theresults ofthepreceding section, wecanobtain animportant estimate fortheabsolute value ofanarbitrary determinant E11 £12 '''Elk £21 £22 '''E21:1): Zkl Ek2 Ekk oforder k.Ifweregard thenumbers 5,2,Z,-2,...,E,-3(i:1,2,... ,k)as thecomponents ofavector x,-with respect toanorthonormal basis ina k-dimensional Euclidean space, then theresult ofSec. 8.74 allows usto interpret theabsolute value ofthedeterminant Dasthevolume ofthe k-dimensional hyperparallelepiped determined bythevectors xl,x2,...,x3. Then, using theexpression forthisvolume interms oftheGram determinant, wehave D2=G(x1, x3,...,x3). Applying Theorem 8.71, weobtain 2:’.-...l\/1*.--{*13:“ D2<(X1. X1)(X2. X2)'''(Xi.-. Xk)= aninequality known asHadamard’s inequality. Moreover, wenote that according toTheorem 8.71, theequality holds ifand only ifthevectors xl,x2,...,x3arepairwise orthogonal. The geometric meaning ofHadamard’s inequality isclear, i.e., the volume ofahyperparallele/epiped does notexceed theproduct ofthelengths ofitssides, anditequals thisproduct andonly'if itssides areorthogonal. 8.8.Incompatible Systems andtheMethod ofLeast Squares 8.81. Suppose wearegiven anincompatible system oflinear equations a11X1 ‘l‘a12X2 ‘l‘'''-lralmxm =b1. a21X1 +a22X2 air'''‘l‘a2mxm =b2. (29) a,,1x1 +a,,3x2 —I----+a,,,,,x,,, =b,,. sEc.8.8 INCOMPATIBLE SYSTEMS AND THEMETHOD orLEAsT sQuAREs 235 Since thesystem isincompatible, itcannot besolved, i.e.,wecannot ﬁnd numbers cl,c2,...,c,,,which satisfy alltheequations ofthesystem when substituted fortheunknowns x1,x2, ...,xm.Thus ifwesubstitute the numbers E1,E2,...,Emfortheunknowns xl,x2,...,x,,,intheleft-hand sideofthesystem (29), weobtain numbers Y1,Y2,...,Y,which diﬁer from thenumbers bl,b2,...,b,,.This suggests thefollowing problem: Given real numbers a,3and b3(j=l,...,m;k=1,...,n)ﬁnd thenumbers E1,E2,...,Emwhich when substituted into(29)give thenumbers Y1,Y2,..., Y,with thesmallest possible mean square deviation 8':=§1(Yt—1».->2 <30) from thenumbers bl,b2,...,b,,,andﬁnd thecorresponding minimum value ofS2. Anexample ofasituation where thisproblem arises inpractice isthe following: Suppose wewant todetermine thecoeﬂicients E,inthelinear relation b:E1a1 Cl‘E2a2 ‘l‘"'‘l‘Errtam connecting thequantity bandthequantities al,a2,...,am,given thermults ofmeasurements ofthea,-(j=l,2,...,m)andthecorresponding values ofb.Iftheithmeasurement gives thevalue a,-,»forthequantity a,»andthe value b,-forthequantity b,then clearly Elail +E2a12 Cl‘'''+Ernaim =bi‘ (31) Thus nmeasurements lead toasystem ofnequations oftheform (31), i.e., asystem oftheform (29). Asaresult ofunavoidable measurement errors, thissystem willgenerally beincompatible, andthen theproblem ofﬁnding thecoeﬂicients Z1,Z2,...,Emdom notreduce totheproblem ofsolving thesystem (29). This suggests determining thecoeﬂicients E,insuch away that every equation isatleast approximately valid andthetotal error isas small aspossible. Ifwetake asameasure oftheerror themean square deviation ofthequantities m Y1=Z1aijzi from theknown quantities b,-,i.e.,ifwetake formula (30) asameasure of theerror, then wearrive attheproblem formulated atthebeginning ofthis section. Moreover, inthiscase, itisalso useful toknow thequantity S2, since thishelps toestimate theaccuracy ofthemeasurements. 8.82. Wecanimmediately solve theproblem juststated, ifweinterpret itgeometrically intherealspace R3.Consider themvectors al,a2,...,am 236 EucL1DEAN SPACES gr-nip, s whose components form thecolumns ofthesystem (29), i.e., a1=(a117 a217 ---7a'!l1)7 a2=(a127 a227 ---7a'!l2)7 am=(alm, a2m, ...,am"). Forming thelinear combinations Elal+E2a2_]_..._I_imam, weobtain the vector YI(Y1,Y2,...,Y,,). Our problem istodetermine thenumbers E1,E2,...,Eminsuch away that thevector Yhasthesmallest possible deviation innorm from thegiven vector b=(bl,b2,...,b,,). Now theset ofalllinear combinations ofthevectors al,a2,...,amforms asubspace L:L(a1, a2,...,am), andtheprojection ofthevector bonto thesubspace Listhevector inLwhich istheclosest tob.Therefore thenumbers E1,E2, ...,Emmust bechosen insuch away that thelinear combination Elal Cl‘E2a2 Cl‘'''Cl‘Emam reduces totheprojection ofbonto L.But, asweknow, thesolution ofthis problem isgiven bythelastequation inSec. 8.53, i.e., (a17 a1) '''(aj—17 a1) (b7a1) (a7'+17 a1) '''(am7 a1) .1ﬁrs: 1.: I’ (ah am) I''(a;i—17 am) (b7 am) (bail-P17 am) -.-(awn am) where DistheGram determinant G(a1, a2,...,am). 8.83. The results ofSec. 8.72 also allow ustoevaluate thedeviation S itself. Infact, Sisjust thealtitude ofthe(m+l)-dimensional hyper- parallelepiped determined bythevectors al,a2,...,am,b,and hence is equal totheratio ofvolumes V[a17 a27 '--7arm V[a17 a27 '-'7aml - Using theGram determinant towrite each ofthese volumes, weﬁnally obtain 82:G(a17 a2»---7am7b) G(a1, a2,...,am) Thus theproblem posed inSec. 8.81 isnow completely solved. sEc. 8.9 ADJOINT OPERATORS AND 1s0METRv 237 8.84. Innumerical analysis thefollowing problem isoften encountered (interpolation with theleast mean square error): Given afunction ﬂ,(t) deﬁned intheinterval a<t<b,ﬁnd thepolynomial P(t) ofdegree k(k<n)for which themean square deviation from thefunction ﬂ,(t), deﬁned by 82/...P)=$011.0.-) —Po.-)1: isthesmallest. Here to,t1,...,t,,arecertain ﬁxed points oftheinterval a<t<b.Using geometric considerations, M.A.Krasnosyelski hasgiven thefollowing simple solution oftheproblem: Introduce aEuclidean space Rconsisting offunctions f(t)considered only atthepoints to,t1,...,t,,, anddeﬁne thescalar product by <1.g)=§0/<r.-)g<r.-)- Then theproblem reduces toﬁnding theprojection ofthevector ﬂ,(t) onto the subspace ofallpolynomials ofdegree notexceeding k.The coeﬂicients of thedesired polynomial P(t) :E0+E1;+...+Eh’).- aregiven bythesame formulas asintheproblem analyzed previously, i.e., (1.1)(1.1) (H-1.1)<1...1)(W51) ---(:51) (1.1) (171) ~-~(/C‘.I)(f...t)0"“.I)---(t"7t) l . . . . . E?‘ZB ‘ 7 <14") (t.1”) (r’C‘.1")(fo71")0"“.1")---(/21") where DistheGram determinant G(l,t,...,t").Theleast square deviation itself isgiven bytheformula k 82’P=G(l,t,...,t,P)_ U0)G(1,t,...,t") 8.9.Adjoint Operators andIsometry 8.91. Adjoint operators with respect totheform (x,y).Wenowapply theresults ofSec. 7.6ontheconnection between linear operators and bilinear forms tothecase where theﬁxed form (x,y)isthescalar product of 238 EucL1DEAN SPACES C]-[A9, s thevectors xandy.LetAandBbelinear operators inaEuclidean space R,,, andusetheformulas A(X7y)=(AX.)/)7 B(X.y)=(X.By) (32) toconstruct bilinear forms A(x, y)andB(x, y).Since anyorthogonal basis isacanonical basis oftheform (x,y),andsince thecanonical coeﬂicients of(x,y) allequal linanysuch basis, itfollows from Sec. 7.61 that the matrix ||a,-3|] oftheform A(x, y)inanyorthonormal basis coincides with thematrix ||a3”|| oftheoperator A,while thematrix ||b,-3|] oftheform B(x, y)isthetranspose ofthematrix ]|bj"’|| oftheoperator B.Conversely, given bilinear forms A(x, y)andB(x, y)inthespace R,,,there exist unique linear operators AandBsuch that theformulas (32) hold (see Sec. 7.62). Moreover, applying Theorem 7.63 totheform (x,y),wegetthefollowing THEOREM. Given anylinear operator Aacting inann-dimensional Euclidean space R2,there exists aunique linear operator A’(theadjoint ofA)acting in Rsuch that" (Ax.y)=(x.Aw) forarbitrary x,y€R,,. Thematrix oftheoperator A’inanyorthonormal basis ofthespace R,isthetranspose ofthematrix oftheoperator A. 8.92. Using theoperation oftaking theadjoint inaEuclidean space, we now introduce thefollowing classes ofoperators: a.Symmetric operators, deﬁned bytherelation A’=A. Asymmetric operator ischaracterized bythefact that transposition does notchange itsmatrix inanyorthonormal basis. b.Antisymmetric operators, deﬁned bytherelation A’=—A. Anantisymmetric operator ischaracterized bythefact that transposition changes thesign ofitsmatrix inanyorthonormal basis. c.Normal operators, deﬁned bytherelation A’A :AA’. The class ofnormal operators obviously contains theclass ofsymmetric operators and theclass ofantisymmetric operators. The study ofthese classes ofoperators willbepursued inSecs. 9.3—9.4. 8.93. We now formulate theresults ofSecs. 7.73-7.76 oninvariant operators forthecase ofaEuclidean space R,,.Consider alinear invertible SEQ3-9 ADJOINT OPERATORS AND ISOMETRY 239 mapping y=Qxofthespace Rnintoitself which does notchange thescalar P’°°'“°" <Q><.Qy)I(x.y)- Amapping ofthiskind, which inSec. 7.73 wassaid tobeinvariant with respect totheform (x,y),willnow becalled isometric. Thus anisometric operator Qischaracterized bytherelation Q’QIE (cf.formula (33), p.201), where Eistheunitoperator andQ’istheoperator adjoint toQwith respect totheform (x,y),i.e.,theoperator adjoint toQ inthesense ofSec. 8.91. The inverse Q_1IQ’ofanisometric operator isitself isometric, andsoistheproduct oftwoisometric operators (seeSec. 7.74). According toSec. 7.75, anisometric operator Qischaracterized bythe factthat itcarriw every orthonormal basis el,...,eninto another ortho- normal basis fl:Qel, ...,f,,:Qe". The matrix QIllqflll ofaniso- metric operator Qinanyorthonormal basis iscalled anorthogonal matrix. Anorthonormal matrix ischaracterized bytheconditions (35), p.202, which inthepresent case take theform iiH)(Hpi1ifj==k, 1:111)" ‘I1:'0ifjik’ orbytheconditions (35’), p.202, which take theform 1- 1if1=ITI (i) (mi 7‘I.4- : 1:21 pL i0 ifjqé nt, i.e.,thesum ofthesquares oftheelements ofanyrow(orcolumn) equals l, while thesum oftheproducts ofthecorresponding elements oftwodifferent rows (orcolumns) equals 0. 8.94. Itfollows from therelation Q“1:Q’that theformulas flIql”e1+---+qi.“@,.. ..................... (33) ftIq§"’e1+---+qi."’e,, forthetransformation from oneorthonormal basis el,...,entoanother orthonormal basisfl, ...,f,,(such atransformation iscalled anorthogonal transformation) are“inverted” bytheformulas 81Iqi“f1+'-'+qi"’f,,. ..................... (34) 8,.Iq§.“fi+---+qL"’fn 240 EUCLIDEAN SPACES CHAP. 8 BySec.5.31, thecomponents 1],,ofavector xwith respect tothebasisfl, ..., f,,areexpressed interms ofthecomponents E,ofthesame vector withrespect tothebasis el,...,e,,bytheformulas mIqi"€1+---+qi."Em ...................... (35) inIqi"’E1+''-+qi!"Z.,, with inverse formulas E1=qinlll +'''+qimﬂn, ...................... (36) E.Iqi."m+"--+qiI"m- 8.95. Given m<nrows ofnumbers qf.“(i=1,...,n;j= 1,...,m) satisfying theconditions 5;(1')on [1ifJ:k’qtqt =i=1 0ifj¢k, consider theproblem ofﬁnding n-—mmore rows ofnumbers qiil(j= m+1,...,n)such thatthen><nmatrix ||q§"ll (i,j=l,...,n)isorthog- onal. This problem iseasily solved byusing ageometrical argument. Suppose thegiven rows qf.”areinterpreted ascomponents ofmvectors ina Euclidean space R,,with scalar product 1| ((511 '''1E'n)9(7i19 ~-'1 =ZEiyii i=1 (recall Example 8.22b). Then ourproblem consists ofaugmenting mgiven orthonormal vectors ql,...,q,,,with further vectors tomake anorthonormal basis forthespace R,,.With thisgeometrical interpretation, theproblem is obviously solvable. For example, wecan augment q1,. ..,q,,, with any other vectors q,,,+1, ...,q,,such that theresulting system ofnvectors is linearly independent, andthenuseTheorem 8.61tomake thewhole system of nvectors orthonormal. 8.96. Wenowconsider some further properties ofsymmetric operators. a.Ifthesubspace R’CRisinvariant under theoperator A,then, bySec. 7.65, theorthogonal complement ofR’isinvariant under theadjoint operator A’.Therefore, inthecaseofasymmetric operator A,ifthesubspace R’ isinvariant under A,then soistheorthogonal complement ofR’. b.THEOREM. Every symmetric operator intheplane (n=2)hasan eigenvector. PROBLEMS Z4| Proof. Inthiscase, theequation determining theeigenvectors isjust anT7‘ a12 _0 a21 a22*7\ Thediscriminant ofthisquadratic equation is (an+a22)2 *4(a11a22 *a21a12) =(a11 "‘a22)2 +4ai2 >0, andhence hasrealroots. | i c.From these considerations andthefactthat every operator inareal space hasaninvariant plane (seeSec. 6.66), itfollows that every symmetric operator inthespace R,,hasanorthogonal basis consisting ofeigenvectors. InSec.9.45 wewilldeduce thisresult inamore general way, without recourse totherealJordan canonical form. PROBLEMS 1.Suppose wedeﬁne thescalar product oftwovectors ofthespace V3asthe product ofthelengths ofthevectors. Istheresulting space Euclidean? 2.Answer thesame question ifthescalar product isdeﬁned astheproduct of thelengths ofthevectors andthecube ofthecosine oftheangle between them. 3.Answer thesame question ifthescalar product isdeﬁned astwice theusual scalar product. 4.Find theangle between opposite edges ofaregular tetrahedron. 5.Find theangles ofthe“triangle” formed inthespace R2(——1, 1)bythe vectors x1(t)=1,x2(t)=r,x3(t)=1—r. 6.Write thetriangle inequalities inthespace R2(a, b). 7.Find thecosines oftheangles between thelineE1=E2- I5,,andthe coordinate axesinthespace R,,. 8.Inthespace R,expand thevector fasthesum oftwovectors, avector g lying inthelinear manifold spamed bythevectors biandavector horthogonal tothissubspace: a)f=(5.2. -2.2). blI(2,1.1. I1). b2I(1.1.3.0); b)fI(—3,5,9,3), blI(1,1,1,1). beI(2,—1.1.1),b3=(2,-7,-1,-1). 9.Prove thatofallthevectors inthesubspace R’,thevector gofSec.8.51 (theprojection offonto R’)makes thesmallest angle with 10.Show thatifthevector gointhespace R’isorthogonal tog(theprojection offonto R’),thengoisorthogonal tofitself. 242 sucunznu SPACES CHAP. 8 11.Show thattheperpendicular dropped from theorigin ofcoordinates onto a hyperplane Hhasthesmallest length ofallthevectors joining theorigin withH. 12.Given thesystem ofvectors x1Ii,x2I2i,x3I3i,x4I4iI2j,x5I Ii+10j,x6Ii+j+Skinthespace V3withbasis i,j,k,construct thevectors y1,y2, ...,yeﬁguring intheorthogonalization theorem. 13.Using themethod oftheorthogonalization theorem, construct anorthogonal basis inthethree-dimensional subspace ofthespace R4spanned bythevectors (1,2.1,3).(4.1.1,1),(3.1,1,0)- 14.Given twosubspaces R’andR”ofaEuclidean space R,letm(R’, R”)denote themaximum length oftheperpendiculars dropped onto R”from theends of theunitvectors e’eR’,anddeﬁne thequantity m(R”, R’)similarly. Then the quantity 6Imax {m(R’, R”), m(R", R')} iscalled thespread ofthesubspaces R’andR”.Show thatthesubspaces R’and R”have thesame dimension if6<1l.(M.A.Krasnosyelski andM.G.Krein) 15.Find theleading coefﬁcient A,,oftheLegendre polynomial P,,(r). 16.Show thatP,,(r) isaneven function foreven nandanoddfunction forodd n.Inparticular, ﬁndP,,(I1). 17.Show thatifthepolynomial rP,,_1(r) isexpanded interms oftheLegendre polynomials, sothat rP,,_1(r) Ia0P0(t) +a1P1(t) +---+a,,P,,(r), then thecoefﬁcients ao,al,...,a,,_3anda,,_1 arezero. 18.Find thecoeﬂﬁcients a,,_2anda,,oftheexpansion ofthepolynomial rP,,_1(r) given inthepreceding problem, thereby obtaining therecurrence formula nP,,(t) I(ZnI1)rP,,_1(r) -(n-1)P,,__.,(r). 19.Find thepolynomial Q(r)I1"+b1r"I1 +---+b,._1t+b., forwhich theintegral iiQ2(t)i1t hasthesmallest value. 71 20.Find thenorm oftheLegendre polynomial P,,(r). 21.LetAbeanylinear operator acting inann-dimensional Euclidean space R,,. Show thattheratio V[Ax1,Ax2, ...,Axn] k(A) I———————————V[x1, x2,...,x,,] isaconstant (i.e., isindependent ofthechoice ofthevectors xl,x2,...,x,,), andﬁndthevalue ofk(A) (the“distortion coefﬁcient”). PROBLEMS 243 22.Show thatk(AB) Ik(A)k(B) foranytwolinear operators AandB. 23.Letxl,x2,...,xk,y,zbevectors inaEuclidean space R.Prove the inequality V[x1, x2,...,x;,,y, 2] V[x1, xi,...,xk,2] < . 3 Vlxu X2,---,Xx’)/l V[X1, X2,---,Xkl V (7) 24.Letxl,x2,...,x,,,bevectors inaEuclidean space R.Prove theinequality m . V[x1, x2,...,x,,,]<H{V[x1, ...,x,,_1, x,,+,, ...,x,,,]}1l""*1l. (38) k=1 What isthegeometric meaning ofthisinequality? 25(Continuation). Prove thefollowing inequalities, which Sharpen I-Iadamard’s inequality: V[x19 X2! '~~9xm] 17! <H{I/[X17 ---,xk—1, Xk+1, ---,xml}1l(m_1) k=1 m ll’ 1-2-1! -2 <~ {V[x1s '''9xk-19 xk-kl! '''9xl-1’ xl+1! '''7xml} (m )(m ) 1<k<l<m <...< H {I/[xsv X52’ __.,xsr]}1'2"'('!|-T)l(1n-1)(m-—2)"'1‘ l<.s1<$2<'"<s,€m <...<1<s11's£<m{V[xs1,x82]}1l(m—1) <11|x8|_ (M.K.Faguet) 26.If|a,-,,|<M,then detHamil <M"n"l2, byHadamard’s inequality. Show that thisestimate cam1ot beimproved for nI2"‘ 27.Show thatifN(A) andT(A) arethenullspace andrange, respectively, of theoperator A,then theorthogonal complements ofthese subspaces arethe range andnullspace, respectively, oftheadjoint operator A’. 28.LetAbeanorthogonal matrix. Show thatA,-,,Ia,-,,detAisthecofactor oftheelement a,-,,ofA. 29.Show thatthesumofthesquares ofalltheminors oforder kappearing in kﬁxed rows ofanorthogonal matrix equals 1.Show thatthesumoftheprod- uctsofalltheminors oforder kappearing inonegroup ofkrows with the corresponding minors inanother group ofkrows equals 0. 30.Alinear operator Qpreserves thelength oievery vector. Show thatQis isometric. 244 EUCLIDEAN SPACES CHAP. 8 31.Anoperator Awhich preserves theorthogonality ofanypairofvectors x andy,i.e.,such that (x,y) I0implies (Ax,Ay) I0,iscalled anisogonal operator. Isometric operators andsimilarity operators (AxIXxforevery x) areisogonal, andsoistheproduct ofanysimilarity operator andanyisometric operator. Show that every isogonal operator istheproduct of1similarity operator andanisometric operator. 32.LetQbealinear operator acting inann-dimensional Euclidean space R,,(n>3).Suppose Qdoes notchange theareaofanyparallelogram, sothat Vix,ylIV[Q-Y, Qy]- Show thatQisanisometric operator. 33.LetQbealinear operator acting inann-dimensional Euclidean space R,,, andsuppose Qdoes notchange thevolume ofanyk-dimensional hyperparallele- piped (k<n).Show thatQisisometric. (M.A.Krasnosyelski ) Comment. ForkIntheassertion ofProblem 33fails tobevalid, since thenevery operator QwithdetQIilwillsatisfy thecondition oftheproblem. 34.LetFI{xl,x2,...,xk}andGI{y1,y2, ...,yk}betwoﬁnite systems of vectors inaEuclidean space R,,.Show thatanecessary andsufﬁcient condition fortheexistence ofanisometric operator Qtaking every vector x,-intothe corresponding vector y,-(iIl,2,...,k)isthattherelations (X.-,x.»>=(y.-.y.~) our=1,2,_-..10 hold. 35(The angles between twosubspaces). LetR’andR”betwosubspaces ofa Euclidean space R.Lettheunitvector e’vary over theunit sphere ofthe subspace R’,andlettheunitvector e”vary (independently ofe’)over theunit sphere ofthesubspace R”.Forsome pairofvectors e’Ie1,e”Iea’,theangle between e’ande”achieves aminimum, which wedenote bycl.Illow lete’vary over itsunitsphere while remaining orthogonal toe{,andletevary over its unitsphere while remaining orthogonal toe1’.With these constraints, theangle between e’ande”achieves aminimum ck>clforsome paire’Ieé,e’Ie;'. Then let”e' vary over itsunitspherehwhile remaining plrthogolnal tpe{dan'd elé, ah_eevary oyer isuni_sperewiiremaining odogo aoe,a,ea” ndlt 't 'th '1 " rt nt n .n isway, wegeanew minimum ange c3>ckananew paire3anest.“ Continuing thisprocess, weobtain asetofangles cl,ck,...,ck,thenumber o which equals thesmaller ofthedimensions ofR’andR”.Theangles cl,ck,..., ckarecalled theangles between thesubspaces R’andR”.Prove thefollowing facts; a)Theangles cl,ck,...,ckareuniquely deﬁned anddonotdepend on thechoice ofthevectors e{,e1’,eé,eg,...ifthese vectors arenotuniquely deﬁned bytheconstruction; b)Theangles cl,ck,...,ckdetermine thesubspaces R’andR”towithin their spatial orientation, i.e.,ifthere aretwopairs ofsubspaces R’,R” andS’,S”such thattheangles between R'and R”arethesame asthose PROBLEMS 245 between S’andS”,then there exists anisometric operator which simul- taneously carries S’intoR’andS”intoR"; c)Given anypreassigned angles cl<ck<---<ck<tr/2,wecancon- struct apairofspaces R’andR”such that cl,ck,...,ckaretheangles between R’andR”. 36.Letyl,y2,...,y,,,betheprojections ofthevectors xl,x2;...,x,,,onto some subspace. Show thatthevolume ofthehyperparallelepiped determined bythe vectors y1,y2, ...,y,,,does notexceed thevolume ofthehyperparallelepiped determined bythevectors xl,x2,...,x,,,. 37(Continuation). InProblem 36suppose thatboth thevectors xl,x2,...,xm andthevectors y1,y2, ...,y,,,arelinearly independent. Show thattheformula V[Y1,)/'2, ---,)/ml IVlxp X2,---9xm] C05 ‘*1C05 ‘*2'''C05 mm holds, where <11,<12,...,amaretheangles between thesubspaces L1I L(x1, x2,...,x,,,)andL2IL(y1,_)/2, ...,y,,,) (seeProb. 35). 38.Asetofkvectors inaEuclidean space Rwillbecalled ak-vector, andwe willsaythattwok-vectors {xl,x2,...,xk}and{\y1,j/2, ...,yk}areequal if 1)Thevolume V[x1, x2,...,xk]equals thevolume V[y1,y2, ...,yk]; 2)Thelinear manifold L(x1, x2,...,xk)coincides with thelinear manifold L(,v1.y2. ---.yi); 3)Thesystems xl,x2,...,xkandy1,y2, ...,ykhave thesame orientation, i.e.,theoperator inthespace L(x1,x2,. ..,xk) carrying thesystem xl, x2,...,xkintothesystem yl,)/'2,...,ykhasapositive determinant. Show that ak-vector {x1,x2, ...,xk} inann-dimensional space R,,is uniquely determined ifweknow thevalues ofalltheminors oforder kofthe n><kmatrix iia§"|| (i=l,2,...,n;jI1,2,...,k) formed from thecomponents ofthevectors xl,x2,...,xkwith respect toany orthonormal basis el,e2,...,e,,ofthespace R,,. 39.Ifthek-vector {xl,x2,...,xk}equals thek-vector {_y1,yk,...,yk}(Prob. 38),show thattheminors oforder kofthematrix formed from thecomponents ofthevectors xl,x2,...,xkequal thecorresponding minors ofthematrix formed from thecomponents ofthevectors y1,y2, ...,yk. 40.Bytheangles b_etween twok-vectors {xl,x2,...,xk}and{J/1,)/'2,...,yk} wemean theangles between thesubspaces L,IL(x1, x2,...,xk)andL2I L(y1,y2, ...,yk)(seeProb. 35)subject, however, tothesupplementary con- dition thatthevectors el,e2,...,ekchosen inthesubspace L1(when con- structing theangles) have thesame orientation asthevectors xl,x2,...,xk (thiscondition plays aroleonlyinconstructing thelastvector ek),andsimilarly forthesubspace L2.Show thattheangles B1,B2,...,Bkbetween thek-vectors andtheangles <11,<12,...,izkbetween thecorresponding subspaces arecon- nected bythefollowing relations: a,~=[i,- (/<k), °‘k=l5k 01' °‘k="”51.-- 246 EUCLIDEAN smczs CI-IAP. s 41.Bythescalar product oftwok-vectors XI{xl,x2,...,xk}and YI {y1,yz,...,yk}, speciﬁed bythematrices XandYmade upofthecomponents ofthevectors x,»andy,withrespect tosome orthonormal basis ofthespace R,,, wemean thesumofalltheproducts oftheminors oforder kofthematrix Xwith thecorresponding minors ofthematrix Y.Show thatthisscalar product equals V[x1, x2,...,xk]V[y1,y2, ...,yk]cosB1cos52-'-cosBk, where B1,B2,...,Bkaretheangles between thek-vectors XandY. 42.Show thatthescalar product ofthetwok-vectors XI{x1,x2,. ..,xk} andYI{_y1,y2, ...,yk}canbewritten intheform (xv)/1) (xv)/'2) (xv)/k) {X Y}: (X21)/'1) (X21)/'2) (X21)/'k) _ (xkwyl) (xk!)/2) (Xk))/Ir) 43.Show thatifthepolynomial [P(t)]" isanamiihilating polynomial ofthe isometric operator A,thensoisthepolynomial P(t). chapter 9 UNITARY SPACES 9.l.Hermitian Forms 9.11. Anumerical function A(x, y)oftwoarguments xandyinacomplex space Ciscalled aHermitian bilinearform orsimply aHermitian form ifitis alinear form oftheﬁrst kind inxforevery ﬁxed value ofyandalinear form ofthesecond kind (Sec. 4.14) inyforevery ﬁxed value ofx.Inother words, A(x,y) issaid tobeaHermitian form inxandyifthefollowing conditions aresatisﬁed forarbitrary x,y,zinCandarbitrary complex ot:T A(x+Z.Y)=A(x,y) +A(Z»Y). A(Qx, Y)I°@A(x» J’). A(x.y+2)=A(x,y)+A(x.2). (1) A(x, oty)Ic2A(x, y). Using induction and(1),weeasily obtain thegeneral formula Ir m Irm _A(2c-Y.-.2 F5.-.Vi)=Zice.-A<x.. y.->. <2) i-I1 l=1 l=1 i=1 where xl,...,xk,y1, ...,y,,, arearbitrary vectors ofthespace Cand oak,...,oak,(-31,...,(-3,,arearbitrary complex numbers. TAsusual, theoverbar denotes thecomplex conjugate. 247 248 UNITARY SPACES CI-IAP. 9 9.12. Examples a.IfL1(x) isalinear form oftheﬁrstkind andL2(x) isalinear form of thesecond kind (Sec. 4.14), then A(x, y)IL1(x)L2(y) isaHermitian form. b.Anexample ofaHermitian form inann-dimensional space C,,with a ﬁxed basis el,e2,...,e,,isthefunction ;l\/l=,..l\/l=is A(x, Y)=__tkzmki (3) where 7| 7| XIZ51¢.-. yIimei. |'=1 lr=l arearbitrary vectors anda,-k(i,k=1,2,...,n)areﬁxed complex numbers. Infact, (3)isthegeneral representation ofaHermitian form inann- dimensional complex space. This isproved inthesame wayastheanalogous proposition forbilinear forms inaspace K,,(seeSec. 7.13). 9.13. AHermitian form A(x, y)issaid tobeHermitian—symmetric (or simply symmetric) if A(y.X)IA(x.y) (4) forarbitrary vectors xandy.Given asymmetric Hermitian form A(x, y) inann-dimensional complex space C,,,suppose weuse(3)towrite A(x, y) interms ofthecomponents ofthevectors xandywith respect tothebasis e1,..., e,,.Then an=A(ei! en)=A(ek! er)=dkir (5) i.e.,thematrix ||a,k|| oftheform A(x, y)inthebasis el,...,e,,iscarried into itself bytransposing thematrix andreplacing allitselements bytheir complex conjugates. Conversely, ifthecoeﬁicients ofaHermitian form A(x, y)satisfy thecondition (5),then A(x, y)issymmetric, since 1| _ 1| _ 1| II- A()C X)=Zainllizn =Zdi|iEi|"l|' I‘Z1ai..-5.11.» =A(x’ Y)-.= = .lr= ~.I-1- -.>1-1- Amatrix ||a,-k\| such that a,~k=ak,(i,k=l,...,n)will henceforth be called Hermitian-symmetric (orsimply Hermitian). 9.14. a.Suppose theHermitian form A(x, y)hasthematrix Am=||a,-k|| inthebasis el,...,e,,ofthespace Cand thematrix Am =||b,-k|| inthe basis fl,...,f,,,where therelation between thetwobases isgiven by f,I§p§"*e, (iI1,...,n). j=1 SEC.9-1 HERMITIAN roams 249 Then, reasoning asinSec.7.15, weﬁndthattherelation between thematrices AmandAmisgiven bytheformula Am=Pump’ (6) whereP =||p}"*|\ isthematrix ofthetransformation from thebasis el,...,e,, tothebasisfl, ...,f,,,andP*isthematrix obtained from Pbytransposing and then replacing elements bytheir complex conjugates. Writing P*= ||p;""'*||, wehave - 113""=i>.-"’(i,.iI1.---.1)- b.Just asinSec. 7.23, itfollows from (6)that therank ofthematrix AmoftheHermitian form A(x, y)isindependent ofthechoice ofthebasis {e}.The form A(x, y)issaid tobenonsingular ifitsrank (i.e., therank of thematrix Aminanybasis {e})equals thedimension nofthespace C,,.If theform A(x, y)isnonsingular, then, given anyvector xo#0, there isa vector yoeC,,such that A(x0, yo)ck0(cf.Sec. 7.l5c). 9.15. a.ByaHermitian quadratic form inacomplex space Cwemean thefunction ofonevariable xECobtained bychanging ytoxinany Hermitian bilinear form A(x, y).Itfollows from Sec. 9.l2b that inann- dimensional complex space C,,with basis el,...,e,,,aHermitian quadratic form canbeexpanded interms ofthecomponents E1,...,5,,ofthevector xbytheformula A(x, x)=Z11111555» (7) i.k=l with complex coefﬁcients a,-k.Conversely, afunction A(x, x)oftheform (7) istheHermitian quadratic form obtained bychanging ytoxintheHermitian bilinear form A(x, Y)=_;_1a1i1Ei7li=- b.IfaHermitian bilinear form A(x, y)issymmetric, sothat a,k=ak,-, then thecorresponding Hermitian quadratic form A(x, x)isalso said tobe symmetric. Asymmetric Hermitian quadratic form A(x, x)canonly take realvalues, since itfollows from (4)that A(x, x)=A(x, x). Unlike thesituation inSec. 7.22, there isaunique Hermitian bilinear form A(x, y)corresponding toagiven Hermitian quadratic form A(x, x). Infact, A(x+y.X+y)IA(x.X)+A(x.y)+A(y.X)+A(y.y). A(x +1’)/1 x+ =A(x>‘x) _1-A(x: +"A(Y» x)+ Y)- 250 UNITARY SPACES CI-IAP. 9 Multiplying theﬁrstequation byiandthen subtracting thesecond equation from theﬁrst, weeasily ﬁndthat 1 . . .A(x.y)I5[A(x+y,X+y)+1A(><+1y.x+11)] I [A(><.>0+A<y.y>1. sothatA(x, y)isuniquely determined interms ofthevalues A(x, x),A(y, y), A(x+y,x-1-y)and A(x +iy,x+iy)ofthegiven Hermitian quadratic form. IftheHermitian quadratic form A(x, x)hastherepresentation §ivi=.5Pi‘ A(><.x)I =E.-E1. insome basis el,...,e,,,then theHermitian bilinear form §ivi=“taA(x. Y)= i;.Ei7]i~ obviously reduces toA(x, x)ifwemake thesubstitution y=x.Moreover, asjust shown, thisistheunique Hermitian bilinear form reducing toA(x, x) under thissubstitution. 9.16. a.Given asymmetric Hermitian quadratic form A(x, x)inan n-dimensional complex space C,,,there exists abasis inCkinwhich A(x, x) canbewritten inthecanonical form A(x. X)=27\k"lkl'lk =27% llliiiz (8)iI1 I.-I1 withrealcoeﬂicients 7.1,7.2,...,7.”. The proof ofthis proposition isanalogous tothat ofTheorem 7.31. Instead ofequation (13), p.186, wehave blmzlgm +b2m£2Em +'''+bm—1,m€m—1E11i +bmmgmém 'i'T71,.,E1Em +'''+Em—1_mE-mI1Zm bm b°m v bm— m :bmm ;E1+;i§2+...+__.1-_gm_1+Em _j_A1(x,x) (b,,,,,, ck0),where A1(x, x)isasymmetric Hermitian quadratic form inthe variables E1,Z2,...,E,,,_1. Instead ofthetransformation (14), p.187, we SEC.9-1 I-IERMITIAN FORMS 251 now have thetransformation E1IE1+Eé. E2IE1+iié. Ea= 5..IE1. which carries thesum a12E1.E2 +12125152 (a12cé0)intotheexpression ((1..+1..)&;E;I»"<a..Iﬁ1.)Eé?.é+---. where atleast oneofthetwo(real) coeﬁicients an+anandi(a12 I12,2)is nonzero. b.Thelawofinertia (Theorem 7.91) continues tohold forasymmetric Hermitian quadratic form A(x, x)inacomplex space, i.e.,thetotal number p ofpositive coeﬂicients andthetotal number qofnegative coeﬂicients among thenumbers 7.1,7.2,...,1,,donotdepend onthechoice ofthecanonical basis. Theproof ofthisproposition istheexact analogue ofthat ofTheorem 7.91. Asintherealcase, thenumber piscalled thepositive index ofinertia andthe number qthenegative index ofinertia oftheform A(x, x). Itshould benoted that thelawofinertia does nothold forquadratic (asopposed toHermitian quadratic) forms inacomplex space C,,.For example, thequadratic form A(x.X)Iii+E3 istransformed into A(x,X)I1'li*vii bythecoordinate transformation 711=E11 712=1-52- c.Given asymmetric Hermitian quadratic form A(x, x)inaspace C,,, acanonical basis canalways befound such that thecorresponding canonical coeﬁicients canonly take thevalues il. Infact, having reduced theform A(x, x)totheform A(x. x)=7\1i"11|2 *1‘'''Ii‘)\ﬂi7l11i2 '“P11l"1|>+1l2 I'''Tl*al"11|+a|2~ where thenumbers 7.1,...,AD,ul,...,pi,areallpositive, wemake another coordinate transformation T1= 7111 ---1T1) =\/Z17l1HT11+1 : 7l11+1# '''!T11+0 =\/Q 7l11+0> 252 UNITARY SPACES CHAP. 9 thereby reducing A(x, x)totheform A(x’ x):iT1i2 +'''+iT1|i2 _iTv+1i2 _'''_iTp+qi2 (cf.Sec. 7.93). 9.17. a.Thevector x1issaidtobeconjugate tothevector ylwith respect totheHermitian bilinear form A(x, y)if A(x1» Y1)=0- Ifthevectors xl,x2,...,xkareallconjugate tothevector yl,then every vector ofthelinear manifold L(x1, x2,...,xk)spanned byxl,x2,...,xk isalso conjugate toyl(cf.Sec. 7.42c). Ingeneral, avector ylconjugate to every vector ofasubspace C’CCissaidtobeconjugate tothesubspace C’. The setC”ofallvectors yleCconjugate tothesubspace C’isobviously a subspace ofthespace C.This subspace C”issaid tobeconjugate toC’. Abasis el,e2,...,e,,ofthespace C,,issaid tobeacanonical basis of theform A(x, y)if A(e,, ek)=0for ickk. Every symmetric Hermitian bilinear form A(x, y)hasacanonical basis. In fact, letel,e2,...,e,,beabasis inwhich thecorresponding quadratic form A(x, x)canbewritten inthecanonical form A(x» X)=Z7115.-E1. {I1 where x=2Zkei. 1:] Then, bySec.9.l5b, thebilinear form A(x, y)takes thecanonical form A(x, Y)=27\1'Et7l¢ i=1 inthisbasis, where y2 ylieis iI1 andhence 7.,ifi=k, A(ei: ek)=0_f_¢ki1 . b.Suppose theprincipal descending minors S1,S2,...,3,,_1ofthe matrix ||a,k|| ofasymmetric Hermitian quadratic form A(x, x)areallnonvanishing. Then, justasinSec.7.52, wecanuseJacobi’s method toconstruct acanonical SEQ9»1 HERMITIAN roams 253 basis forA(x, x),andthecanonical coefﬁcients ofA(x, x)aregiven bythe same formulas 8 8x,=8,, x,=§,...,x,,=S—"- 1 nl (8,,=det||a,-k||) asonp.195. c.Asymmetric Hermitian bilinear form A(x, y)issaid tobepositive deﬁnite ifA(x, x)>0forevery xcé0.Just asintherealcase (Sec. 7.94), anequivalent condition isthat allthecanonical coeﬁicients ofA(x, x)be positive, oralternatively, thatp =n,wherep isthepositive index ofinertia oftheform A(x, x). Just asinTheorem 7.96, anecessary and sufﬁcient condition forthe form A(x, y)tobepositive deﬁnite isthat S,>0,S,>0,...,8,,>0 (Sylvester’s conditions). The proof given onp.209 carries over without change tothecomplex case. 9.18. a.Given anonsingular symmetric Hermitian bilinear form (x,y), wecanintroduce theconcept oftheadjoint ofalinear operator (with respect totheform (x,y)), just asinSec. 7.6.First wenote that ifAand Bare linear operators inthespace Ck,then theforms A(x.y)I(AX.y). B(x,y)I(X.By) areHermitian bilinear forms, whose matrices arerelated tothematrices of theoperators AandB(inanycanonical basis oftheform (x,y)with canonical coefficients 2,)bytheformulas ajm=Email)» btm=51173;) (thenotation isthesame asinSec. 7.61). Conversely, given twoHermitian bilinear forms A(x,y) and B(x,y), then, just asinSec. 7.62, there exist unique linear operators AandBsuch that A(x.y)I(Any). B(x,y)I(x»By)- b.ltfollows, justasinSec.7.63, thatgiven anylinear operator Aacting inthespace C,,,there exists aunique linear operator A*acting inC,,such that (Ax.y)I(X.A*y) -forarbitrary x,yeC,,.The matrices ||a§,f)|| and ||aff’)|| oftheoperators A andA*inanycanonical basis oftheform (x,y)with canonical coeﬂicients s,arerelated bytheformula aflm) :itaw 91' 254 UNITARY SPACES CHAP. 9 Theoperator A*iscalled theadjoint (orHermitian conjugate) oftheoperator Awithrespect totheform (x,y). c.The operation leading from anoperator Atoitsadjoint A*hasthe following properties (cf.Sec.7.64): 1)(A*)* IAforevery operator A; 2)(A-1-B)*IA*+B*forevery pair ofoperators AandB; 3)(7.A)* I7.A* forevery operator Aandevery number 7.6C; 4)(AB)* IB*A* forevery pair ofoperators AandB. 9.19. a.AsinSec. 7.71, twocomplex spaces C’and C”equipped with nonsingular symmetric Hermitian bilinear forms A(x’, y’)and A(x”, y”), respectively, aresaidtobeA-isomorphic ifthespaces C’andC”areisomorphic regarded aslinear spaces over theﬁeld C(seeSec.2.71) andif A(x’.y’)IA(x".y") forallcorresponding pairs ofelements x’,y’eCandx",y”eC". b.THEOREM. Twoﬁnite-dimensional complex spaces C’andC”,equipped with nonsingular symmetric Hermitian bilinear forms A(x’, y’)andA(x”, y"), respectively, areA-isomorphic andonly theyhave thesame dimension and theindices ofinertia p’,q’oftheform A(x’, y’)coincide withthecorresponding indices ofinertia p”,q”oftheform A(x”, y”). Proof. Precisely thesame asthat oftheanalogous proposition forreal spaces (Theorem 7.93). I c.lnparticular, twon-dimensional complex spaces C;andCZ,equipped with positive deﬁnite forms A(x’, y’)andA(x", y"),respectively, arealways A-isomorphic (cf.Sec. 7.97). 9.2. The Scalar Product inaComplex Space 9.21. ltwillberecalled from Sec. 8.21 that thescalar product oftwo vectors xandyinarealspace istaken tobeanyﬁxed symmetric positive deﬁnite bilinear form (x,y). The corresponding quadratic form (x,x) is then positive forevery nonzero vector x,and can beused todeﬁne the length ofx(see Sec. 8.31). lnacomplex space, any symmetric positive deﬁnite Hermitian bilinear form hastheanalogous property (seeSec.9.l7c). This leads tothefollowing deﬁnition: Acomplex linear space Cissaidtobe aunitary space ifitisequipped with asymmetric positive deﬁnite Hermitian bilinear form (x,y),called the(complex) scalar product ofthevectors xand sec.9.2 THESCALAR PRODUCT 1NACOMPLEX SPACE 255 y,i.e.,ifthere isaruleassigning toevery Pair ofvectors x,y6Cacomplex number (x,y)such that a)(y,x)I(x,y)forevery x,y6C; b)(x,y +2)I(x,y)+(x,2)forevery x,y,zeC; c)(7.x,y)I7.(x,y)forevery x,yeCandevery complex number 7.; d)(x,x)>0forevery xI0and(x,x)I0forxI0. Axioms a)—c) imply thegeneral formula PE‘ 3 _l\/1*’___[\/is(§1¢ix1'-glﬁayi) :,= _=‘°‘1'Ei(xi» Y1)» where xl,...,xk,y1, ...,y,k arearbitrary vectors ofthespace Cand al,..3,ak,(-31,...,i-},karearbitrary complex numbers. 9.22. Examples a.Inthen-dimensional space Ck(Sec. 2.15b) wedeﬁne thescalar product ofthe vectors x=(Z1,Z2,...,Zk)andyI(111,1,2,...,'Y]k)bytheformula (xa :glﬁl +€2ﬁ2 +---+inﬁn- The reader caneasily verify that axioms a)—d) aresatisﬁed inthiscase. b.Inthespace C(a,b)ofallcontinuous complex-valued functions onthe interval a<t<_b(Sec. 2.15d) wedeﬁne thescalar product ofthefunctions x=x(t)andy=y(t)bytheformula b __ (X.y)=lX(t)y(1) dt- Axioms a)—d) arethen immediate consequences ofthebasic properties ofthe integral. 9.23. Basic metric concepts. Next weintroduce various metric concepts inaunitary space C,justaswasdone inthecase ofarealEuclidean space (Sec. 8.3). a.Thelength ofavector. Asintherealcase, bythelength (ornorm) ofa vector xinaunitary space Cwemean thequantity |x|=+\/(x, x). Every nonzero vector hasapositive length, andthelength oftheZero vector equals 0.Foranycomplex 7.,wehave theequality itxiI~/(M.M)Iﬁe.x)Iin~/<x.x)Iiii|x|. 256 UNITARY SPACES CHAP. 9 which shows thatthelength ofavector xmultiplied byanumerical factor 7. equals theabsolute value of7.times thelength ofx. Avector xoflength 1issaid tobeauni'tvector. Every nonzero vector canbenormalized, i.e.,multiplied byanumber 7.such that theresult isa unitvector. Infact, weneed only choose 7.such that 1IllII,_ ixi justasonp.217. Thesetofallvectors x6Csuch that |x|<1iscalled theuni'tballinC, while thesetofallx6Csuch that |x|I1iscalled theunitsphere. b.TheSchwarz inequality. Theinequality |(x.J/)|IlxlI)/I (9) holds forevery pairofvectors xandyinC.Theidea oftheproofis thesame asintherealcase (Sec. 8.33), except that wemust now becareful about complex numbers. The inequality (9)isobvious if(x,y)=0. Thus let (x,y) :,é0.Clearly, (MIy.MIy)>0 forarbitrary complex 7..Expanding theleft-hand side, weget I71’(x,x)Ito.1)IX(x.1)+(1.1)>0- (10) LetYbethelineinthecomplex plane determined bytheorigin andthecomplex number (x,y),andletY’bethelinesymmetric toYwith respect tothereal axis. Suppose 7.varies over thelineY’,sothat 7.Itzo,where tisrealand Z0Z (X: |(><.)/)1 istheunitvector determining thedirection ofY’.Then 7(x.y) It|(x.y)| isreal, andhence X(>7)I70¢.y). sothat theinequality (10)becomes 1’(x,X)IZ1|(X.y)| +(LY) >0- (11) The same argument asinSec. 8.33 now leads tothedesired inequality (9). Ifequality holds in(9),then thetrinomial intheleft-hand side of(11) hasaunique realroot to(ofmultiplicity two). Replacing tzoby7.,weﬁnd thatthetrinomial intheleft-hand sideof(10)hastheroot 7.0=tozo. Therefore (%xIy.7.XIy)I0 and hence y=7.0x, sothat thevectors xandydiﬁer only bya(complex) numerical factor. SEC.9.2 TI-IESCALAR PRODUCT 1NACOMPLEX SPACE 257 c.Orthogonality. Although theconcept oftheangle between twovectors isnotintroduced inaunitary space, westillconsider thecase where two vectors xandyareorthogonal, which means, just asintherealcase, that (x,y)I0. Ifxandyareorthogonal, then obviously (y.X)I(X.y)I0- Itiseasily veriﬁed thattheanalogues ofLemmas 8.36aIb andthePythagorean theorem (Sec. 8.37) remain valid fororthogonal vectors inaunitary space. Moreover, theanalogue oftheexpansion theorem ofSec. 8.51 also holds, i.e.,given aﬁnite-dimensional subspace C’CCandavectorfwhich isin general notanelement ofC’,there exists aunique representation fIg+h where g6C’andhisorthogonal toC’.The setofallvectors horthogonal tothesubspace C’isitself asubspace, which wecalltheorthogonal comple- ment ofthesubspace C’anddenote byC”.Just asinSec. 8.51, weseethat theoriginal space Cisthedirect sum ofthesubspace C’anditsorthogonal complement C”. d.Thetriangle inequalities. Ifxandyaretwovectors inaunitary space C,then, bySchwarz’s inequality (9), |x+ﬂ“%x+xx+ﬁ=%&ﬂ+%mﬁ+%&ﬁ+%xﬁ I(X.X)+2|(x.y)| +(y.y)I(IXI+|y|)*, >(X,X)I2|(x.)/)| -1-(Y1)/) >(|x|II)/I)’. Ix+)/I Ilxl-1-1)/I. (12) |x+y|>1|x|—|y|l- (13) Asintherealcase, these inequalities arecalled thetriangle inequalities. 9.24. Orthogonal bases inann-dimensional unitary space Ck.According toSec.9.l6a, thesymmetric Hermitian bilinear form (x,y)hasacanonical basis el,e2,...,ekinthen-dimensional space Ck, and inthis case the condition (er,ek)=0 (195 k) forthebasis tobecanonical reduces totheorthogonality condition. More- over, theorthogonal basis vectors el,e2,...,ekcanberegarded asnor- malized, sothat |@.lIie.iI---I|@..lI1. 253 UNITARY SPACES ¢H,|_p_ 9 Let 1| 1| X=2Ek¢’i~» Yzzllkeu kI1 k=1 beany two vectors inCk,with components Ek,1],,(k:1,___,,1)with respect tothebasis el,e2,...,ek.Wethen getthefollowing formula for thescalar product (x,y)interms ofthecomponents ofxandy: 7| (xiY)I215.1171):-,_= 9.25. a.Asshown inSec.9.l8a, theformula A(x.y)I(Ax.y) establishes aone-to-one correspondence between Hermitian bilinear forms A(x, y)and linear operators Aacting inthespace Ck.Inanyorthonormal basis el,e2,...,ekofthespace Ck,thematrix ||a,-,k|| oftheform A(x, y)and thematrix l|a§’"’|| oftheoperator A,where aim :A(eJ's em)’ T’ ('1Ae,Ila ’e,,,,_. i»=1 arerelated bytheformula aim Zair?- b.LetAbeanylinear operator acting inthespace Ck.Then, asshown inSec. 9.l8b, there isaunique operator A*,theadjoint ofAwith respect tothescalar product (x,y),such that (Ax,y) I(X.A*y) forarbitrary x,yeCk.Since anyorthonormal basis isacanonical basis for theform (x,y),with canonical coefficients i-:,.Il,thematrices ||af,fl|| and ||a:1"*|l oftheoperators AandA*arerelated bytheformula a;i=(m) :J5 Inother words, thematrix oftheoperator A*isobtained from that ofthe operator Aby“Hermitian transposition,” i.e.,bytransposition followed by replacing allelements ofthematrix bytheir complex conjugates. Corre- spondingly, wecallthematrix ofA*theHermitian conjugate (oradjoint) of that ofA. c.AsinSec. 8.96a, thesubspace C’CCisinvariant under theoperator A,then theorthogonal complement ofC’isinvariant under theadjoint opera- torA*. SEC. 9.3 NORMAL OPERATORS 9.26. Acoordinate transformation inann-dimensional unitary space Ck leading from one orthonormal basis toanother iscalled aunitary trans- formation. Unitary transformations areanalogous toorthogonal trans- formations inaEuclidean space (seeSec. 8.94). Ifel,...,ekandfl, ...,fk areorthonormal bases inCkand ifUI||u§j*|l isthematrix ofthecorre- sponding unitary transformation, sothat '7| fi:Elli.-’)@i.. kI1then obviously 1-_—-_— lif1=1,</.-,f.~>I21i§."1il" I <14)1.-I1 0ifiIj. Conversely, ifthenumbers ufc”satisfy theconditions (14), then thematrix ||u,§“|| isaunitary matrix, i.e.,thematrix ofaunitary transformation. The linear operator Ucorresponding toaunitary matrix iscalled a unitary operator. Just likeanisometric operator inareal space, aunitary operator inacomplex space does not“change themetric.” Inother words, if 7| X= Y=271191,I 1I1 ,..'l\/l=J\\s.5-“ then u.M3. ..'-M=.3 (U79, UY) =:Ei7l,~(U@1, U35) ::Eiﬁ;'(f1',fa') ziglzﬁlj =(x)Y)- The matrix Voftheinverse transformation from thebasisfl, ...,fkto thebasis el,...,ekisalso unitary. Moreover, ifV= |lv§,“||, wehave ( . uh’) Z ek): Uk’) Z(eisflr) Zugl- Thus theinverse ofaunitary matrix isobtained byﬁrsttransposing andthen going over tocomplex conjugate elements. Therefore UI1 =U* foraunitary operator U,orequivalently, U*U IUU* IE. 9.3.Normal Operators 9.31. Deﬁnition. Anoperator Aacting inann-dimensional unitary space Ckissaidtobenormal ifitcommutes with itsown adjoint, i.e.,if A*AIAA* (15) 260 UNITARY SPACES ct-iAi>_ 9 (cf.Sec. 8.92c). Anexample ofanormal operator isgiven byanyoperator Awhose eigenvectors el,...,ek,satisfying therelation Ael:)‘:iea' (]'=1,---J1). form anorthogonal basis inCk.Infact, thematrix oftheoperator Ainthe basis el,...,ekisthen oftheform 7.,0...0 07.2 0 (16) 00 7.k But, bySec.9.25, thematrix oftheoperator A*inthesame basis el,...,ek isjust 1,0...0 0X, 0 . (17) 00 X, from which isisobvious that theoperators AandA*commute. 9.32. THEOREM. Everyeigenvector xofanormal operator Awitheigeni-alue 7.isaneigenvector oftheoperator A*with eigenvalue X. Proof. LetPCCkbethesubspace consisting ofalleigenvectors ofthe operator Awith eigenvalue 7..Ifx6P,then AA*xIA*AxIA*(7.x) I7.A*x, which implies A*xeP. Hence Pisinvariant under the operator A*. Moreover, (A*x.y) I(X.Ar)I(x.Ky)I(ix.y) forarbitrary x,yeP,andhence A*xIix. I 9.33. a.THEOREM. Given any normal operator Aacting inaunitary space Ck,there exists anorthonormal basis el,...,ekinCkconsisting of eigenvectors ofA. Proof. Thenormal operator A,likeevery linear operator inthespace Ck, hasaneigenvector (see Sec. 4.95b). Lete1beaneigenvector ofAwith eigenvalue 7.,andletPCCkbethesubspace consisting ofalleigenvectors ofAwith thiseigenvalue 7..IfPisthewhole space Ck,then weneed only arbitrarily augment e1with vectors ek,...,ektomake anorthonormal basis sEc.9.3 NORMAL OPERATORS 261 forCk,thereby proving thetheorem. Thus suppose PICk,and letQbe theorthogonal complement ofPinCk.Thesubspace Pisinvariant under the operator A*, asintheproof ofTheorem 9.32 (infact, A*carries every vector xePinto thevector 7.x). Itfollows that Qisinvariant under the operator Aitself, because ofSec. 9.25c and thefact that (A*)* IA(see Sec. 9.18c). Wecannow prove thetheorem byinduction. Infact, suppose thetheorem istrue forevery space Ckofdimension n<k.Then itisalso true forCk+1, since togetanorthonormal basis forCk+1 consisting of eigenvectors ofA,weneed only choose such abasis inthesubspace Q(such exists bytheinduction hypothesis, since thedimension ofQ is<k) andthen augment thisbasis byanyorthonormal basis inP.Theproof isnow complete, since thetheorem isobviously truefortheone-dimensional space C1. I b.Itfollows from Theorem 9.33a that every normal operator Ais diagonalizable (seeSec.4.72f). Infact, Ahasthediagonal matrix 1,0 ...0i 07.2 0 A: 00 7.k intheorthonormal basis constructed inTheorem 9.33a, consisting of eigenvectors ofA.Theeigenvalues ofAlieontheprincipal diagonal ofthis matrix, each appearing anumber oftimes equal tothedimension ofthe corresponding characteristic subspace (cf.p.110). Hence thecharacteristic polynomial det||AII7.E||oftheoperator A,which asweknowis independent ofthechoice ofbasis (seeSec.5.53), hastheform det||AI7.E||I1"’[(i,I1)", ﬁr,II1, (18) I.-I1 kI1 where 7.1,...,7.k,arethedistinct eigenvalues oftheoperator Aandrl,..., r,karethedimensions ofthecorresponding characteristic subspaces. c.Ontheother hand, suppose itisknown that anormal operator Ahas acharacteristic polynomial oftheform 8 K 116111/1 IXEIIIlT(i1i. IA)”. Z171.I'1. (19) kI1 lrI1 where ul,...,ti,aredistinct complex numbers andpl,...,pkarecertain positive integers (multiplicities). Then itcanbeasserted that theoperator A hasanorthonormal basis consisting ofeigenvectors witheigenvalues ul,...,u,, where thedimension ofthecharacteristic subspace corresponding tothe eigenvalue pi,isjust pj.Infact, thepolynomials (18)and(19)must coincide, bytheuniqueness ofthecharacteristic polynomial. Butthen ourassertion 262 UNITARY SPACES gr-i,|_p_ 9 follows from thefamiliar theorem ontheuniqueness ofthefactorization ofa polynomial. 9.34. Self-adjoint operators. Anoperator Aacting inaunitary space C issaidtobeself-adjoint ifA*IA,i.e.,if (Ax.Y)I(X.A)/) (20) forarbitrary vectors x,yEC. Note that Aisself-adjoint ifandonly ifthe bilinear form (Ax, y)corresponding toAisHermitian—symmetric.T Accord- ingtoSec. 9.25, thematrix ofaself-adjoint operator Ainanyorthonormal basis coincides with itsown Hermitian conjugate, i.e., with thematrix obtained from that ofAbytransposition followed bytaking complex conjugates ofallelements. Conversely, every operator Awith aHermitian- symmetric matrix (i.e., amatrix equal toitsown Hermitian conjugate) in some orthonormal basis isself-adjoint. Since aself-adjoint operator Aisobviously normal, itfollows from Theorem 9.33a thatthere exists anorthonormal basis el,...,ekinthespace Ckinwhich thematrix oftheoperator Atakes theform (16)andthat ofA* takes theform (17). Hence 2,»I7.,(jI1,...,n),since A*IA,i.e.,the numbers 7.,areallreal. This proves thefollowing THEOREM. Given anyself-adjoint operator Ainaunitary space Ck,there exists anorthonormal basis el,...,ekconsisting ofeigenvectors ofAwith eigenvalues thatareallreal. Conversely, every linear operator Ainthespace Ckwith theindicated property isself-adjoint. Infact, Aisnormal bySec. 9.31, andcomparing (16)and(17)weﬁndthat A*IA,since thenumbers 7.,areallreal. 9.35. Antiself-adjoint operators. Anoperator Aacting inaunitary space Ckissaid tobeantiself-adjoint ifA*II-A. The matrix ofanantiself- adjoint operator Ainanyorthonormal basis el,...,ekhasthefollowing characteristic property: av.=(Ael, ek)=(ei.A*6’k) :(en "Aek) :I(Aek, er):“dict (i,kI l,...,n). Anantiself-adjoint operator Aisobviously normal. Applying Theorem 9.33a, weﬁndthatthere exists anorthonormal basis el,...,ekinthespace Ckinwhich thematrix oftheoperator Atakes theform (16)andthat ofA* takes theform (17). Hence X,II7.,- (jI1,...,n), since A*II-A, Tln fact, thecondition (Ay, x)I(Ax,y) isequivalent to(20). For thisreason, a self-adjoint operator might also becalled Hermitian—symmetric. SEC_ 9.4 APPLICATIONS TO OPERATOR THEORY IN EUCLIDEAN SPACE i.e.,thenumbers 7.,areallpurely imaginary. This proves thefollowing THEOREM. Given anyantiself-adjoint operator Ainaunitary space Ck, there exists anorthonormal basis el,...,ekconsisting ofeigenvectors ofA with eigenvalues thatareallpurely imaginary. Conversely, every linear operator Ainthespace Ckwith theindicated property isantiself-adjoint. 9.36. AsinSec.9.26, anoperator Uacting inaunitary space Ckissaid tobeunitary ifU*U IUU* IE.Inparticular, every unitary operator is normal. Applying Theorem 9.33a, weﬁnd that there exists anorthonormal basis el,...,ekinthespace Ckinwhich thematrix oftheoperator Utakes theform (16)andthatofU*takes theform (17). Hence 7T,-7.,Il(jI1,..., n),since U*U IE,orequivalently, l)\jl:1 (jI1.---.11) This proves thefollowing THEOREM. Given any unitary operator Uinaunitary space Ck, there exists anorthonormal basis e1,...,ek consisting ofeigenvectors ofthe operator Uwith eigenvalues thatareallofabsolute value 1. Conversely, every linear operator Uinthespace Ckwith theindicated property isunitary. 9.4.Applications toOperator Theory inEuclidean Space 9.41. Embedding ofaEuclidean space inaunitary space. AsinSec.8.21, letRbea(real) Euclidean space with scalar product (x,y).Consider the complex space Cconsisting oftheformal sums x+iywhere x,yeR,with thefollowing natural operations ofaddition andmultiplication byarbitrary complex numbers: (x1"1'1’Y1) 'i'(x2*1‘1-Y2) =(xi+x2)'i'1’(Y1 +Y2), (1+il1)(x+fr)I(axI111)+i(¢y+ﬁx)- Then itiseasily veriﬁed that Chasalltheproperties ofacomplex linear space. Wenow identify thevectors x+i0with thevectors xeR,calling them realvectors ofthespace C.Thevectors 0+iywillbedenoted simply byiy andcalled purely imaginary vectors. Bythecomplex conjugate ofthevector x+i'y,written x+iy,wemean thevector xIIiy. Next weintroduce ascalar product inthespace C,deﬁned bytheformula (x1*1‘‘Y1’x2*1‘1Y2) Il(x1» x2)Ii‘(Y1,Y2)l ‘l‘1’l(Y1, x2)"(xl,Y2)l- 264 UNITARY SPACES CHAP. 9 Itiseasily veriﬁed thatthisscalar product satisﬁes axioms a)—d) ofSec.9.21. Inparticular, (X+1'y.X +1'y)I (X.X)-5-(1.1)- Thus thespace Ccontains thespace Rasasubset, equipped with thesame scalar product, andsubject tothesame operations ofaddition and multi- plication byrealnumbers. Note that every orthonormal system (orbasis) el,...,ekinthespace Risalso anorthonormal system (orbasis) inthe space C. 9.42. Every linear operator Aspeciﬁed inthespace Rcanbeextended intothespace Cbytheformula A(x+iy)IAx+iAy, (21) where theoperator Aisobviously alin'ear operator inthespace C.The matrix oftheoperator Ainthespace Crelative toabasis el,...,ekER coincides with thematrix oftheoperator Ainthespace Rrelative tothesame basis, since, according to(21), Ae,-IAe,. (j=l,...,n). This extension from AtoApreserves algebraic relations between linear operators, i.e.,ifA+BIDinthespace R,then A+BIBinthespace C,while ifABIDinthespace R,then ABI1)inthespace C.This follows forexample from thefactthat matrices arepreserved under theextension from AtoA. 9.43. LetA’betheadjoint oftheoperator Aintherealspace R(seeSec. 8.91). Then theextension A’oftheoperator A’intothespace Cisjustthe operator A*adjoint totheextension AofA.Infact, given arbitrary vectors z=x—l—iy,w=u—l—iveC,wehave (A’(x + u+ Z(A’x1 u)+ u)ti(A’x1 U)+(A1,)/'9 U) I(X.A")-1-1'()/. A")—1'(x. Av)-1"O’,Av) I(x+iy.Ra+iv», asrequired. Inparticular, theextension ofasymmetric operator (A’IA)isaself- adjoint operator (A*IA),theextension ofanantisymmetric operator (A’I—A) isanantiself-adjoint operator (A*IIIA), andtheextension of anisometric operator (U’IUI1) isaunitary operator (U*IUI1). Finally, theextension ofanormal operator (A'A IAA’) isagain anormal operator (A*A IAA*). SEC.9.4 APPLICATIONS TOOPERATOR THEORY INEUCLIDEAN SPACE 265 9.44. Structure ofarealnormal operator. Letcsand1berealnumbers. Then theeasily veriﬁed matrix equality = = 2(Z2)cs1 cs-1 cs—1 61 02-1-1" 0 ——1 cs 1 cs 1 cs -1 cs 0 02-1-1 G.Tshows thatthematrix commutes with itsown transpose (and hence afortiori with itsown adjoint); more generally, thesame istrue ofthequasi-diagonal (real) matrix ——11 01 02 12 —12 0261 T1 GmTm (23) —1,,, cm [Z THEOREM. Given anynormal operator Ainareal Euclidean space R,,, there exists anorthonormal basis fl,...,f,,eR,, inwhich thematrix ofA isoftheform (23), with m+r-n,where thenumbers A,=0,»+i1, (j=1,...,m)and )\,,,+1, ...,X,areuniquely determined byA.Infact, these numbers aretheroots ofthecharacteristic equation det||A-xE||=0, (24) andeach root of(24) appears inthematrix (23)anumber oftimes equal to itsmultiplicity.oforder2m+r—m=m+r. Proof. AsinSec. 9.41, weconstruct theunitary space C"whose scalar product istheextension ofthescalar product (x,y)deﬁned inthespace R,,. 266 UNITARY SPACES CHAP. 9 Wethen use(21) toextend theoperators AandA’into thespace C”.As shown inSec. 9A.43, theextensions ofAandA’arethenormal operator A anditsadjoint A*. Let||a,-,,|l denote thematrix oftheoperator Arelative to anyorthonormal basis el,...,eninthespace Rn(thenumbers amarereal).T Then theoperator Ahasthesame matrix relative tothebasis el,...,enin thewhole space C".Since thecharacteristic equation (24)hasrealcoeﬁicients, ifA,isanimaginary root of(24), then soisthecomplex conjugate 1,.Bearing thisinmind, wewrite thesequence ofdistinct roots of(24) intheform )\1,)\1,. ..,hp,hp,)\,,+1, ...,ha, where theroots A1,...,)\,,areimaginary and theroots ADM, ...,7,‘are real. Then, bySec.9.33b, thespace C"canberepresented asadirect sum of orthogonal subspaces A1,A1,. ..,A,, A,,A,,+1, ...,A,,, where A,»consists ofalleigenvectors oftheoperator Acorresponding tothe eigenvalue A,and K,consists ofalleigenvectors ofAcorresponding tothe eigenvalue X,-,while _ __ An+1 =Ap+1’ '‘‘’Ac:Au" Ifz=x+iyeA,,then theequation Az=)\,~zbecomes zaikck =)‘jC:/' k=1 incomponent form (with respect totheoriginal basis el,...,e,,),where z=@hM@w4a+mW@;+mh Taking thecomplex conjugate andrecalling that thenumbers a,-karereal, weget 7| _ __ zaikck =7\jCi k=1 This means that thevector 2=(Z1,...,in)isalso aneigenvector ofthe operator Awith eigenvalue Xi.Itfollows that theoperation oftaking the complex conjugate carries thespace A,intothespace -Aj- Now letA1=01+i11,where 11¢0since X1¢X1,andletglbeanyunit vector inA1,sothat §1eA1.Moreover, let f1=ﬁ(g1 +gr), f2=\71i—l_(g1 _gr), sothat a=ﬁm+m a=ﬁm-m TInthecourse oftheproof, wewillconstruct aneworthonormal basis fl,...,f,,for R,inwhich Ahasamatrix oftheform (23). SEC.9.4 APPIJCATIONS TOOPERATOR THEORY INEUCLIDEAN SPACE 267 where thevectors flandflareobviously real, and moreover orthonormal, since itfollows from (g1,g1)=(§1,§1)=l, (g1,§1)= 0 that (flifl)=(Ah)=E[(g1ago+<z1,§1>1=1, . l _ _ 1 __ (f1af2)= _Z(g1 +81’Z1—Z1)=— [(81, E1)—(Z1, 31)] =0- Since A 1 A A 1 — A =A =—_ A +A" =—_ )1 +)1' f1 f1 \/2( E1 81) \/2( 131 181) =§[(61+iT1)(.f1+112)+(61-1+l><fl-it/2)]=on-ms. Af2=Afz =_{_ (A31 "K51) = (A131 —11$-$1) =T1f1 +°1f2,(/2i (/2l weseethat theoperator Atransforms theplane ofthevectors fl,flinto itself andhasthematrix G T1 1 (25) —1l cl inthebasisfl,f2. Ifthedimension ofAlisgreater than 1,wechoose another unit vector gleAl orthogonal togl,with complex conjugate £26Al(the latter isautomatically orthogonal togl). Repeating theabove construction forglandgl,wegetanew pair ofrealvectors fa,flwhich arelinear com- binations ofgl,gland hence orthogonal tothevectors fl,f2(themselves linear combinations ofgl,gl).Clearly Atransforms theplane ofthevectors f3,f, into itself andhasthesame matrix (25). Continuing thisconstruction, weeventually get2morthonormal realvectors fl,fl,...,f2,,,_l,f2m, where misthesum ofthedimensions ofthesubspaces Al,...,Al,andAtrans- forms theplane ofthevectors fl,-_l, fl,intoitself ,with either thesame matrix (25) ortheanalogous matrix obtained byreplacing cl,1lbyck,1,, (k=2,...,p). Next consider thesubspace A,,+l corresponding totherealroot )\,,+l = Xl,“. The operation oftaking thecomplex conjugate obviously carries the subspace A,,+l intoitself. Letg beanyvector inA,,+l, andletgbeitscomplex conjugate. There arejusttwopossibilities, namely, thevectors gandgare either linearly independent (inC,,)orlinearly dependent. Ifgandgare linearly independent, then soaretherealvectors /-L-(g+ g)f'=—1—<g -:1,/2 ’ (/2i Like gandg,these vectors belong toA,,+l,andhence areeigenvectors ofthe 268 UNITARY svxcss CHAP. 9 operator Awith thesame eigenvalue 7\,,+l. Ontheother hand, ifgandgare linearly dependent, then g=emg (0<cp<11:), since gandghave thesame length. Therefore ewg :e-tog =ewg’ sothatthevector f=@‘“’g isreal. Moreover, since fbelongs toA,,+l, likegitself, fisaneigenvector of Awith thesame eigenvalue )\,,+l. Thus, inanyevent (continuing thiscon- struction ifnecessary), wecanalways ﬁndabasis inA,,+l consisting ofreal vectors. Applying theorthogonalization theorem (Theorem 8.61) tothis basis, weﬁnally getﬁrst anorthogonal andthen anorthonormal basis in A,,+l. Clearly theoperator Atransforms A,,+l intoitself andhasthediagonal matrix AMI 0 __p0 01,+l 0 (26) 0 0 "@+1 intheorthonormal basis. Repeating this construction fortheremaining subspaces AMl,...,Al,weeventually obtain asetoforthonormal vectors f2,,,+l,f2,,,+2, ...,f,,,which together with thepreviously constructed vectors fl,f2, ...,fm form afullorthonormal basis forR,,.Tocomplete theproof, weneed only take account ofthespecial form ofthetypical blocks (25)and (26), compensating forthesomewhat different indices in(23) which refer toroots which arenotnecessarily distinct. I The geometric meaning ofanormal operator canbededuced from this theorem. First weobserve that theoperator with matrix G T —1cs inthebasis fl,flcanbeinterpreted asarotation accompanied byanexpansion intheplane ofthevectors fl,fl.Infact, weneed only note that G T 61 _i \/02+12 \/62+12 cosoc sinoc =(/62 _|_T2 =M _ , -1 6 1 G —-sin ozcosoz —\/62+12 \/62+12 ii 6 . 1M=\/¢;*+-r2, COSoc=€, s111ot=?, \/6+1 \/0+1 SEC. 9.4 APPLICATIONS TO OPERATOR THEORY IN EUCLIDEAN SPACE cosonsinozwhere theeﬁect ofthematrix ll—sin ozcosoz istorotate every vector inthefl,f2 plane through theangle ex,while Mis clearly theexpansion coefﬁcient. Recalling (23), wenow seethat thetotal eﬁect ofthenormal operator Aistoproduce rotations accompanied by expansions inmmutually orthogonal planes andexpansions only (byfactors of7\,,,+l, ...,71,,respectively) inther—mdirections orthogonal tothese planes andtoeach other.T 9.45. The structure ofarealsymmetric operator. LetAbeasymmetric operator acting inarealspace R",sothat A’=A.Then theextension A oftheoperator Aintotheunitary space C"isself-adjoint, i.e.,A*=A.The eigenvalues kl,...,71,,ofaself-adjoint operator areallreal(seeSec. 9.34). Hence there arenoblocks oftheform (25) intherepresentation (23), and allthat remain arediagonal elements. This proves thefollowing THEOREM. Given anysymmetric operator AinarealEuclidean space R,,, there exists anorthonormal basis inR,,consisting ofeigenvectors ofA. Geometrically, asymmetric operator produces expansions (byfactors of kl,...,kn,respectively) along each ofnorthogonal directions. The num- bers )\l,...,)\,, aretheroots ofthecharacteristic equation (24). Hence thecharacteristic equation corresponding toasymmetric matrix A= ||a,-,,|| must have n(notnecessarily distinct) realroots andnoimaginary roots atall. 9.46. The structure ofareal antisymmetric operator. IfAisananti- symmetric operator acting inR",sothat A’=—A, then theextension A oftheoperator Ainto thespace C"isantiself-adjoint, i.e.,A*=—A. The eigenvalues kl,...,A"ofanantiself-adjoint operator areallpurely imaginary (see Sec. 9.35). Hence theblocks (25) intherepresentation (23) take the special form 01, _ (1=1,2,---,m), —1l 0 while thenumbers )\m+1, ).m+2, ...,A,must allbe0.Thisproves thefollowing TThe expansion isactually acontraction if0<\/0’+-:2<1orif0<1,,<1. Moreover, expansion byafactor 7.,<0isactually anexpansion accompanied bya reﬂection. 270 UNITARY SPACES CHAP. 9 THEOREM. Given anyantisymmetric operator AinarealEuclidean space R,,,there exists anorthonormal basis inR,,inwhiclz thematrix ofAtakes the quasi-diagonalform 0 1l +1l 0 —12 0 0 Till ~1m0 (27) 0 W B Conversely, ifthematrix ofanoperator Aisoftheform (27) insome orthonormal basis, then Aisantisymmetric (Sec. 8.92b). Geometrically, anantisymmetric operator produces rotations through 90°followed byexpansions (byfactors of1l,...,1m, respectively) inin mutually orthogonal planes, while mapping into 0allvectors orthogonal to these planes. 9.47. The structure ofareal isometric operator. IfAisanisometric operator acting inR,,,sothatA’:A11, then theextension Aoftheoperator Ainto thespace C"isunitary, i.e.,A*:A1‘. The eigenvalues ).l,...,7." ofaunitary operator areallofabsolute value l(seeSec. 9.36). Hence the blocks (25) intherepresentation (23)take thespecial form cosex,sinoi, ,ll-sin oz,cosax, andthenumbers ).m+l, ...,Armust allbej;l.This proves thefollowing THEOREM. Given anyisometric operator AinarealEuclidean space Rn, there exists anorthonormal basis inR,,inwhich thematrix ofAtakes the PROBLEMS 271 quasidiagonal form cosoilsin£11 —sin oilcosoil cosoi,sinoi, —sin oi,cosoi, cos1,, sinum —sin im cosat :51 E E Geometrically, anisometric operator Aproduces arotation through a certain angle (with noaccompanying expansion) ineach ofmmutually orthogonal planes, andacts liketheoperator Eor—E ineach ofther—m directions orthogonal tothese planes andtoeach other. However, wecan combine every pair ofsuch directions with identical expansion coefficients (both +1orboth —1)intoaplane inwhich theoperator Aalsoproduces a rotation (through 0°or180°). Making allsuch combinations, weﬁnd that ifnisodd, then some lastdirection hasthecoeﬁicient +1or—l,while ifn iseven, there may betwoungrouped directions with coeﬁicients +1and+1. The presence of+1among these remaining coeﬁicients shows that besides theindicated rotations there isanadditional reﬁection with respect tosome coordinate plane, forexample, theplane orthogonal tothebasis vector e,,. Wethen have detA=—l,whereas detA:+lifthere isnosuch reﬁection. PROBLEMS 1.Aself-adjoint operator acting inaunitary space C"issaidtobenonnegative (orpositive) ifallitseigenvalues xl,...1,,arenonnegative (orpositive). Show thatthesquare ofevery symmetriC operator isnonnegative. 2.Show thatgiven anyself-adjoint nonnegative (orpositive) operator A,we canﬁndaunique nonnegative (orpositive) operator B,the“square rootofthe operator A,”such thatB2=A. UNITARY SPACES CHAP. 9 3.Take thesquare rootoftheoperator Aspeciﬁed bythematrix 13 14 4 A=142418 41829 inanorthonormal basis el,el,el,. 4.LetAbeanarbitrary linear operator acting inaunitary space C”,andlet A*beitsadjoint. Prove thatA*A isanonnegative operator. Prove thatA’A isapositive operator ifAisnonsingular. 5.Given thatalinear operator Aistheproduct SQofaself-adjoint operator Sandaunitary operator Q,prove thatS2=AA*. 6.Show thatevery nonsingular linear operator Acanberepresented asthe product SQofaself-adjoint operator andaunitary operator. 7.Prove thattherepresentation oftheoperator Aasaproduct SQinProblem 6isunique. 8.Alinear operator Vacting inC,,issaidtobenonexpanding if|Vx| <[x|for every x.Prove thatevery linear operator Acanberepresented astheproduct ofaself-adjoint operator andanonexpanding operator. 9.Show thattwoself-adjoint operators AandBcommute ifandonly ifthey have acommon system ofnmutually orthogonal eigenvectors. 10.Given alinear operator Aacting inthespace C,,,ﬁndanorthonormal basis inwhich thematrix ofAhasthetriangular form (1) (1) (1)a1 a2 an 12) (2)A= 0a2 an 00 a‘,,'” chapter I0 QUADRATIC FORMS INEUCLIDEAN AND UNITARY SPACES l0.I. Basic Theorem onQuadratic Forms inaEuclidean Space 10.11. We begin with thefollowing theorem concerning symmetric bilinear forms inaEuclidean space: THEOREM. Every symmetric bilinear form A(x,y)inann-dimensional Euclidean space Rnhasacanonical basis consisting oforthogonal vectors. Proof. Consider thelinear operator Acorresponding tothegiven sym- metric bilinear form (see Sec. 8.91). The operator Aisalso symmetric. According tothetheorem onsymmetric operators (Theorem 9.45), thespace R,,hasanorthonormal basis consisting oftheeigenvectors oftheoperator A,andthematrix ofAisdiagonal inthisbasis. Since thismatrix isalsothe matrix ofthebilinear form A(x,y), theorthonormal basis just found isa canonical basis ofA(x, y). I 10.12. Wenow apply thisresult tothestudy ofquadratic forms. Given aquadratic form A(x, x)=2 aikziak (am=at.-1'), (1) z.k=1 wewillregard thenumbers El,Z2,...,inasthecomponents ofavector x inann-dimensional Euclidean space R,,,with ascalar product deﬁned by theformula (X.y)=g1€mt. 273 274 QUADRATlC roRMs INEUCLIDEAN AND UNITARY SPACES CHAP. 10 where y=(nl,hl,...,nu). Thebasis el= (l,O,...,O), e2=(0,l,...,0), e"=(0,0,...,l) isanorthonormal basis inR",andclearly _'M=1.“5*’3 x= Y=Z1ﬁi@t- Now consider thebilinear form '¥M=“taA(x» Y):_ikgink corresponding tothequadratic form (1).ByTheorem 10.11, thisform has anorthonormal basisfl,f2, ...,f,,. Ifthecomponents ofthevectors xand yare1l,12,...,1,,andBl,02,...,0",respectively, inthisbasis, then we canwrite thebilinear form A(x, y)as A(x, Y):Z7\tTt0¢ i=1 andthequadratic form A(x, x)as A(x,x)=ME». (2) The transformation from thebasis el,el,...,e,,tothebasis fl,f2, ...,f,, isgiven by t.=§1q£”e. <i"=1,2.....~>, where Q:||q§"’|| isanorthogonal matrix (Sec. 8.93). According tothe formulas (36), p.240, therelation between thecomponents 1l,12,...,1,, and El,Z2,...,5,,isgiven bythesystem ofequations n E;=Zq‘,-"11 (j=1.2,---,'1), (3) =1». involving thetransposed matrix Q’.Thus wehave proved thefollowing important THEOREM. Every quadratic form (1)inann-dimensional Euclidean space R"canbereduced tothecanonical form (2)bymaking anisometric coor- dinate transformation (3). SEC. 10.1 BASIC THEOREM ON QUADRATIC FORMS lN AEUCUDEAN SPACE 10.13. The sequence ofoperations which must beperformed inorder to construct thecoordinate transformation (3)andthecanonical form (2)of thequadratic form (1)canbededuced from theresults ofSecs. 4.94 and 9.45. Wenow give thissequence ofoperations inﬁnal form: a)Usethequadratic form (1)toconstruct thesymmetric matrix A=||a,,,||. b)Form thecharacteristic polynomial A(7\) :det(A~7.E)andﬁnd its roots. BySec. 9.45, this polynomial hasn(not necessarily distinct) real roots. c)From aknowledge oftheroots ofthepolynomial A01), wecanalready write thequadratic form (1)incanonical form (2); inparticular,,we can determine itspositive andnegative indices ofinertia. d)Substitute theroot klintothesystem (28), p.I10. Forthegiven root kl,thesystem must have anumber oflinearly independent solutions equal tothemultiplicity oftheroot kl.Find these linearly independent solutions byusing therules forsolving homogeneous systems oflinear equations. e)Ifthemultiplicity oftheroot klisgreater than unity, orthogonalize theresulting linearly independent solutions byusing themethod ofSec. 8.61. f)Carrying outtheindicated operations forevery root, weﬁnally obtain asystem ofnorthogonal vectors. Wethen normalize them bydividing each vector byitslength. Theresulting vectors fl:(q1ns 1111’, ---9(firm): f2Z(q12)9 11;”, ~~~11122,)» f.=(q1"’.111"’.---.112"’) form anorthonormal system. g)Using thenumbers ql”, wecanwrite thecoordinate transformation (3). h)Toexpress thenew components 1l,12,...,1" interms oftheold components El,E2,...,E,,,wewrite T;":2qi'j)€t' (.1-:‘192s'--an)» /‘=1 recalling thattheinverse oftheorthogonal matrix Qisthetransposed matrix Q’. 10.14. InSec. 7.33a wesaw that neither thecanonical form northe canonical basis ofaquadratic form isuniquely deﬁned inanaﬁine space; ingeneral, anypreassigned vector canbeincluded inthecanonical basis ofthequadratic form. Thesituation isquite dilferent inaEuclidean space, provided that only orthonormal bases areconsidered. Thepoint isthat the matrix ofthequadratic form andthematrix ofthecorresponding symmetric linear operator transform inthesame way, asalready noted inSec. 8.91. QUADRATIC FORMS lN EUCLIDEAN AND UNITARY SPACES CHAP. l0 Thus acanonical basis forthequadratic form isatthesame time abasis consisting oftheeigenvectors ofthesymmetric operator, and theco- efﬁcients ofthequadratic form relative tothecanonical basis (the“canonical coefficients”) coincide with theeigenvalues oftheoperator. Buttheeigen- values oftheoperator Aaretheroots oftheequation det(A——XE)=0,an equation which does notdepend onthechoice ofabasis andisaninvariant oftheoperator A.Hence thesetofcanonical coeﬂicients oftheform (Ax, x) isuniquely deﬁned. Asforthecanonical basis ofthequadratic form (Ax, x), itisdeﬁned with thesame arbitrariness asinthedeﬁnition ofacomplete orthonormal system ofeigenvectors oftheoperator A,i.e., apart from permutations oftheeigenvectors, wecanmultiply anyofthem by-1, or more generally, wecansubject them toanyisometric transformation inthe characteristic subspace corresponding toaﬁxed eigenvalue 7.. l0.2. Extremal Properties ofaQuadratic -Form 10.21. Next, given aquadratic form A(x, x)inaEuclidean space R,,, weexamine thevalues ofA(x, x)ontheunitsphere (x,x)=1ofthespace R,,,andinquire atwhat points oftheunit sphere thevalues ofA(x, x)are stationary. Itwill berecalled that bydeﬁnition adiﬁ‘erentiable numerical function f(x), deﬁned atthepoints ofasurface U,takes astationary value atthepoint xl,eUifthederivative ofthefunction f(x) along anydirection onthesurface Uvanishes atthepoint xl,.Inparticular, thefunction f(x) is stationary atthepoints where ithasamaximum oraminimum. The problem ofdetermining thestationary values ofaquadratic form ontheunitsphere isaproblem involving conditional extrema. One method ofsolving theproblem istouseLagrange’s method,T asfollows: Wecon- struct anorthonormal basis inthespace R,,anddenote thecomponents of thevector xinthisbasis byEl,E2,...,E,,.Inthiscoordinate system, our quadratic form becomes '1M=H§A(x,X)= 'k€iE.l-s andthecondition (x,x)=lbecomes 'ie=1. i=1 Using Lagrange’s method, weconstruct thefunction 7.1-"M=H§$1T E2F(€1»E2» ---9an) : :iZ.1,-'_ A ’rSeee.g., R.Courant, Differential andIntegral Calculus, Vol.ll(translated byE.J. McShane), lnterscience Publishers, lnc., New York (1956), p.190. SEQ 10-2 EXTREMAL PROPERTIES orAQUADRATIC FORM 277 andequate tozeroitspartial derivatives withrespect toE,(i=l,2,...,n), recalling that a,-k=al,,-: z§a,,,z,,_z1z,=o (i=1,2,...,n). 7:=1 After dividing by2,weobtain thefamiliar system (a11 _7\)€1 ‘l‘a12€2 -l"'''+anti" =0, a21€1 +(1122 —7\)E.2 -l‘‘"-l‘again =0, a1i1€1 +a1-1252 +'''+(amt '_)‘)€n =0 (cf.p.110), which serves todeﬁne theeigenvectors ofthesymmetric operator corresponding tothequadratic form A(x, x).Itfollows that thequadratic fvrm A(x, x)takes stationary values atthose vectors oftheunitsphere which areeigenvectors ofthesymmetric operator Acorresponding totheform A(x, x), 10.22. Wenow calculate thevalues which theform takes atitsstationary points. Todothis, weintroduce thecorresponding symmetric operator A andwrite thequadratic form as A(x, x)=(Ax, x). Suppose thatA(x, x)takes astationary value atthevector e,.Since wehave just shown that e,isaneigenvector oftheoperator A,i.e.,Ae,=).,e,, we have A(ei~ ea):(Aer: ea)=7\r(eo er)=7%- Hence thestationary value oftheform A(x, x)atx=e,-equals thecorre- sponding eigenvalue oftheoperator A.Since theeigenvalues oftheoperator Aarethesame asthecanonical coefficients oftheform A(x, x),wecan conclude that thestationary values oftheform A(x, x)coincide with its Canonical coefﬁcients. Inparticular, themaximum oftheform A(x, x)on theunitsphere isequal toitslargest canonical coefficient, andtheminimum OfA(x, x)ontheunitsphere isequal toitssmallest canonical coefficient. 10.23. Quadratic forms and bilinear forms canboth beconsidered not only onthewhole n-dimensional space R,,,butalso onak-dimensional subspace RkCR",andwecanthen look foranorthonormal canonical basis inR1.Letthequadratic form A(x, x)have thecanonical form A(x,X)=Mil+MEZ+'''+ME: (4) 278 QUADRATIC roams lNEucuoeim AND UNITARY srxces CHAP. to inthewhole space R,,,andthecanonical form A(x» xi:l*1Ti ‘l‘P-2T; ‘l‘'''+l*kTi~ inthesubspace Rk.Wenow ﬁnd therelation between thecoefficients ul,ul,...,ul,and thecoefficients kl,7.2,...,7.".For convenience, we assume that thecanonical coeﬁicients arearranged indecreasing order, i.e., that 7*1>7\2>"'>7\n» l‘L1>l‘L2>"'>l‘Lk- Asweknow, thequantity klisthemaximum value ofthequadratic form A(x, x)ontheunit sphere ofthespace R";similarly, it,isthemaximum value ofA(x, x)ontheunit sphere ofthesubspace Rk.This implies that p.l<kl.Moreover, wealso have pLl>).,,_,,,l. Toseethis, letel,el,...,e,, bethecanonical basis inwhich A(x, x)takes theform (4).Consider the (n-k+1)-dimensional subspace R’spanned bythevectors el,el,..., e,,_,,+l. Since k+(n—k+1)>n,then, byCorollary 2.47c, thesubspaces R’andRkhave atleast onenonzero vector incommon. Letthisvector be x0=(&l°’.....51°11...0.....0). and assume that xl,isnormalized, i.e.,that |x0|=l.According to(4),we have A(-E0, xo)=l1(€l°’)2 ‘l"'''‘l‘)‘n—lr+1(€lt0lk+1)2 ->)‘n—k+1(£lm)2 +'''+)‘rt—k+1(€:'t‘Bk+1)2 =7\1t-k+1- This implies thatul,themaximum value ofthequadratic form A(x, x)onthe unit sphere ofthesubspace Rk,cannot belessthan ).,,_,,+l, asasserted. Thus thequantity tilsatisﬁes theinequalities A1>l*1>7\1t-k+1- (5) 10.24. Naturally, thequantity p.ltakes diﬁerent values fordiﬁerent k-dimensional subspaces. We now show that there exist k-dimensional subspaces forwhich theequality signs hold in(5).Let_R' bethesubspace spanned bytheﬁrstkvectors el,el,...,el,ofthecanonical basis oftheform A(x, x).Then A(x, x)isjust A(x» X)=Aizi ‘l‘A25; ‘l‘'''+Altai inthebasis el,el,...,el,ofR’.Inparticular, A(el, el)=kl=max A(x, x). lrl=_1 Thus thequantity ,ek :11=lL1(Rk) =1113A(x.X)‘;”.;,, takes itsmaximum value klforRl.:R’. sec. 10.2 EXTREMAL PROPERTIES orAQUADRATIC FORM 279 Next letR”bethesubspace spanned bythelast kvectors e,,_,,+l, e,,_,,+2,. ..,e,,ofthecanonical basis oftheform A(x, x).Then A(x, x)isjust A(x» x)=)‘n—lr+1€:—k+1 +'''‘l‘Aug: inthebasis e,,_,,+l, ...,e,,ofR".Inparticular, A(@1t-k+1» e1t—lr+1) =)‘n—k+1 =max A(x, X),‘.34.?! and, justasbefore, weconclude that 11.1takes itsminimum value ).,,_,H_l for Rk=R".Thus weobtain thefollowing new deﬁnition ofthecoefficient ).,,_,,+l: Thecoeﬂicient 7.,,_,,+l inthecanonical representation ofthequadratic form A(x, x)equals thesmallest value ofthemaximum ofA(x, x)ontheunit spheres ofallpossible k-dimensional subspaces ofthespace R,,. 10.25. Using thisresult, wecanestimate theother canonical coefﬁcients ofthequadratic form A(x, x)onthesubspace Rk.Forexample, ifthesub- space Rkisﬁxed, then 1.12isthesmallest value ofthemaximum ofA(x, x)on theunit spheres ofallthe(k-1)-dimensional subspaces ofR2,while ).,,_,,+2 isthesmallest value ofthemaximum ofA(x, x)ontheunit spheres ofallthe(k+1)-dimensional subspaces ofthewhole space R".Hence we have 1.12>7.,,_,,+2, andsimilarly l*z>)\7l—'k'l'37 P-4>7\1t-k+4i ---iP-it>)‘n' Ontheother hand, 7.2isthesmallest value ofthemaximum ofthequadratic form A(x, x)ontheunitspheres ofallthe(n+1)-dimensional subspaces of thewhole space R,,.But, according toCorollary 2.47c, theintersection of every (n—l)-dimensional subspace with thesubspace Rkisasubspace ofno lessthan (n—l)+k—n=k-ldimensions, sothat7.2isnolessthan the smallest value ofthemaximum ofA(x, x)ontheunit spheres ofallsuch subspaces; inparticular, 7.2isnolessthan 1.12,thesmallest value ofthe maximum ofA(x, x)ontheunit spheres ofallthe(k-l)-dimensional subspaces ofRk.Therefore wehave 7.2>1.12,and similarly 7.3>1.13,..., Al,>ul,.Thus thecanonical coefﬁcients ul,1.12,...,1.1,,satisfy theinequalities A1>PL1>)‘n—k+1s X2>PL2>)‘n+k+2» 7\k>'l1-it? >*=. Fork=n-l,theinequalities (6)become A1>l*1>A2. A2>PL2>A3: (7) )‘n—1 >l‘Lrt—1 >)‘n' 280 QUADRATIC FORMS lNEUCLIDEAN AND uNi'rARY SPACES CHAP. 10 *10.26. Consider thebehavior ofthequadratic form ..._.l\/11?’_;'v<_.no A(x, x)= inthe(n+l)-dimensional subspace R,,_l speciﬁed bytheequation <=<.£.+<=<.£.+-~-+<=<.i.=0 (<=<i+<=<§+---+<ti=1). (8) Assuming that allthecoefficients ).l,).2, ...,7." arediﬁerent, wecan calculate thecoefficients ul,1.12,...,i.t,,_l byusing amethod duetoM.G. Krein. Atleast one ofthecoefﬁcients otl,<12,. ..,at"isnonzero. For example, suppose at"qé0.Then (8)implies rt—1l an Z_‘__ Za‘J'El'ot,,i=1 Substituting thisexpression forE"into A(x, x),weﬁndthat A(x, x)hasthe form n—1 2 v2 g2 v )‘nA(x, -Y)Z7\1€1 +Z222 -l-'''-l"7\.t_1Q.i-1 ‘l‘_2(2°‘t€t)ani=1 inthesubspace R,,_l, interms ofthevariables El,E2,...,E,,_l. The canonical coefficients ofthisquadratic form arethesame asitsstationary values ontheunitsphere ofthesubspace R,,_l (Sec. 10.22). Inthevariables El,E2,...E,,_l thissphere hastheequation 1 11-1 2B(x,x) =if+£2+---+El.+-2(§<..,.£,) =1. ot,,i=1 Just asbefore, wedetermine these stationary values byusing Lagrange’s method. Thus weform thefunction n—1 A __A11-1 2A(x.x>-1B(x.x> =Z10.-ma?+ _;"1<=t,z.), and equate tozero itspartial derivatives with respect toEl,(k=1,2,... n-1),obtaining x,,—7."1! €kO‘k T‘7‘)+12i<2.”-iii-i) 11¢:0- (9) an 7:1 The required coefficients ul,5.12,...,i.t,,_l aretheroots oftheequation obtained byequating tozero thedeterminant D().) ofthesystem oflinear equations (9).Thecoefficient matrix ofthissystem isclearly thesum oftwo SEC. 10.2 EXTREMAL PROPERTIES OF AQUADRATIC FORM LOI matrices; theﬁrstmatrix isdiagonal with thenumbers 7%~A(k=1,2,..., n——1)along thediagonal, while thesecond matrix hastheform °‘1°‘1 @211 '''°‘n~1°‘1 )\"-_)\ oc1oc2 oc2oc2 '''oc,,_1oc2 2an . - . °‘1°‘n-1 °‘2°‘n—1 '''°'-1»-1%-1 Bythelinear property ofdeterminants (Sec. 1.44), thedeterminant D(7\) is thesum ofthedeterminant oftheﬁrst matrix and allthedeterminants obtained byreplacing oneormore columns ofthedeterminant oftheﬁrst matrix bythecorresponding columns ofthesecond matrix andtaking account ofthefactor (kn—X)/aft. Since anytwocolumns ofthesecond matrix are proportional, weneed only consider thecasewhere oneofthecolumns ofthe determinant oftheﬁrst matrix isreplaced bythecorresponding column of thesecond matrix. Inparticular, ifthekthcolumn oftheﬁrstmatrix isreplaced bythekth column ofthesecond matrix, theresulting determinant hastheform X1—A O ''' O ockocl O O 0 X2——X-' O ockoc2 0 0 )\n_)\ 0 0 7\k_1 —Xockoc‘._1 O 0 <11 0 0 0 <><,_,¢k 0 0 0 0 O oc;5xk.+1 AR-_\‘1 —X 0 0 0 0 oc,cv.,,_1 0 --)\,,_1 — 2H(M~X)_oz‘.1-=1 _ <13 X,‘—X ’I'3"Iv-1/\>’>.-Denote thedeterminant oftheﬁrstmatrix by andlet Go)=0%—A). 282 QUADRATlC roams mEUCLlDEAN AND UNITARY SPACES CHAP. 10 Then therequired determinant D(75) becomes l "*1 <11?D(75) =F(75) +T,G(~75) Z———-— , (10) oz’, 1.»=1751.-75 Solving theequation D(75) =0,weﬁnd thequantities 5&1,5&2,...,5&,,_1 in which weareinterested. Note thatthese quantities depend onthesquares of thenumbers oakrather than onthenumbers oakthemselves. Thus changing thesign ofoneormore coeﬁicients in(8)does notchange thecanonical coeﬁicients oftheform A(x, x)inthesubspace R,,_1. *10.27. Equation (10) isofparticular interest inthat itallows usto construct from given numbers 5&1,5&2,...,5&,,_1 satisfying theinequalities (7) asubspace R,,_1 inwhich theform A(x, x)hasthecanonical coeﬂicients 5&1,5&2,...,5&,,_1. (Again itisassumed that thenumbers 751,752,...,75,,are distinct.) Wenow show how thisisdone. First wenote that (l0)canbewritten intheform ~ n—1 2 n 2(X:D(/5) =‘X2F05) +Z oak :Z oak I (11) G(75) GO.) 1¢=1751.——75l'=1751.——75 Thus thenumbers oti,mg,...,aftareproportional tothecoeﬁicients obtained when weexpand therational function D(75)/G(75) inpartial fractions. Now suppose wearegiven numbers 5&1,5&2,...,5&,,_1 satisfying theinequalities 751>5&1>752, 752>5&2>753, (12) )‘n—1 >p'rt—1 >)‘n' Let n~1vw—Hw—n1_ lr=-1 andexpand therational function D1(75)/G(75) inpartial fractions &@:iL+iL+m+;L_ M, G(75) 751—75 x2~x x,,~x Thecoefﬁcients c1,c2,...,c,,aregiven bythefamiliar formulaT c_ Mm __Mm krr rr , » (M“7%)'‘'(7%-1 “7%l(7%+1 '"7%)‘''(7%“7%) G(7%) 1'Seee.g., R.A.Silverman, Modern Calculus andAnalytic Geometry, The Macmillan Co., New York (1969), p.861. SEC. 10.3 SIMULTANEOUS REDUCTION OE TWO QUADRATIC FORMS andallhave thesame sign. Toseethis, wenote that thenumbers D1(751), D1(752), ...,D1(75,,) alternate insign, since, byhypothesis, theroots ofthe polynomial D1(75) altemate with theroots ofthepolynomial G(75). Thus the numbers D1(75k)/G'(75k), andhence thecoeﬁicients ck(k=l,...,n),allhave thesame sign. Bysupplying anextra factor, wecanassume that theckare allpositive andaddupto1.Wecanthen deﬁne thenumbers a1,a2,...,an bytheformulas 1 1%ZC19 (X;ZC21 ~--9aftZcna where each oakcanhave either sign. Finally weshow that thesubspace R,,_1 deﬁned bytheequation °l1£1‘l“°l2€g‘l“""l“°@,,€,,:0 istherequired subspace, inwhich thequadratic form A(x, x)hasthecanonical coefﬁcients 5&1,5&2,...,5&,,_~1. Infact, asproved above, thepolynomial D(75) whose roots arethecanonical coeﬁicients ofA(x, x)inthesubspace R,,_1 isgiven byformula (10)ortheequivalent formula (ll). Comparing (11) with (13)andusing (l4), weﬁndthat thepolynomial D(75) diﬁers only bya numerical factor from thepolynomial D1(75) just constructed. Butthen the roots ofD(75) coincide with thenumbers 5&1,5&2,...,5&,,_1, asrequired. Remark. Itcanbeshown thatthenumbers a1,...,oz”depend continuously onthenumbers 751,...,75,,,5&1,...,5&,,_1. Using this fact, wecan verify that theproblem canstillbesolved ifthenumbers 5&1,...,5&,,_1 satisfy the inequalities (7)instead of(12) orifthenumbers 751,...,75,,arenolonger distinct. l0.3. Simultaneous Reduction ofTwo Quadratic Forms 10.31. Thefollowing question plays animportant roleincertain problems ofmathematics andphysics: Given twoquadratic forms A(x, x)andB(x, x) deﬁned inann-dimensional aﬂine space R,,,howdoes oneﬁnd abasis inwhich both A(x, x)andB(x, x)arereduced tocanonical form (i.e., tosums ofsquares ofthecomponents ofxwith certain coeﬁicients)? Thefollowing example in theplane (n=2)shows that thisproblem does notalways have asolution: Consider thetwoforms A(x,x) =ii—ii, B(x,-Y)=5552- Finding acommon canonical basis forthese twoforms isthesame asﬁnding acommon pair ofconjugate vectors forthehyperbolas A(x, x)=land B(x, x)=1(seeSec. 7.42). Since these areequilateral hyperbolas, weknow from analytic geometry that theconjugate directions ofthehyperbolas are 284 QUADRATIC roams 1NEUCLIDEAN AND UNITARY SPACES CHAP. 10 symmetric with respect totheir asymptotes. Therefore thepolar angles <91 and <92corresponding tothepair ofconjugate directions satisfy therelation <P1‘l‘<P2=% fortheﬁrsthyperbola andtherelation ‘P1+492:0 forthesecond hyperbola (both relations hold only towithin anintegral multiple ofTC).Since thetworelations aremutually exclusive, there does not exist acommon pair ofconjugate vectors inthiscase. Itturns outthattheproblem ofsimultaneous reduction oftwoquadratic forms does have asolution ifwemake thesupplementary assumption that oneoftheforms, sayB(x, x),ispositive deﬁnite, i.e.,that B(x, x)>0for xqé0.Inthiscase, theexistence ofasolution iseasily proved asfollows: LetB(x, y)bethesymmetric bilinear form corresponding tothequadratic form B(x,x), and introduce aEuclidean metric intheaﬁine space R,,by writing (X,y)=B(x,y)- ThefactthatB(x, y)issymmetric andpositive deﬁnite guarantees that (x,y) satisﬁes theaxioms forascalar product. BySec. 10.11 there exists anortho- normal basis (with respect tothismetric) inwhich A(x, x)takes thecanonical form A(x,K)=Mii+ME?+---+M51, (15) where Z1,Z2,...,2,,denote thecomponents ofthevector xinthebasisjust found. Inthesame basis, thesecond quadratic form B(x, x)becomes B(x.x>=(x.x> =ni+n§ +---+ni. byformula (17), p.222. Hence, asasserted, there exists abasis inwhich both A(x, x)andB(x, x)have canonical form. 10.32. Toconstruct thecomponents ofthevectors e1,...,e,,ofthebasis which issimultaneously canonical forboth quadratic forms, weusethe extremal properties ofquadratic forms. Asshown inSec. 10.21, thevectors e1,...,e,,oftherequired basis arethevectors obeying thecondition (x,x)=B(x, x)=l forwhich theform A(x, x)takes stationary values. Suppose A(x, x)and B(x, x)aregiven by '-'M='l'M=..,,,5'-‘ A(x: x) :1 giika B(x, x): bmgiik SEC. 10.3 SIMULTANEOUS REDUCTION orTWO QUADRATIC FORMS 285 intheoriginal basis. Using Lagrange’s method, weform thefunction ..em:3 F(€1, £2,---,E")= ailcgdzk "'11-gbikitik, 1‘>1 >- andthen equate tozero itspartial derivatives with respect toalltheE,~: n wtZane, -51.2171-;¢€k =0 (i=1,2,...,n). (16) k=1 Ic=1 The resulting system ofhomogeneous equations (an *"l*b11)€1 ‘l‘(a12 ““l*b12)€z +‘‘‘‘l‘(ant _l‘l'b1n)€n :0’ (a21"‘P~b21)€1 +(1122—l"'b22)€2 -l-‘‘‘-l-(1121. —p'b2n)€n =0, (17) (anl —l‘l'bn1)€1 +(an2 —l‘l'bn2)€2 +'''+(arm '_l‘l'bnn)€n :0 hasanontrivial solution ifandonly ifitsdeterminant vanishes: an_P1711 a12”"P1712 ‘''ant“‘P171" (121”"P-I721 a22_P1722 ---1121»_P172" :0 (18) anl _l‘l'bn1 an2 —_l‘l'bn2 ‘''amt '_“bun Solving (18), weﬁndnsolutions 5&=5&k(k=1,2,...,n).Then substituting 5&kinto thesystem (17), weﬁnd thecomponents Z1"), Kg”, ..,Eff”ofthe corresponding basis vector ek.The results ofSec. 10.31 guarantee that (18) hasnrealroots andthatevery root ofmultiplicity rcorresponds torlinearly independent solutions ofthesystem (17). 10.33. Tuming tothecalculation ofthecanonical coeﬁicients, wenow show that thecoefficients 751,752,...,75,,inthecanonical representation (15) oftheform A(x, x)coincide with thecorresponding roots 5&1,5&2,...,5&,, ofthedeterminant (18). Wecould useanargument likethat given inSec. 10.22, butweprefer tocarry outadirect calculation. Given theroot 5&,,,,we multiply theithequation ofthesystem (16) byZ?" (the ithcomponent of thesolution corresponding to5&,,,)fori=l,2,...,nandthen addallthe resulting equations, obtaining A(e,,,, e,,,)=iaik€§’"’€§,'"’ Iumibi-1.€l’"’€l,'"’ It*mB(@1,., em)=Hm,(19) i.k=1 i.I5-=1 since B(e,,,,e,,,) I1.Ontheother hand, ifhi”), hf"), ...,njffl arethe canonical components ofthevector e,,,,then obviously “q§'"l =0ifi2%m 286 QUADRATIC roams mEUCLIDEAN AND UNITARY SPACES CHAP. 10 while 'qf,’,"l =l,andhence A(e...e...)=Zt.(ni'"’)’ IA... (20) i=1 Comparing (19) and(20), weget5&,,,I75,,,,asasserted. This result allows ustowrite A(x, x)incanonical form, without calculating thecanonical basis. 10.34. The problem posed inSec. 10.31 ofsimultaneously reducing two quadratic forms A(x, x)andB(x, x)tocanonical form, where oneofthe forms, sayB(x, x),ispositive deﬁnite, wassolved inarather strong form, i.e., wereduced B(x, x)toasumofsquares with coefﬁcients equal to1.Ingeneral, thisisnotrequired, andhence thecoefﬁcients ofthecanonical forms arenot uniquely determined. Nevertheless, aswenow show, theratios ofthe corresponding canonical coeﬂicients arestill independent ofthemeans used tosimultaneously reduce A(x, x)andB(x, x)tocanonical form. Suppose that A(x, x)andB(x, x)have been simultaneously reduced to canonical form intwo diﬁerent ways, i.e., suppose that inthevariables E1,Z2,...,Z,,wehave 7! TL A(x,X)IZlhii, B(x,X)IZlvi-ii, 7= Z: while inthevariables n1,'q2,...,“q,,wehave A(x, x)I 9,-hi, B(x, x)Z 'r,~‘q€. 2=1 'L=1 Since theform B(x, x)ispositive deﬁnite, thenumbers v,and-r,(iI1,2, ...,n)areallpositive. Consider thenewcoordinate transformation gt= ii, ‘hi= ‘ﬁt- Then theforms A(x, x)andB(x, x)become A(x, x): B(x, x)= ~2_< ;M=J\‘ inthevariables Z,and A(x.x)=iﬁii,Bo.x)=Eiti=1T, =1 inthevariables $1,.Lete1,e2,...,e,,bethebasis corresponding tothe variables Z,,andletf1,ﬂ, ...,f,,bethebasis corresponding tothevariables $1,».Both these bases areorthonormal inthemetric determined bytheform B(x, x).Moreover, according toSec. 10.14, thesetofcanonical coefficients ofthequadratic form A(x, x)isuniquely determined. Hence thetwosetsof SEC. 10.4 REDUCTION OE TI-IE GENERAL EQUATION OF AQUADRIC SURFACE numbers 751/v1, 752/v2, ...,75,,/v,, and p1/T1, p2/T2, ...,p,,/-r,, must coincide, except possibly fororder, andourassertion isproved. l0.4. Reduction oftheGeneral Equation ofaQuadric Surface 10.41. Inthisandsubsequent sections, wewillcalltheelements ofthe n-dimensional linear space R,,points rather than vectors (cf.Sec. 2.17), which ismore inkeeping with thegeometry ofthesituation. Byaquadric (orsecond-degree) surface inR,,ismeant thelocus ofthepoints x= (Z1,Z2,...,En)ER,,which satisfy anequation oftheform Zaikgigk ‘l‘22brat -l‘6'=0 (21) i'.k=L i=1 OI‘ A(x, x)+2L(x) —l—cIO, where M»o=ia@n 1.1‘:-1 isaquadratic form inthecomponents oftheradius vector ofthepoint x, uo=§mc isalinear form, andcisaconstantfr Wewill assume that thespace R,,isEuclidean and that thenumbers E1,E2,...,2,,arethecoordinates ofthe point xwith respect toanorthonormal basis. Theproblem ofthissection isthen tochoose aneworthonormal basis inR,,such that ourquadric surface isspeciﬁed byaparticularly simple equation, called thecanonical equation ofthesurface. Subsequently, we willusethecanonical equation tostudy theproperties ofthesurface. 10.42. First ofall,asinSec. 10.12, wemake anorthogonal coordinate transformation e=Zwm» v=t1 so (H)i=1 inR,,,reducing thequadratic form A(x, x)tothecanonical form A(x,x)I2,72-n2,~. 1'Inthecase n-2,thegeometric object deﬁned by(21) iscalled asecond-degree curve. However, wewillhenceforth always usetheword “surface,” despite thefactthat, strictly speaking, itshould bechanged to“curve” whenever n=2. 288 QUADRATIC roams [NEUCLIDEAN AND UNITARY SPACES CHAP. 10 Substituting (22) into (21), weget ZMi-l"2Zlmt+CI0, (33) i=1 i=1 where theI,(i=1,2,...,n)arethenew coeﬁicients ofthelinear form L(x). If75,¢0forsome iin(23), wecaneliminate thecorresponding linear term byappropriately shifting theorigin ofcoordinates. For example, if 751960,wehave I2I2 7\1‘0i ‘l‘21101 :7\1('01 ‘l‘ ”"xi- Wethen set I1 1 .I= +_1’41 ‘ﬁt X1 which isequivalent toshifting theorigin tothepoint (IlI,O,O,...,O). 7\1 Asaresult ofthissubstitution, thepair ofterms 751nf +2l1'q1 ischanged to ,k.2L: 11 s i.e.,thequadratic term hasthesame coeﬁicient asbefore, thelinear term disappears, and lf/75f issubtracted from theconstant term. After making allsuch transformations, theequation ofthesurface becomes xlnf +A20: +'--+Ann: +2lr+1nr+1 +--'+Zlrmn +CI Here, forsimplicity, wehave dropped theprimes onthevariables *q:.,and wehave renumbered thevariables insuch awaythatthevariables appearing inthequadratic form come ﬁrst, i.e.,751,752,...,75,arenonzero and75,,I0 fork>r.Ifr:norifthenumbers l,+1,l,+2, ...,l,,allturn outtobezero, weobtain theequation mi+mi+---+mi+cI0, (24) called thecanonical equation ofacentral surface. Aquadric surface is said tobenondegenerate ifallnvariables appear initscanonical equation, anddegenerate iflessthan nvariables appear initscanonical equation. A nondegenerate central surface, with canonical equation mi+m§+---+mi+c=0 (25) (i.e., such that r=n),issaid tobeaproper central surface ifcqé0anda conical surface ifc:0.The meaning ofthisterminology willbeapparent later. SEC,10.5 GEOMETRIC PROPERTIES orAQUADRIC SURFACE 289 Now suppose atleast oneofthenumbers l,+1,l,+2, ___,1”isnonzero, and carry outanew orthogonal coordinate transformation byusing the formulas T1I'01, T2I'02, T.Im. (26) 1 TH-1 I“T_];(lr+1nr+1 +I'-+Irma)» where Misapositive factor guaranteeing theorthogonality ofthetrans- formation matrix. Since thesum ofthesquares oftheelements ofevery row ofanorthogonal matrix must equal l,wehave M2:l:+1+l:+2+"'+l€t- The remaining rows (i.e., rows r+2,r+3,...,n)can bearbitrary, provided only that theresulting matrix isorthogonal (see Sec. 8.95). Asa result ofthetransformation (26), theequation ofthesurface takes theform A1¢f+---+mi=2M¢,,1 —c. Ifc¢0,another shift oftheorigin given bytheformula , cT =T ———— ,r+1 7+1 O1‘ 2M¢;+1 =2M¢,+1 ~C, allows ustoeliminate theconstant term. Then, dropping theprime on-r,'+,, weobtain theequation A1121 ‘l‘'''‘l‘7‘??? :2MTr+1» (27) called thecanonical equation ofanoncentral surface. l0.5. Geometric Properties ofaQuadric Surface 10.51. Thecenter ofasurface. Byacenter ofasurface ismeant apoint X0I(E‘L E3,---.59.) with thefollowing property: Ifthepoint (€g_l_€1s€g_l_€2>~'-s€‘3;_l_En) 290 QUADRATIC roams 1NEUCLIDEAN AND UNITARY SPACES cuAr>_ i0 liesonthesurface, then thepoint €21- - in), which issymmetric with respect toxo,also liesonthesurface. Asurface with thecanonical equation (24) hasatleast onecenter, since every point forwhich mImI~=m=0 0% isobviously acenter. This explains why such surfaces arecalled central surfaces. Wenow show that asurface with thecanonical equation (24) hasno centers other than thepoint (28), afactthat willbeused later. Toseethis, let(if,Z3,...,Z2)beacenter ofthesurface. Then therelation ME?+am+MEL’+at+---+ME?+if+c=0 implies mi?—£92+>~2(€2~£2)’+---+ME?~€,)’+c=0. Subtracting theﬁrstequation from theﬁrst, weobtain theequation M3155 +Azigiz ‘l‘'''+NEE, :0, (29) satisﬁed forarbitrary Z1,Z2,...,2,,corresponding topoints onthesurface (24). lfthepoint (Z?+Z1,Z2+Z2,...,Z‘),+Zn)liesonthesurface (24), then sodoes thepoint (IE? IZ1,Z3+Z2,...,Z3+En). But Ii?IE1Iii+(I23 IE17, andhence wehave '_£1) +7‘2ag€2 +'''+xrigir I0’ aswell as(29). Subtracting (29') from (29), weget ZM€‘i(€5 +ii)I0, which implies Z1=—-if ifZ‘;¢0.Butsince Z1canbereplaced byIE1, wealso have IE1 =Iif. This, together with Z1=IE1‘, contradicts the assumption that Z‘;¢0,thereby proving that Z1: 0.Similarly, weﬁnd that ___ _T_ oi asrequired. 10.52. Proper central surfaces. Consider aproper central surface, i.e., asurface with canonical equation (25), where c¢0.Dividing byc.we transform (25)into theform 2 2 2i@i%i~i¥=L a1 a2 a;, SEC. 10.5 GEOMETRIC PROPERTIES orAQUADRIC SURFACE 291 where thenumbers a,aredeﬁned by aﬁ=+Jﬁ u=1J,H,m, andarecalled thesemiaxes ofthesurface. Renumbering thecoordinates in such away that thepositive terms appear ﬁrst, weget 2 2 2 ~2 2 %+%+~+%~%ﬂ-~_%=r (ma1 a2 ak ak,,1 a,, Itisnatural toexclude thecase k=0from consideration, since there areno realvalues -q1,'q2,...,'q,,satisfying (30)ifk=0.(Inthiscase, onesome- times says that (30)deﬁnes an“imaginary” surface.) This leaves ndiﬁerent types ofproper central surfaces, corresponding tothevalues k:1,2,...,n. a.Inthetwo-dimensional case (n=2),wehave k=l,k:2,and equation (30) leads tothetwocurves _ ﬁ¢_(k—1) 2I—_,-_1 (ahyperbola), a1 a2 2 2 (k=2) 3%,+3%:1 (anellipse), ai 45 familiar from analytic geometry. b.Forn =3wehave k=1,k=2,k=3,andthecorresponding proper central surfaces inthree-dimensional space aregiven bytheequations 2 _2 2Wzn m_@_m:1,ai a2 a§ 2 2 2u=m %+%~%=naf a2 a2 _2 2 2w=m %+%+@=i "i "5 "5 Wenow remind thereader oftheconstruction ofeach ofthese three surfaces. Consider thesections ofeach ofthesurfaces made bythehorizontal planes ~q2:Ca2(Ioo<C<00).These sections arerespectively hyperbolas 2 2¢=n Q-%=1+d at a§ 292 QUADRATIC roams 1NEUCLIDEAN AND UNITARY SPACES CI-IAP. I0 with the'/51-axis astransverse axis, ellipses 2 2 (/<=2) 1;+3§=1+c* a1 a2 deﬁned forallvalues ofC,andellipses 2 2(/<=3) “~”—;+1§=1-c2 a1 "2 deﬁned only for|C|<l.Tolocate thevertices ofthese sections, we construct thesections ofeach surface made bythecoordinate planes “n1=0, “n2=0.Inthecase k=l,only thecoordinate plane “n2=0gives areal section, i.e.,thehyperbola Iii_13=1 dd The vertices ofthehyperbola formed bythehorizontal sections lieonthis curve, and asaresult oftheconstruction weobtain thesurface shown in Figure 2,called ahyperboloid oftwosheets. 5.1 5 if FIGURE 2 Inthecase k:2,thesections made byboth planes “n1=0and“n2=0 arehyperbolas dd jﬁ_"2II=1’ 2I2I1a2 aa a1 aa with the“n2-axis astransverse axis. Thesetofellipses formed bythehorizontal sections have vertices lying onthese hyperbolas, andform thesurface shown inFigure 3,called ahyperboloid ofonesheet. Finally, inthecase k:3,the sections made bythecoordinate planes “n1:0,“n2==0areellipses. Drawing theellipses made bythehorizontal sections, weobtain anellipsoid (see Figure 4). SEC. 10.5 GEOMETRIC PROPERTIES orAQUADRIC SURFACE 293 ll: ;:§1fi:i;.ll2'€"5- _-.,_.,_.,_,__ ___.,,_|._._,_.,,.,... FIGURE 3 c.Quadric surfaces inspaces ofmore than three dimensions arenot easily visualized. Nevertheless, even inthemultidimensional case, wecan show essential diﬁerences between thetypes ofproper central surfaces corresponding tothediﬁerent values k=l,2,...,n.Webegin bypointing outdiﬁerences which aregeometrically obvious inthree dimensions. On thehyperboloid oftwo sheets (k=1),there exists apair ofpoints which cannot bemade tocoincide byacontinuous displacement ofthepoints along thesurface; toobtain such apair ofpoints, weneed only take the ﬁrst point ononesheet andthesecond point ontheother sheet. Onthe hyperboloid ofonesheet (k=2),anytwopoints canbemade tocoincide bymeans ofacontinuous displacement along thesurface; however, there exists aclosed curve, e.g., acurve going around the“throat” ofthehyper- boloid, which cannot becontinuously deformed intoapoint. Ontheellipsoid, (k=3),any closed curve canbedeformed into apoint. These facts can 773 \inB!§;(§$§N\l 6'?-€A\\:_::.!//' FIGURE 4 QUADRATIC FORMS 1N EUCLIDEAN AND UNITARY SPACES CHAP. 10 serve asthestarting point forclassifying thegeometric diﬁ‘erences between proper central surfaces inann-dimensional space, aswenow show. Weintroduce thefollowing deﬁnitions: Ageometric ﬁgure Aissaid to behomeomorphic toaﬁgure Bifthere exists aone-to-one, bicontinuousT mapping ofthepoints oftheﬁgure Ainto thepoints oftheﬁgure B.A ﬁgure Alying onasurface Sissaid tobehomotopic toaﬁgure Blying on thesame surface iftheﬁgure Acanbemapped into theﬁgure Bbymeans ofacontinuous deformation, during which theﬁgure Aalways remains on thesurface S. Using these deﬁnitions, wecan formulate thegeometric diﬁerences between theproper central surfaces corresponding todiﬁerent values ofk asfollows: Fork=1wecanﬁndapair ofpoints onthesurface which are not homotopic toeach other. For k=2every point onthesurface is homotopic toevery other point, butthere exists acurve which ishomeo- morphic toacircle andnothomotopic toapoint. Fork=3every curve which ishomeomorphic toacircle ishomotopic toapoint, butthere exists a part ofthesurface which ishomeomorphic toasphere (inthree-dimensional space) andnothomotopic toapoint. Continuing inthisway, wecanformulate thefollowing distinguishing property oftheproper central surface cor- responding toagiven value ofk:Every part ofthesurface which ishomeo- morphic toasphere in(k—l)-dimensional space ishomotopic toapoint, butthere exists apart ofthesurface which ishomeomorphic toasphere in k-dimensional space and not homotopic toapoint. Inparticular, this implies that theproper central surfaces inn-dimensional space (which are obviously homeomorphic toeach other forequal values ofk)arenot homeomorphic toeach other fordistinct values ofk.The proof ofthese facts willnotbegiven here, and canbefound inacourse onelementary topology. 10.53. Conical surfaces. Next weconsider aconical surface, i.e., a surface with canonical equation (25), where c=0.Inthiscase, equation (25) becomes homogeneous, i.e.,ifthepoint (n1,“n2,...,'q,,)satisﬁes (25), then sodoes thepoint (t“q1,t'q2,. ..,t'q,,) foranyt.This means that the surface ismade upofstraight lines going through theorigin ofcoordinates.I Just asbefore, wecanwrite thecanonical equation ofaconical surface in theform 2 2 2 235+---+1;‘—'ZI‘ —%"=°- <31)a1 at an-1 an 1'Equivalently, continuous inboth directions, i.e., continuous with acontinuous inverse. 1Except when alltheterms in(25) have thesame sign, inwhich case (25) deﬁnes a single point, namely theorigin. SEC. 10.5 GEOMETRIC PROPERTIES orAQUADRIC SURFACE 295 Wenow ﬁnd thenumber ofdiﬁerent types ofconical surfaces corre- sponding toagiven value ofn.Ifthenumber ofnegative terms m n-—k inthecanonicai equation (31) isgreater than n/2, then, multiplying the equation by—1,weobtain anequation describing thesame surface butwhich now hasanumber ofnegative terms lessthan n/2. Therefore itissufﬁcient toconsider thecases corresponding tothevalues m<n/2. Ifmiseven, then, excluding thecase ofapoint (m=0),weobtain n/2dilferent types of conical surfaces, corresponding tothevalues m 1,2,...,n/2. Ifnis odd, there are(n—1)/2 diﬁerent types ofconical surfaces, i.e., those corresponding tothevalues m=1,2,...,(n—1)/2. a.Intheplane (n=2),besides apoint, there isonly oneother type of conical surface (m=1),with thecanonical equation 2 2W1 W2_7-GI“a1 a2 The corresponding geometric ﬁgure isapair ofintersecting straight lines with theequations 1:12 111 a2 Inthree-dimensional space (n=3),besides apoint, there isalso only one other type ofconical surface, corresponding to n—1 3-1mzizizl‘ 2 2 with canonical equation 5 NJU-lb’!:5 .->acneNJ___m:02 . §>-§ml Qav The corresponding geometric object isacone. Intheparticular case where a1=a2,thisisaright circular cone (seeFigure 5). b.Tovisualize theform ofaconical surface inthegeneral case, we consider itsintersection with thehyperplane n,,=Ca,, (~00 <C<oo). (32) Substituting (32)into (31), weget 2 2 2 _2g+H.+%_%aLH_%@:g a1 ak ak_,1 a,,_1 This istheequation ofaproper central surface inan(n-—1)-dimensional space. The surfaces corresponding todiﬁerent values ofCareallsimilar to 296 QUADRATIC roams 1NEUCLIDEAN AND UNITARY SPACES crmr. 10 "1 ~~ /// ’ I1”,,,,///// ?“\e~'= / v 1",’ _. '5 \\3\-,5gigg,\\\\ FIGURE 5 each other, with semiaxes proportional tothevalue ofC.Thus every conical surface inthen-dimensional space R,,canbeobtained from acentral surface inthe(n—l)-dimensional space R,,_1 bydisplacing thecentral surface along aperpendicular toR,,_1 andatthesame timeproportionately stretching thesurface inalldirections. Moreover, toobtain allpossible types ofconical surfaces inthisway, weneed onlyusethecentral surfaces inR,,_1forwhich thenumber ofnegative terms inthecanonical equation does notexceed (n—1)/2. 10.54. Nondegenerate noncentral surfaces (paraboloids). Just asinSec. 10.52, wecanreduce thecanonical equation ofanondegenerate noncentral surface totheform _2 2 2 2%+~+%-I’ nF=%. caa1 an ak+1 an—1 Wenow ﬁnd thenumber ofdiﬁerent types ofnondegenerate noncentral surfaces. Ifthenumber ofnegative terms intheleft-hand side of(33) is greater than (n—1)/2, then, multiplying (33)by—l,weobtain theequation ofthesame surface, butwith anumber ofnegative terms intheleft-hand sidewhich islessthan (n—1)/2andwith achange ofsignoftheright hand side. The sign oftheright-hand side isrestored bythemirror reﬂection ‘Q;—-'q,,. Thus, ifwedonotcount surfaces obtained from each other by mirror reﬂections asbeing ofdilferent types, thenumber ofdiﬁerent types sec.10.5 GEOMETRIC PROPERTIES orAQUADRIC SURFACE 297 ofnondegenerate noncentral surfaces isequal tothenumber ofintegers satisfying theinequality 0<m<(n—1)/2. This number equals n/2ifn iseven and(n+1)/2ifnisodd. a.Intheplane (n=2)there isonly onenondegenerate noncentral curve, i.e.,theparabola with canonical equation niIZaim. (mI0)-i b.Inthree dimensions there aretwonondegenerate noncentral surfaces ,,:3'i1:3;r1:2’2 2 " 2 2%+%=m. w=ma1 a2 2 2%—%=m. m=ua1 a2 Intheﬁrst case (m-=0),thesections ofthesurface made bytheplane ‘Q3=C>0isanellipse. Toﬁndtheposition ofthevertices ofthisellipse, weconstruct thesections ofthesurface made bythecoordinate planes n1=0and “n2=0.Each ofthese sections isaparabola, and theinter- sections ofthese parabolas with theplane n2=Clocate thevertices ofthe ellipse. Theresulting surface, shown inFigure 6,iscalled anelliptic parabo- loid(acircular paraboloid inthespecial case where a1=a2). Inthesecond case (m=1),thesection ofthesurface made bytheplane “qa=C>0isahyperbola with the'q1—axis asitstransverse axis. Toﬁnd ll: ” Illlo"“-//////////////~ _|.:;._-;_-,,..~.-:'-:-.- ti‘— -';A;r;;_ 'I1-"_-'--'_~.2:_-j-;;'.;.-*.'1"" ‘s FIGURE 6 1'Notethatnowm=n-—1—k. 298 QUADRATIC roams 1NEUCLIDEAN AND UNITARY SPACES CHAP. 10 0’ FIGURE 7 theposition ofthevertices, wenote that thesection ofthesurface made by thecoordinate plane “n2=0istheparabola Bi::zaiylib whose intersection with theplane “qk=Cgives theposition ofthevertices ofthehyperbola. Thesection made bytheplane “n2=C<0isahyperbola with then2-axis asitstransverse axis. The vertices ofthishyperbola lieon theparabola ﬁg='_2a€7l:; intheplane n1=0.The section made bytheplane -qa=0isapair of straight lines, which serve asasymptotes fortheprojections ontheplane “n2=0ofallthehyperbolas lying inhorizontal sections ofthesurface. The surface itself iscalled ahyperbolic paraboloid (seeFigure 7). c.Tovisualize theform ofthesurface (33) inthegeneral case, we investigate thewaythesections made bythehyperplanes 'q,,=Cchange when Cvaries from 0to+oo.Every such section isacentral surface inn—1 dimensions. Allthese surfaces aresimilar toeach other, andtheir semiaxes (unlike thecase ofconical surfaces) vary according toaparabolic law, i.e., areproportional tothesquare root ofC.For C=0thecentral surface sac. |0.5 GEOMETRIC PROPERTIES orAQUADRIC SURFACE 299 becomes conical. For C<0thecentral surface goes into theconjugate surface, i.e., thepositive and negative terms inthecanonical equation exchange their roles. lnthespecial case where theterms of(33) have the same sign, which, tobeexplicit, wetake tobepositive, thesurface exists onlyinthehalf-space n,,>0. d.Thereason forcalling thisclass ofnondegenerate surfaces noncentral isthat such surfaces actually have nocenters. For n=3this isobvious from Figures 6and7.Toprove theassertion inthegeneral case, assume the contrary, i.e., suppose that thesurface (33) hasacenter (hf,ng,...,nﬂ). Since, inparticular, thiscenter must beacenter ofsymmetry forthesection 'q,,='q‘j,, which represents anondegenerate central surface inn—1 dimensions, wemust have n‘lIn2— —ﬂ‘2._5I0 (cf.Sec 10.51). Thus thecenter must lieonthe'q,,-axis. Now ifwegofrom anarbitrary point (n1,...,'q,,_1, hf,+S)lying onthesurface tothesym- metric point (—'q1, ...,—*q,,_1, hf,—S),equation (33)must stillbesatisﬁed. Buttheleft-hand sideof(33)remains thesame when wemake thistransition, andhence itsright-hand sidecannot change. Itfollows thatS=0,andhence thatthere arenopoints onthesurface forwhich 'q,,¢nﬂ.But(33)obviously hassolutions “n1,“n2,...,'q,,with ‘q,,¢‘Q2.This contradiction shows that oursurface cannot have acenter. 10.55. Degenerate surfaces. AsinSec. 10.42, byadegenerate surface wemean asurface whose canonical equation contains lessthan ncoordinates. For example, suppose that thecoordinate 'q,,isabsent inthecanonical equation. Then allthesections ofthesurface made bythe(n—1)-dimensional hyperplanes “q,,=C(—00 <C< 00)give thesame surface inn—1 dimensions. Therefore every degenerate surface inthen-dimensional space R,,isgenerated bytranslating aquadric surface inthe(n—1)-dimensional space R,,_1 along aperpendicular toR,,_1. a.Wenow ﬁndtheappropriate curves intheplane (n=2).Inthiscase, thecanonical equation contains only onecoordinate andhence isjust 2if,=c. "1 ForC>0weobtain apair ofparallel lines, forC=0apair ofcoincident lines, andforC<0an“imaginary curve.” b.Toconstruct degenerate surfaces inthree-dimensional space (n=3), wemust translate allthesecond-degree curves inthe'q1'q2-plane along the '/13-21X1S. When thisisdone, ellipses, hyperbolas andparabolas give elliptic. hyperbolic andparabolic cylinders, respectively (see Figure 8),while pairs 300 QUADRATlC roams 1NEUCLIDEAN AND UNITARY SPACES CHAP. 10 ll: 5.1‘-i:l:E:Ei1ié£5é==€5'r -;..-$3; -- "3 1;. 'I»-E:1§i="..;::;33:::§:2::2:11;;;§T-‘,.;;=:===;;-_. :~ -=;::;::;;::::.-::::........ ~:-"'1':-1: ' . .~--I--‘:1:;:§:=5:-::Z122.....-I1' *--:;3 ---_::,;,£5::3i5,5 --I»-1:-:;3=§:E:::::::::::1;;;;1~-- gii -I{;.§5;;',=‘_55§ ~»1»;;::-_1:=::S:5;::;:::::::2-.- -~: --:->1:;:.-1'5. -'::EE1:==55'-='.-: ..:1:I-- .:,:=»-»-I-:=_::=' ->:2;;-- -‘-..2.:;--1;-' . ‘ ' /4. FIGURE 8 ofintersecting, parallel andcoincident lines lead tointersecting, parallel and coincident planes (seeFigure 9). *l0.6. Analysis ofaQuadric Surface from ltsGeneral Equation 10.61. Wehave justdescribed allpossible types ofquadric surfaces inan n-dimensional Euclidean space, where thetype ofthesurface wasdetermined from itscanonical equation. However, thesurface isoften speciﬁed byits general equation (21) rather than byitscanonical equation, anditissome- times important todetermine thetype ofthesurface, i.e.,construct itsllll1I7 SEC. 10.6 ANALYSIS OFAQUADRIC SURFACE FROM ITS GENERAL EQUATION canonical equation, without carrying outallthetransformations described inSec. 10.42. Ittums outthat towrite down thecanonical equation ofthe surface speciﬁed byequation (21), weneed only know thefollowing two quantities: a)The roots ofthepolynomial an“7‘ a12 au- a a-I75--- a1 21 22 27! anl an‘: ''arm 7-)‘ ofdegree n; b)Thecoefﬁcients ofthepolynomial anT7‘ a12 ' an bl a21 a22”A''' a21| I72 A105) : . . . . 11.1 11.2 ''11....I7\b7! b1 b2 b,, c ofdegree n. Toobtain explicit expressions forthecoefﬁcients ofA1(75), weusethe linear property ofdeterminants (Sec. 1.44). Every column ofthedeterminant A1(75), except thelastone, canbewritten asasum oftwocolumns, theﬁrst consisting ofthenumbers a,,(i==l,2,...,n;jﬁxed) andthenumber b,, thesecond consisting ofnzeros andthenumber I75. Asaresult, thedeter- minant A1(75) canbewritten asasum ofdeterminants, each ofwhich is obtained byreplacing certain columns (except thelastone) in-thematrix an a12 '''am bl "21 "22 '''“21. b2 A1: . . .. . (34) am anz '''ann bu b1 b2 ---b,, c bycolumns consisting ofnzeros andthesingle element -75,with thenumber -7.appearing ontheprincipal diagonal ofthematrix. After expansion with respect tothecolumns containing thenumber -75, each ofthese determinants becomes ('_)‘)kA/Ir|+1~kv 302 QUADRATIC roams INEUCLIDEAN AND UNITARY SPACES CHAP. 10 where kisthenumber ofcolumns containing theelement -75, andM,,+1_k isaminor oforder n+l-kofthematrix A1.This minor ischaracterized bythefact that ifituses theithrow (i=l,2,...,n)ofA-1.,italso uses theithcolumn, andmoreover, itmust usethelastrow andcolumn ofA1. Minors with thisproperty willbecalled bordered minors. Itisobvious that every bordered minor ofthematrix A1appears intheexpansion ofthe determinant A1(75). From thisweimmediately conclude that thecoeﬂicient of(-75)" intheexpansion ofthedeterminant A1(75) inpowers of-75equals the sum ofallthebordered minors oforder n+1-k.Itisconvenient towrite theexpansion ofA1(75) intheform A1(75) ==ot,,+1 -ot,,75 +ot,,_1752 -l-ot1(-75)", where thecoefficient otkisthesum ofallthekth-order bordered minors of thematrix A1. 10.62. Aswealready know, theroots ofthecharacteristic polynomial A(75) giveusthecoefﬁcients ofthesquared variables inthecanonical equation. Toﬁnd theremaining term, which isofdegree 0ifthecanonical equation hastheform (24) andofdegree lifithastheform (27), wemust examine thebehavior ofthepolynomial A1(75) under coordinate transformations. Thus consider thequadratic form A1(X, x)=,%'1a7lk€l-E)’: +2z1bt€i€,,+1 +¢’€:+1 (35) inthe(n+l)-dimensional Euclidean space R,,+1, where Z1,Z2,...,K,,,Z,,+1 arethecomponents ofthevector x6R,,+1 with respect tosome orthonomial basis e1,e2,...,e,,,e,,+1. The operator corresponding to(35) isthesym- metric operator A1which hasthematrix (34) inthebasis e1,e2,...,e,,, e,,+1; wewillalso denote thismatrix byAm. Besides thisoperator, consider theoperator E1deﬁned bytherelations E1915 :elc (k<")1 E1e,,+1 ==0. This operator hasthematrix 100---00 010---00 001---00 E1I (36) 000---10 000---00 SEC. 10.6 ANALVSIS OF AQUADRIC SURFACE FROM ITS GENERAL EQUATION inthesame basis e1,e2,...,e,,,e,,+1. LetR,,denote thesubspace with the vectors e1,e2,...,e,,asabasis. Then theoperator E1isobviously the identity operator inthissubspace. Now suppose wearegiven anisometric operator Qinthespace R,, Then Qcarries theorthonormal basis e1,e2,...,e,,into another ortho- normal basis f1,f2,...,f,,. Weconstruct anew isometric operator Q1in thespace R,,+1 bysetting Q1e,,Iﬁ. (k<'1), Q1en+1 :er-+1 :f1-+1’ Ifthematrix oftheoperator Qhastheform 911 912 '''911- 921 922 '''921-Q: 91-1 91-2 '''91-1- inthespace R,,,then thematrix oftheoperator Q1justconstructed hasform 911912"'91110 921 922 '''92» 0 Q1_.: . . .... 91-1 91-2 '''91-15 0 0 0---0l inthespace R,,+1. This matrix corresponds tothefollowing coordinate transformation (seeSec. 8.94): £1:91101 ‘l‘92102 l'''l91-1%, E.Iqum+qzmz+-'-+q..m.., ....................... (37) E...Iqimi+qzmi+-''+q....m, E1-+1 =01-+1~ Inthenew basisf1,f2, ...,f,,,f,,+1 theoperator Ahasthematrix Am:QFIAWQ (see Sec. 5.51), while theoperator E1hasthesame matrix (36) asbefore. Moreover, according toSec. 5.52, det(A5,, -75E1) ==det(A5,, -75E1). 304 QUADRATIC roams INEUCLIDEAN AND UNITARY SPACES CHAP. 10 Wenow assume that (37) isthetransformation (see Sec. 10.42) which reduces thequadratic form A(x,x): w £1515‘QM-H§$5- tothecanonical form 7| A(x, x)I 75,-hi. 1-I1 Itfollows from (37) that Q1transforms thequadratic form (35) inn+1 variables into ,g17‘I’0I ‘l‘2I_§1lmmn+1 ‘l‘¢”03|+1- (38) After thistransformation, thematrix oftheoperator A1,which, asweknow, transforms inthesame way asthematrix ofthecorresponding quadratic form, becomes 7510 0 0 0 [1 0752 0 0 0 [2 QQ...;5,Q...Q], Am: QQ Q Q 017+, ’ 00...()()...0]" I1 I2 Ir lr+1 In 9 andthepolynomial A1(75) =det(A5,, -75E1) equals thedeterminant 7.1-75 0 --- 0 0 --- 0 l1 0 7.2-75--- 0 0 - 0 l2 0 0 1,-1 0 -01, 0 0--0 -1 -0l1‘+1 O O -- O O -75 l,, I1 /2 Ir lr+1 In 9 The coefﬁcients ofthispolynomial canbecalculated byusing thebordered minors ofthematrix A5,,, just asthey were calculated before byusing the bordered minors ofthematrix A5,,=A1. SEC. 10.6 ANALVSIS OF AQUADRIC SURFACE FROM ITS GENERAL EQUATION Wenote that forr<nallthebordered minors ofthematrix A5,,which areoforder higher than r+2must vanish, since they contain two pro- portional columns. Thus forr<nthecoefﬁcients ot,+3, ot,+2, ...,ot,,+1 vanish. Moreover, forr<nthenonvanishing minors oforder r+2must usetheﬁrst rrows andﬁrst rcolumns ofthematrix A5,,. Ingeneral, the bordered minors oforder r+lneed notusethese rrows and columns. However, wenote thefollowing twocases where abordered minor oforder r+1must infactusetheﬁrstrrows andcolumns: 1)r==n,inwhich case itisobvious that thematrix A5,, hasonly one minor oforder r+l(i.e., ofordern +l),namely itsdeterminant, made up ofalltherows andcolumns ofA5,,; 2)r<n,l,.51Il,+2I----l,,I0,inwhich case there isonly one nonvanishing bordered minor oforder r+l,made upofelements from the rows andcolumns with numbers l,2,...,r,n+l. 10.63. Next weshow how thenext stepinthetransformation ofequation (38), made with theaimofeliminating thequantities l1,I2,...,l,,alfects thematrix oftheoperator A1.First consider thetransformation r ll I W12'01+T01-+1, 7\1 t]kI1]k (kI2,3,...,n—l—l), carrying thematrix A5,,into thematrix X1 0 0 0 0 0 0;52 0 0 0 12 0 0 )5, 0 0 [T Al§lI 00 00 01,,1 0 0...0 0 ...0 In 22_L1 7\1 This operation onA5,, canbedescribed asfollows: The ﬁrst column is multiplied byl1/751 andsubtracted from thelastcolumn, andthen theﬁrst row isalso multiplied byl1/751 andsubtracted from thelastrow. The sub- sequent transformations required toeliminate thequantities l2,la,...,l,0 [2 If l,__5_1 In 306 QUADRATIC roams 1NEUCLIDEAN AND UNITARV SPACES CHAP. 10 can bedescribed similarly. Asaresult ofallthese transformations, the matrix A5,,goes intothematrix . )5, 0 0 0 0 0 0)52 0 0 0 0 Amﬁ 0 0 )5, 0 0 0 (/)'_ '0 0 0 Q 017+, 00 0 0 0 In QQ 017+, In cl Moreover, these transformations donotchange thevalues ofthebordered minors ofthematrix A5,,which usetheﬁrst rrows andcolumns ofA5,,. Next consider thepolynomial det(A{;’,-75E1)3:?A‘{’(A) 11~1 0--0 0 00 0752-75 0 0 00 0 0 1,-1 0 00 0 0-- 0-A---0l,+1 0 0 '- 0 0 '---75 l,, 0 0 ... Q [H4 ...In C =u',,+1-011,75+ot§,_1752 ----+ot1(-75)", where wehave dropped theprime onc’.Thecoefﬁcients ofthispolynomial arecalculated byusing thebordered minors ofthematrix Am injust the same way asthecoefficients ofthepolynomial A1(75) arecalculated byusing thebordered minors ofthematrix A5,,. Since thebordered minors oforder r+2(where r<n)areinvariant under thetransformation leading from A5,,toAm, asshown above, weﬁndthat 01,22, Ioc,+2. Inthesame way, we have 111,1 I05,21 inthetwospecial cases noted above. 10.64. First weconsider thespecial case rIn.Here thecoefficient 112+, ofthepolynomial A{’l(75) isobviously equal totheproduct 751752---75,,c, sec.10.6 ANA1.vs1s orAQUADRIC SURFACE mom ITSGENERAL EQUATION 307 sothat thequantity cinthecanonical equation (25), p.288isjust I at-+1 °‘n+1cI-IiI—II——. )51)52 ...)5" X052 ...)5“ 10.65. Next suppose that r<n.Then wemust determine thecoefficient ot,+2 ofthepolynomial A{”(75), which willbeneeded inamoment.1' The nonvanishing bordered minors ofA5}:oforder r+2have theform I-75175.2’--75,1?" (mIr+1,...,n), andtheir sum, which equals thecoefficient 111+, Iot,+2, isgiven by __)51)52 ...7570:“ _5_[L2 _5_..._5_ Werecall thatthecondition forreducing equation (21)tothecanonical form (27) isthat atleast oneofthecoefﬁcients l,.51, l,+2, ...,l,,benonvanishing. We can now formulate this condition equivalently intheform ofthe inequality _I at‘-1-2 ;é01 andatthesame time givethefollowing formula forcalculating thecoeﬁicient Mofthecanonical form (27): M1=,3H+,;+2+...+,f,=___&i__ )51)52...)5,_ However, if01,11I0,then l,+1Il,+2=---Il,,I0,and(21)reduces to thecanonical form (24). Thus wehave arrived atanother special case. In thiscase, thecoefficient oz;+1I11,51isobviously equal totheproduct 751752---75,c,sothat thecoefficient cofthecanonical form (24)isjust °@I+1 = “H-1 751752 ---75, 7.1752 '''75, 1Itiseasily veriﬁed thatinthiscaseallthecoeﬁicients <x,,,ofthepolynomial A{"(7.) with m>r+2vanish. 308 QUADRATIC roams 1NEUCLIDEAN AND UNITARY SPACES CHAP. 10 10.66. Wenow summarize these results intheform ofatable. Asbefore, weagree toarrange theroots 751,752,...,75,,ofthecharacteristic polynomial A(75) insuch awaythatthenonzero roots 751,752,...,75,come ﬁrst, denoting theproduct 751752---7.,byA,. Data Canonical Equation on75,,;éO 7,1-15%-5\-7,2-y5;+..._5_;\n.,,:+_1_'£i1=() 7| 7.,,IO x,,_1I0 75,,_1 IO O 7‘n-2 7L0°‘1-+1 if0 0101 +A2102 +' ot,,_H :O 7511): +7.21); +- 05,,‘-#0 ot,,I0X1111+X2112+' Mi+M2+'+7‘n-1-'l:_1 1')‘n-171:-1 +7‘ﬂ—2"l:-2 +791-27151-2°‘1-+1 2_i : 1/A aﬂ A,,_1 aﬂ2_I = \/ A1|~ "F1 <x,,_1 ‘ An-271.. O O 752IO ot3IOon 7‘1"li+2\/Tiaylzzo1 x1¢0 02:0°‘2752-I 1711+)“ O l0.7. Hermitian Quadratic Forms 10.71. Many ofthetheorems ofthepreceding sections carry over tothe case ofquadratic forms inacomplex space. Webegin with thefollowing basic THEOREM. Every symmetric Hermitian bilinear form A(x, y)inann- dimensional unitary space C,,hasacanonical basis consisting ofnorthogonal vectors. Proof. According toSec. 9.34, thelinear operator Aassociated with the form A(x, y)bytheformula A(x, y)I(Ax, y)isself-adjoint. Hence by Theorem 9.34, there isanorthonormal basis e1,...,e,,inthespace C,, consisting ofeigenvectors oftheoperator A.The matrix oftheoperator A isdiagonal inthisbasis, and hence soisthematrix oftheform A(x, y), since theoperator andtheform have thesame matrix inanyorthonormal .0.7 SEC 1 HERMITIAN QUADRATIC roams 309 basis ofthespace C,,.Therefore e1,...,e,,isacanonical basis oftheform A(x.y)-I 10.72. Itfollows from this theorem that every Symmetric Hm-mitian quadratic form A(x, x)canbereduced tothecanonical form 7| A(X.X)IZ7,»ll,-l21-1 byaunitary transformation. The sequence ofoperations leading todeter- mination ofthecoefficients 75,-andthecomponents ofthevectors ofthecanon- icalbasis isthesame asintherealcase (seeSec. 10.13). 10.73. Next welook forthestationary values ofasymmetric Hermitian quadratic form A(x, x)ontheunitsphere ﬁle.-1*=1 III inC,,,recalling from Sec. 9.l5b that A(x, x)takes only real values. Let e1,...,e,,beanorthonormal basis oftheform A(x, x).Then inthisbasis wehave A(x,X)=2%lilzI275,-(6? +T27), .’! .’! 7 7 fI1 J-I1 TL 7|(X.x)=2,12.-1*=,Z1(@?+-1),- - (Z,Ics,+i-r,). Using Lagrange’s method, weequate tozero thepartial derivatives ofthefunction A(x, x)-75(x,x)with respect toeach ofthe2n realvariables 0,,-r,(jIl,...,n).This gives 275,o,—275o,IO, 275,1, —275'r,I O l,...,n). These equations aresatisﬁed foravector xwith |x|Ilifand only if75 coincides with one ofthenumbers 751,...,7.,,.Suppose 75I75k.Then a solution oftheequations isgiven bythevector xwith components Z,I cs,+i-t,»I0forj Ikand liklIl.Hence, just asinthereal case (Sec. 10.21), theHermitian quadratic form A(x, x)takes stationary values atthose vectors oftheunitsphere which belong toitscanonical basis e1,...,e,,,in other words attheeigenvectors ofthecorrmponding self-adjoint operator A. The values oftheform atthese points coincide with thecorresponding canonical coefficients. Inparticular, themaximum oftheform A(x, x)isthe largest ofthecoefficients 75,,andtheminimum ofA(x, x)isthesmallest of these coeﬁicients. 10.74. Next consider theproblem ofthesimultaneous reduction to canonical form oftwo symmetric Hermitian quadratic forms A(x, x)and QUADRATIC FORMS [N EUCLIDEAN AND UNITARV SPACES CI-IAP. I0 B(x, x),oneofwhich, sayB(x, x),ispositive deﬁnite. Tosolve thisproblem, wechoose theHermitian bilinear form B(x, y)asthescalar product. Then, bySec. 10.72, there exists anorthonormal canonical basis fortheform A(x, x),inthesense ofthegiven scalar product. Inthisbasis wehave A(x,x)=2»12.12. Bo.x)=_ Q.-'11’.!\xQ.:1-1 asrequired. The calculation ofthecoefﬁcients 75,andthecomponents ofthevectors ofthecanonical basis (with respect toanarbitrary original basis) iscarried outinthesame wayasintherealcase(Sec. 10.32), after ﬁrstwriting the forms A(x,x) and B(x,x) asreal functions ofthereal variables 0,,-r, (jI1,...,n),where Z,Ics,+i-r,.Weleave thedetails asanexercise forthereader. PROBLEMS 1.Use anorthogonal coordinate transformation totransform each ofthe following quadratic forms tocanonical form: 3) + —45.152 _45253;b)2&1+5&3+5&3+4&1&2—4&1&,-s&2&,; c)2&1+2&2+2&3-4&1&.+2&1&,+2&.&2—4&,&.; d)25152 45‘25153 _25154 _25253 +25254 +25354- 2.What arethestationary values ofthequadratic form A(x, x)Ixi+ex;+§x§ onthesphere |xl=l,where x=(x1,x2,x3),andofwhat typearethey(mini- mum, maximum, etc.)’? 3.Show thateach ofthequantities 5&1,5&2,...,5&kcanactually attain theupper andlower bounds indicated informula (6),p.279. 4.Two quadratic forms A(x, x)andB(x,x)inR,,aresaidtobecomparable if theinequality A(x, x)<B(x,x)holds foreveryx eR,,.Let751>752>~~~>75,, bethecanonical coefﬁcients oftheform A(x, x),andlet5&1>5&2>~~~>5&,,be those oftheform B(x, x).Show thattheinequality 7%<l*k holds forevery k=1,2,...,n.(This isobvious inthecase where A(x, x) andB(x, x)have acommon canonical basis.) 5.Find acommon pairofconjugate directions forthecurves x2 2 I+‘% I1, 2x_y I1. PROBLEMS 3|I 6.Construct thelinear transformation which reduces both quadratic forms A(x,x) =E?+25,352 +25;—25,53 +3E,§, B(x, X) = +25152 +3€gE3 _ + tocanonical form. What arethecorresponding canonical forms? 7.Show thatthebasis inwhich thequadratic forms A(x, x)andB(x,x)both takecanonical form, with canonical coeﬁicients X1,X2,...,1,,andv1,v2,..., v,,,respectively, isuniquely determined towithin numerical factors, provided thattheratios 1112 ‘_» v1,v2,...,vﬂ aredistinct. 8.Prove thatthemidpoints ofthechords ofaquadric surface parallel tothe vector y=(1)1,1)2,...,1),)lieonan(n—1)-dimensional hyperplane (the diametral plane conjugate tothevector y). 9.What quadric surfaces inthree-dimensional space (with coordinates x,y,z) arerepresented bythefollowing equations: x2 ‘Y2 Z2 x2 ‘Y2 Z2 2 ﬁ)z—§+T—1, b)I—§—T——1, C)-Y—)/2+1, d)y=x2+z2+1; e)y=xz'? 10.Simplify thefollowing equations ofquadric surfacm inthree-dimensional spaoe, andgivethecorresponding coordinate transformations: a)5x2+6y2+7z2—4xy+4yz —lOx+8y+ 142-6 =0; b)x2+2y2—z2+l2xy —4xz —8yz+ l4x+ l6y— 122-3 =0; c)4x2+y2+4z2 —4xy+8xz —4yz —12x -12)/+62 =0- 11.Show thattheintersection ofanellipsoid withsemiaxes a1>a2>~~~>a,, with ak-dimensional hyperplane going through thecenter oftheellipsoid is another ellipsoid withsemiaxes bl>b2>~'~>bk,where al>bl>an—k+1: az>be>an—k-1-2: ak>bk>a,,. *chapter II FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS Il.l.More onAlgebras 11.11. Theconcept ofanalgebra wasintroduced inSec. 6.21, thisbeing thename given toalinear space (over aﬁeld K)equipped with a(commutative ornoncommutative) operation ofmultiplication ofelements, obeying axioms l)—3), p.136. The algebras considered inChapter 6were forthemost part commutative, but, inpassing, wementioned animportant example ofa noncommutative ﬁnite-dimensional algebra, namely, thealgebra B(K,,) of alllinear operators acting inann-dimensional space Kn.This chapter is devoted tothestudy ofB(K,,) anditssubalgebras. Butﬁrst wewillﬁnd it convenient toconsider abstract ﬁnite-dimensional algebras. 11.12. Not every algebra hasaunit, asshown bytheexample ofthe trivial algebra, i.e.,anyalgebra such that xy=0forallelements xandy (Example 6.22a). Nevertheless, every algebra canbeextended toanalgebra with aunit inthefollowing standard way? Given anyalgebra A,letA+bethesetofallformal sums a+)\,where aeAandAisanumber from theﬁeld K.Then A+isobviously alinear space with operations @+u+@+w=w+w+o+w and u(a—l-7\)= ua+)\p. 3I2 SEC.11.2 REPRESENTATIONS orABSTRACT ALGEBRAS 3I3 (a,beA;7.,iteK).Moreover, A+isanalgebra with respect tothemulti- plication operation (11+%)(b+l*)=(l1b +lb+iw)+M». The algebra A+certainly hasaunit, i.e.,theformal sum ofthezero element ofAandthenumber l.Wenow need only note that theoriginal algebra A canberegarded asasubset ofA+bysimply identifying each element a6A with theformal sum a-l-0eA+. ll.2. Representations ofAbstract Algebras 11.21. LetAbeanabstract algebra overaﬁeldK,andletB(K) bethe algebra ofalllinear operators acting inalinear space Kover thesame ﬁeld K.Wenow consider morphisms ofthealgebra Ainto thealgebra B(K), henceforth indicated bynotation oftheform T:A —>B(K). a.Deﬁnition. Amorphism T:A ->B(K) iscalled arepresentation ofthe algebra Ainthespace K.Arepresentation iscalled trivial ifTa=0for every aEA and exact (orfaithful) ifTisamonomorphism, i.e., ifthe operators T,andT,,corresponding todistinct elements aandbofthealgebra Aarethemselves distinct elements ofthealgebra B(K). The setofallelements aeAwhich arecarried into thezero operator bytherepresentation Tiscalled thekernel oftherepresentation T.The kernel ofthetrivial representation isthewhole algebra A,while thekernel ofanexact representation consists ofasingle element, namely thezero element ofthealgebra. Inthegeneral case, thekernel ofanyrepresentation isatwo-sided ideal ofthealgebra A(seeExample 6.25d). b.Deﬁnition. Two representations T’:A->B(K’) and T";A ->B(K") ofanalgebra Aaresaidtobeequivalent ifthere isanisomorphism U:K’->K” between thelinear spaces K’andK”such that UT;=Tgu forevery aeA.Obviously, inthecase ofﬁnite-dimensional spaces K’and K”,equivalence oftherepresentations T’andT”means that theoperators T;and T:(a6A)have identical matrices insuitable bases ofthespaces K’andK”. c.LetT:A->B(K) bearepresentation ofthealgebra A.Asubspace K’CKiscalled aninvariant subspace oftherepresentation Tifitisinvariant with respect toalloperators Tu,aEA. Byconsidering theoperators T, only onthespace K’,weobviously getanewrepresentation TK':A->B(K’), called therestriction oftherepresentation Tonto K’. 3'4 FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS CHAP. ll d.Finally letT;A —>B(K) bearepresentation ofthealgebra Asuch that Kisthedirect sum ofsubspaces Kk(l<k<n)invariant with respect tothe representation T,andletT"denote therestriction oftherepresentation T onto K,,(l<k<n).Then wesaythattherepresentation Tisthedirect sum oftherepresentations T"(l<k<n). 11.22. Toevery algebra Awecanassign inanatural wayarepresentation T:A —>B(A) inthelinear space Aitself which associates with each element a6Atheoperator ofleftmultiplication bya,i.e.,theoperator T,deﬁned by theformula Tab=abforevery beA.This representation iscalled theleft regular representation ofthealgebra A.The invariant subspaces oftheleft regular representation areobviously leftideals inA(Sec. 6.23a). Using thisconcept, wecanestablish thefollowing important THEOREM. Every algebra isisomorphic toasubalgebra ofthealgebra B(K), forasuitable choice ofK. Proof Itiseasy toseethat thetheorem isequivalent totheassertion that every algebra hasanexact representation. LetAbethegiven algebra. Asshown inSec.l1.12,there exists analgebra A+withaunitewhich hasA asasubalgebra. LetT:A+ —>B(A+) betheleftregular representation ofthis algebra. Then Tisexact, since Tue=ae=a¢0forevery aeA+,a¢0. Hence therestriction ofthemorphism Tonto thesubalgebra ACA+isan exact representation ofthealgebra Ainthespace K=A+. I ll.3. Irreducible Representations andSchur’s Lemma 11.31. Among allrepresentations ofagiven algebra wenow distinguish those with thesimplest structure inacertain sense. Every representation T:A ->B(K) ofanalgebra Ahasatleast twoinvariant subspaces, Kitself andthesubspace {0}consisting ofthezeroelement alone. Any other invariant subspace issaid tobeproper. Proper invariant subspaces which contain no other such subspaces arecalled minimal invariant subspaces oftherep- resentation T. Deﬁnition. Anontrivial representation T:A ->B(K) issaid tobe irreducible ifithasnoproper invariant subspaces. 11.32. Given any vector 26K, itiseasy toseethat thesetK,= {Taz eK:a eA}isaninvariant subspace oftherepresentation T.Avector zeKissaidtobecyclic (with respect totherepresentation T)ifK,=K. This deﬁnition, together with thedeﬁnition ofirreducibility, immediately implies thefollowing THEOREM. Arepresentation acting inthespace Kisirreducible ifandonly ifevery nonzero vector zeKiscyclic. sac. 11.4 BASIC TVPES orFINITE-DIMENSIONAL ALGEBRAS 315 Despite itssimplicity, thisresult willsubsequently befound very useful. 11.33. The irreducible representations ofalgebras over theﬁeld Cof complex numbers havethefollowing important property: THEOREM (Schur’s lemma). LetT:A ->B(C) beanirreducible represen- tation ofthealgebra Aover theﬁeld C.Then every operator inCwhich commutes with alltheoperators Ta,aeA,isamultiple oftheidentity operator E. Proof LetSbeanoperator which commutes with allT“,a6A,andlet xbeaneigenvector ofS(Sec. 4.9). Then Sx=Axforsome complex 7.,and hence ST,,x =T,,Sx =).T,,x forevery aEA. But therepresentation Tis irreducible, andhence, byTheorem 11.32, every vector y6Ccanberepre- sented intheform y:Tax, aeA.Itfollows that S=7.E. | Itshould benoted thattheproof makes essential useofthefactthatevery linear operator ina(ﬁnite-dimensional) complex linear space hasaneigen- vector (see, Sec. 4.95b). Inview ofthedecisive role ofSchur’s lemma, we will henceforth conﬁne ourselves toaconsideration oflinear spaces and algebras over theﬁeld ofcomplex numbers. ll.4. Basic Types ofFinite-Dimensional Algebras Beginning with thissection, unless thecontrary isexplicitly stated, we will consider only ﬁnite-dimensional algebras (i.e., algebras which are ﬁnite-dimensional regarded aslinear spaces) over theﬁeld Cofcomplex numbers. What isthestructure ofﬁnite-dimensional algebras andtheir represen- tations? Most ofthis chapter will bedevoted toresults along just these lines. Inparticular, wewill distinguish some classes ofalgebras whose structure canbestudied completely, i.e.,wewillsucceed indescribing all such algebras (towithin anisomorphism) andalltheir representations. We refer totheclasses ofsimple andsemisimple algebras. Thevarious classes ofalgebras arise when weconsider speciﬁc properties oftheir ideals andrepresentations. 11.41. Deﬁnition. Anontrivial algebra iscalled simple ifitcontains no proper two-sided ideals (Sec. 6.23a). Anexample ofasimple algebra is-the algebra B(C,,) ofalllinear operators inaﬁnite-dimensional space. Infact, letJbeatwo-sided ideal inthealgebra B(C,,), andletA=Hajkn 6Jbea nonzero matrix such that an¢0,say. Then, asshown inSec. 4.44, by multiplying thematrix Afrom theright andfrom theleftbycertain matrices, i.e.,byperforming operations thatdonotleave theideal J,wecangeta matrix E"whose only nonzero element 1appears intherthrow andsth 316 FlNlTE—DlMENSlONAL ALGEBRAS AND THEIR REPRESENTATIONS CHAP. I1 column. Moreover, byfurther multiplying E"from theright andfrom the leftbycertain matrices, wecangetanymatrix Ejk(j,k=1,...,n)without leaving theideal J.Butlinear combinations ofthematrices E”,give the matrix ofanyoperator inB(C,,), andhence J=B(C,,). Aswewillseelater (Sec. 11.64), this example isunique intheclass ofallﬁnite-dimensional algebras over thecomplex number ﬁeld. THEOREM. Every simple algebra hasanexact irreducible representation. Proof. LetAbeasimple algebra, andconsider itsleftregular repre- sentation T:A->B(A). Itfollows atonce from thefact that Aisﬁnite- dimensional that among theinvariant subspaces oftherepresentation T there isaminimal subspace A’.The restriction Toftherepresentation T onto A’isnontrivial. Toshow this, weneed only prove that forevery b6A’,thesetAb={abta eA} ¢{O}, resorting tothefollowing simple proof (due toA.S.Nemirovski): Suppose, tothecontrary, that Ab={O}. Then, asiseasily seen, thesetbA={bza 6A}isatwo-sided ideal inA,and hence, since Aissimple, either bA=AorbA={O}. ButifbA=A,then Ab:{0}implies that every product inAequals zero, while ifbA={O},the set{).b:7i 6C}isatwo-sided ideal inAsince Ab={O}, and hence must coincide with thewhole algebra since Aissimple. Thus, inboth cases, the algebra Aturns outtobetrivial, andhence cannot besimple. Thus therepresentation T:A ->B(A’) isnontrivial. Butthen, ontheone hand, itisirreducible, bytheminimality ofA’,while ontheother hand, its kernel, being atwo-sided ideal distinct from thewhole simple algebra A, consists ofthezero element alone. Therefore T(like anyirreducible rep- resentation ofA)isatthesame time exact. I Itturns outthat theconverse theorem isalso true, i.e., every ﬁnite- dimensional algebra with anexact irreducible representation issimple. This willbeshown attheendofSec. 11.64. 11.42. Anarbitrary algebra may nothave exact irreducible representa- tions. Butitisnatural tosingle outthose algebras whose properties canbe described interms oftheir irreducible representations. This leads tothe following wider class ofalgebras: Deﬁnition. Analgebra Aiscalled semisimple if,given any nonzero element aeA,there exists anirreducible representation mapping ainto a nonzero operator. Inother words, theintersection ofthekernels ofall theirreducible representations ofasemisimple algebra consists ofthezero element alone. Itfollows from Theorem 11.41 that every simple algebra isalso semi- simple. Ontheother hand, consider then-dimensional (n>l)algebra C", sEc.11.4 BASIC rvrss orFINITE-DIMENSIONAL ALGEBRAS 317 consisting oftheelements a:(Oil,...,oz")where oc,-6C,with multiplication component bycomponentfr This algebra isobviously commutative. More- over, thesetofalla=(Oil,...,Oi”)such that at,=0,say, isatwo-sided ideal inC”,sothat thealgebra C,,isnotsimple. Suppose that with every element a=(otl,...,an)weassociate thecomplex number otk(1<k<n), orequivalently theoperator ofmultiplication byotkintheone-dimensional space C1.Then wegetanirreducible representation ofthealgebra C"which maps every element ofC”with oak¢0into anoperator distinct from zero. Since every nonzero element aeC”hasatleast onenonzero component, there exists anirreducible representation mapping aintoanonzero operator. Therefore thealgebra C,,issemisimple. Inthisexample, C”isadirect sum ofsimple (one-dimensional) algebras. Theexample caneasily begeneralized byconsidering adirect sum ofsimple noncommutative algebras. Then, aswillbeshown inSec. 11.77, wegetthe general form ofaﬁnite-dimensional semisimple algebra over theﬁeld of complex numbers. 11.43. Next weintroduce algebras whose properties are, inacertain sense, theopposite ofthose ofasemisimple algebra; Deﬁnition. Analgebra Aiscalled aradical algebra ifevery nontrivial representation ofAhasaproper invariant subspace. Inother words, a radical algebra hasnoirreducible representations atall. Asanexample, consider thealgebra Aofpolynomials P(z) =clz—l- ---+c,,z" with theusual operations butsubject tothecondition 2"“ =0. Then every element ofthealgebra Avanishes when raised tothe(n+l)th power, sothat noelement ofAhasaninverse. The algebra Ahasnonon- trivial one-dimensional representations, since every nonzero linear operator inaone-dimensional space isinvertible. LetTbeanontrivial (and hence multidimensional) representation ofthealgebra A,andletZbetheoperator corresponding totheelement 2.Since Z(like zitself) isnoninvertible, there exists avector e#0 such that Ze=0.But then P(Z)e :0forevery P(z) eA.Thus wehave found anontrivial invariant subspace (the straight linedetermined bythevector e)oftherepresentation T.Itfollows that A isaradical algebra. _11.44. Deﬁnition. Bytheradical ofanalgebra Aismeant theintersection ofthekernels ofallirreducible representations ofAifsuch representations exist, orthewhole algebra Aifnosuch representations exist. ‘i’I.e.,ifa=(al,,...,on"),h=([3,,...,B"),then ah=(1113,, ...,<x,,,’5,,). FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS CHAP. 11 Since thekernel ofevery representation isatwo-sided ideal ofthe algebra A(see Sec. l1.2la), theradical ofA,being anintersection oftwo- sided ideals ofA,isitself atwo-sided ideal ofA. The study ofalgebras with nontrivial radicals (inparticular, radical algebras) involves substantial difﬁculties, with results that, asarule, arenot indeﬁnitive form (some ofthese results willbefound attheend ofthis chapter). Ontheother hand, semisimple algebras andtheir representations canbestudied incomplete detail. Infact, aswewillseebelow, thestudy ofsemisimple algebras reduces tothat ofsimple algebras. Wenow turn tothedetailed study ofsimple algebras andtheir represen- tations. ll.5. TheLeftRegular Representation ofaSimple Algebra 11.51. Thus letAbeasimple algebra, andletT:A-> B(X) beaﬁxed exact irreducible representation ofA(the existence ofTfollows from Theorem 11.41). This representation willhenceforth becalled standard. THEOREM. LetT:A —>B(A) betheleftregular representation ofasimple algebra A,andletIbeaminimal invariant subspace ofT.Then a)Therestriction T‘oftherepresentation Tonto Iisequivalent toT; b)Thesubspace I,regarded asasubalgebra ofA,hasaright unit. Proof. First weﬁxanelement a6I,a¢0.Since therepresentation T isexact, Tax¢0forsome xeX.Consider thelinear operator U:I->X deﬁned bytheformula Ub=T,,xforevery beI.Itiseasy toseethat the kernel oftheoperator Uisaleftideal inA(orequivalently aninvariant subspace oftherepresentation T)contained inIbutnotcoinciding with I. Hence thekernel ofUconsists ofthezero element alone. Ontheother hand, theimage ofUisobviously anonzero invariant subspace oftheirreducible representation T,andhence coincides withthewhole space X.Thus Uisan isomorphism ofIonto X.Moreover, forarbitrary beIandceA, UTlb=U(cb)=T,,,x=T,(T,,x) =T,Ub, andhence UT:=T,U, which shows thattherepresentations T‘andTareequivalent (seeSec. 11.21b). Furthermore, since Umaps Ionto allofX,there exists anelement eGI such that Ue=Tex=x.Itfollows that U(be) =T,,,x =T,,(T,,x) =T,,x=Ub SEC.11.5 Tl-IELEFT REGULAR REPRESENTATION orASIMPLE ALGEBRA 319 forevery b6I.ButUisaone-to-one mapping, andhence be=b.Thus eis aright unit inthealgebra. I Itshould benoted that anyexact irreducible representation ofasimple algebra canbechosen asthestandard representation. Therefore anauto- matic consequence ofthis theorem isthefact that allexact irreducible representations ofasimple algebra areequivalent. 11.52. LEMMA. Given anarbitrary algebra A,letI1andI2beleftideals ofAwith right units e1ande2,respectively, where ael=0forevery aeI2 Then there exists aright unite2inI2such thatbe2=0forevery beI1. Proof Lete2=e2—e1e2. Then forevery aeI2wehave ae2=ae2—ae1e2 =a, since ae;=aandael=0.Moreover, be2=be2—be1e2 =be2—be2=0 forevery beI1.I 11.53. THEOREM. Theleftregular representation ofasimple algebra A isthedirect sumofitsirreducible representations. Proof Wewillconstruct thedesired setofminimal invariant subspaces oftherepresentation T:A ->B(A) byinduction, proving ateach step that, asanalgebra, thedirect sum ofthesubspaces already found hasaright unit. For theﬁrst subspace wetake any minimal invariant subspace I1ofthe representation T.According toTheorem 11.51, I1has aright unit e1. Suppose wehave already found minimal invariant subspaces I1,...,I2such thattheleftideal J2=I1+---+I2hasaright unitek.IfJ2=A,wehave succeeded inconstructing thedesired invariant subspaces. Otherwise, let J;={a€A:aek =0}. Then itiseasy toseethat J;isaninvariant subspace oftherepresentation T,whose intersection with J;isempty. Moreover, since every element aeA can berepresented intheform a=aek+(a~—aek), where aek€J,Q and a—aekeJZ,thealgebra Aisthedirect sum ofJ;andJZ. Theﬁnite-dimensional invariant subspace J;contains aminimal invariant subspace, which wedenote byI,,+1. According toTheorem 11.51, Ik+1 contains aright unitelm, where aek=0forevery a6I,,+1since 12+,CJ2. Itfollows from Lemma 11.52 that I,,+1 contains aright unit exsuch that be;=0forevery beJ2.Lete,,+1=ek+el,’.Then, asiseasily veriﬁed, ek+1 isaright unit intheideal Ji+1 =I1-1"''‘‘l‘Ik+Ik-I-1* 320 FINITE-DIM|;NS1uNAL ALGEBRAS AND THEIR REPRESENTATIONS CHAP. 11 This proves thelegitimacy ofmaking theinduction from ktok+1.The algebra Aisﬁnite-dimensional, andhence atsome stage wegetthesetof minimal invariant subspaces I1,...,I,,,oftherepresentation Twhose direct sum isthewhole algebra A.Hence theleftregular representation ofAis thedirect sum ofitsirreducible representations. I 11.54. Wenote that itwasshown inthecourse oftheproof that every simple algebra hasaright unit. Actually, wehave thefollowing stronger THEOREM. Every simple algebra hasaunit. Proof LetAbeasimple algebra, andletebearight unitofA.Consider theoperator T,inthestandard representation T:A ->B(X). Then T0(T,x —x)=T,,,x —Tax=0 forevery x6XandaeA.Since Tisirreducible, every nonzero vector must becyclic (Theorem 11.32). Itfollows that T,x—x=0.Inother words, T,istheidentity operator inthespace X.Butthen TaT¢ :TeTa :Ta forevery aeA,andhence ae=ea=abytheexactness oftherepresentation T.Therefore eisaunit inA.I 11.6. Structure ofSimple Algebras Attheendofthissection wewillsolve theproblem ofthestructure of simple algebras. Insodoing, wewillﬁndthefollowing concept very useful: 11.61. LetXbealinear space, andletA0beasubalgebra ofB(X). The subset ofB(X) consisting oftheoperators which commute with alloperators inA0willbecalled thecommutator ofthealgebra A0,denoted byA0. Itiseasy toseethat A0isitself asubalgebra ofB(X). The commutator ofthisnew subalgebra, denoted byK0,willbecalled thesecond commutator ofthealgebra A0.Obviously wehave A0CK0. 11.62. Given anyalgebra A,every element a6Adeﬁnes twooperators inB(A), theoperator ofleftmultiplication T0,speciﬁed bytheformula T0b=ab,andtheoperator ofright multiplication R0,speciﬁed bytheformula R0b=ba.Itiseasy toseethat thesetofalloperators ofleftmultiplication andthesetofalloperators ofright multiplication form subalgebras inB(A), which wedenote byAgandA3,respectively. SEC. 11.6 STRUCTURE orSIMPLE ALGEBRAS 321 LEMMA. Ifthealgebra Ahasaunit,thenAl)=A3andA3=Al). Proof. IfSeAl),then S(ab) =ST,,b :T,,Sb =aSb. Setting b=e,where eistheunit inA,wegetSa=aSe. Therefore Sisthe operator ofright multiplication bytheelement SeeA,i.e.,SeA3.Itfollows that Al)CA3, and hence that Af):A3, since obviously A3CAl).The formula A3=Al)isproved injustthesame way. I 11.63. THEOREM. Given asimple algebra Awith standard representation T:A —>B(X), letA0bethealgebra ofoperators ofT.Then A0=A0. Proof. The algebra A0deﬁned above canobviously beregarded asthe algebra ofoperators oftheleftregular representation T:A ->B(A) ofthe algebra A.According toTheorem 11.53, this representation isthedirect sum ofcertain irreducible representations TVA->B(I,) (1<i<m), where, byTheorem 11.51, each representation isequivalent tothestandard representation. This means thefollowing: Wecanﬁnd abasis xl,...,x,, inthespaceXand abasis fl“), ...,fflin eachofthe subspacesli (1<i<m) such that forevery a6A,thematrix oftheoperator T0inthebasis flu’, fl",...,ff")ofthewhole space Ahasthequasi-diagonal form Zfl T'.= * . (1) FT? where each block along theprincipal diagonal isthematrix oftheoperator T0inthebasis xl,...,x,,andthe“off-diagonal” blocks consist entirely of zeros. Itfollows from theruleformultiplication ofblock matrices (Sec. 4.51) that every matrix commuting with allmatrices oftheform (1)isamatrix oftheform S11 ‘''Slm . . . , (2) Sml 'A' Smm where each block Si,isann><nmatrix commuting with allthematrices T0, a6A. FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS CHAP. ll Now letPbeanoperator inA0,and letPbeitsmatrix inthebasis xl,...,x,,.Then thequasi-diagonal matrix ZE P’Z . I5 obviously commutes with allmatrices oftheform (2),andhence determines inthebasis ff", fl“, ...,f7§"" ofthespace Aanoperator belonging tothe second commutator ofthealgebra A0.ByTheorem 11.54, every simple algebra hasaunit, andhence, byLemma 11.62, which means that thematrix Pdetermines inthebasis flu’, fl", ...,ff") an operator P,equal toT0forsome beA.Butthen P=T0forthesame b, andhence Pbelongs tothealgebra A0.Theproof isnow complete, since P isanarbitrary element ofA0. I 11.64. Wearenow inaposition toprove thebasic theorem onsimple algebras: THEOREM (First structure theorem). Every simple algebra isisomorphic tothealgebra ofalllinear operators acting insome ﬁnite-dimensional space X. Proof LetAbeasimple algebra, andletT:A—> B(X) bethestandard representation ofA.Itissuﬁicient toprove that thealgebra A0ofoperators oftherepresentation Tcoincides with B(X). Since thereprmentation Tis irreducible, itfollows atonce from Schur’s lemma (Theorem 11.33) that the commutator A0ofthealgebra A0consists ofjustthose operators which are multiples oftheidentity operator. But then thesecond commutator A0 coincides with thewhole algebra B(X). Atthesame time A0=A0,by Theorem 11.63, andhence A0=B(X). I Itshould benoted that behind alltheconsiderations leading totheﬁrst structure theorem liesthefact that every simple algebra hasanexact ir- reducible representation. Hence wehave incidentally proved that every algebra with anexact irreducible representation isisomorphic tothealgebra B(X). Itfollows atonce that theconverse ofTheorem 11.41 holds: Every algebra with anexact irreducible representation issimple. sEc. 11.7 STRUCTURE orSEMISIMPLE ALGEBRAS 323 ll.7. Structure ofSemisimple Algebras 11.71. Inthissection wewillshow that theproblem ofthestructure ofa semisimple algebra reduces completely totheproblem ofthestructure ofa simple algebra (already studied above). Tothisend, wewillﬁnd ituseful tointroduce some new concepts. Deﬁnition. Byanormal series ofanalgebra Aismeant achain ofalgebrasl" A=I02 I12 ---21,2 I,,+1={0} inwhich each algebra isatwo-sided ideal ofthepreceding algebra. Bya composition series ofanalgebra Aismeant anormal series ofAinwhich each ideal ismaximal (i.e., isnotcontained inanylarger two-sided ideal) and Incontains noproper two-sided ideals. Itiseasy toseethat every ﬁnite-dimensional algebra hasacomposition series. Infact, among the(proper) two-sided ideals ofaﬁnite-dimensional algebra Athere isamaximal ideal I1,say.Similarly, thealgebra I1contains amaximal two-sided ideal I2,I2contains amaximal two-sided ideal I3, and soon.Since theoriginal algebra Aisﬁnite-dimensional, after aﬁnite number ofsteps weﬁnally arrive atanalgebra Inwhich contains nofurther proper ideals. The chain ofalgebras A=I0DI1D ---DI,,DI,,+1={0} soobtained isobviously acomposition series ofthealgebra A. 11.72. Before turning tothespecial properties ofnormal andcomposition series ofsemisimple algebras, weprove thefollowing LEMMA. Given anyelement aofasemisimple algebra A,there exists an element beAsuch thatevery power oftheelement baisnonzero. Proof Bythedeﬁnition ofasemisimple algebra, there exists anirreducible representation T:A —>B(X) such that T0¢0.Then forsome xeX,x¢0, thevector y=Taxisnonzero andtherefore, byTheorem 11.32, isacyclic vector oftheirreducible representation T.Hence there isanelement beA such that T0y=x,i.e.,such that Tux =Ta(T¢x) =Ta)’=x- Itfollows thatevery power oftheoperator T00,andhence every power ofthe element baeA,isnonzero. I ‘IHere andintherestofthissection (only) wewrite AEB(equivalently, B2A)to mean thatAisasubset ofB, reserving thenotation ACB(equivalently, BDA)tomean that Aisaproper subset ofB(i.e., AEBbutA¢B). 324 FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS CHAP. ll 11.73. THEOREM. Anormal series ofasemisimple algebra cannot contain nonzero trivial algebras. Proof. LetAbeasemisimple algebra, andlet A=I02I12 ::'2I,,2I,,+1={0} beanormal series ofA.Itcanbeassumed without lossofgenerality that thealgebra 1,,contains anelement adistinct from zero. Obviously, toprove thetheorem, weneed only ﬁndanelement c6Insuch that ca¢0. ByLemma 11.72, there exists anelement b6Asuch that every power of baisnonzero. ¢,,=(ba)2"+‘-‘b (k=0,1,...,n_1). Then induction onkshows that ck6I,,+1. Infact, fork=0wehave c0= bab6I1since a6I1,andthepossibility ofcarrying outtheinduction follows atonce from theobvious relation c,,+1=ckack and thefact that aeIk+2. Thus weseethattheelement c=c,,_1belongs tothealgebra I",andmoreover ca=(ba)2”—1ba =(ba)2“ ¢0, asrequired. I 11.74. Next weprove three simple propositions: LEMMA. LetA2I12I2D{O}beanormal series ofanalgebra A,where thealgebra I2issimple. Then I2isatwo-sided ideal inA. Proof. ByTheorem 11.54, thealgebra I2hasaunit e.Since e611, the elements aeandeabelong toI1forevery aeA.Butthen ab=a(eb) =(ae)b eI2, ba=(be)a =b(ea) eI2 forevery beI2.I 11.75. LEMMA. LetAbeanarbitrary algebra, andletIbeatwo-sided ideal ofAwith aunit. Then Ahasatwo-sided ideal Jsuch thatAisthedirect sum ofIandJ. Proof LetJ:{a6A:ae :0},where eistheunit ofthealgebra I. Then obviously Jisaleftideal inA.Moreover, Aisthedirect sum ofIand J,sinceb= be+(b—be) andb —beeJ. Wemust stillprove that Jisaright ideal inA.Clearly ab:abe+a(b—be) forarbitrary a6JandbeA.Butbe:ebesince be6I,andhence abe=(ae)be :0 SEC. 11.7 STRUCTURE OF SEMISIMPLE ALGEBRAS since ae=0.Therefore ab=a(b—be), sothat abistheproduct oftwo elements ofJ.Itfollows that ab6J.I 11.76. LEMMA. Let Iand Jbetwo-sided ideals ofanalgebra A,and suppose Aisthedirect sum ofIandJ,with Ithemaximal two-sided ideal inA.Then thealgebra Jcontains noproper two-sided ideals. Proof LetJ’beatwo-sided ideal ofJwhich does notcoincide with J. Then thealgebra J”=I+J’isatwo-sided ideal inA.ButIismaximal, andhence J”=I.Itfollows that J’:{O}. I 11.77. Wearenow atlastinaposition toprove thebasic theorem on thestructure ofsemisimple algebras: THEOREM (Second structure theorem). Every semisimple algebra Aisa direct sumoftwo-sided ideals ofA,each ofwhich isasimple algebra. Proof Asshown inSec. 11.71, wecanconstruct acomposition series A:IoDI1D "’DI1.DIn+1:{0l forA.Our theorem isthen obviously aspecial case ofthefollowing Assertion. For every k(0<k<n)thealgebra Inrk isadirect sum of two-sided ideals ofI,,_,,, each asimple algebra, andmoreover I,,_,,hasaunit. Wenow prove thisassertion byinduction onk.The algebra 1,,hasno proper two-sided ideals, and moreover isnontrivial, byTheorem 11.73. Hence thealgebra Inissimple and, inparticular, hasaunit (byTheorem 11.54). This proves theassertion fork:0. Suppose now that theassertion istrue forsome k(0<k<n—1). This means, inparticular, that thealgebra I,,_,, hasaunit, andhence, by Lemma 11.75, I,,_k_1 isadirect sum I,,_k-1-Jwhere Jisatwo-sided ideal in I,,_k_1. Since I,,_k isamaximal two-sided ideal inI,,_k_1, itfollows from Lemma 11.76that thealgebra Jcontains noproper two-sided ideals. Atthe same time, applying Theorem 11.73tothenormal series A=I0DI1D“‘DIri*k-1DJD{0l» weﬁnd that Jisnontrivial andhence simple. Bytheinduction hypothesis, thealgebra I,,_,, isadirect sum oftwo-sided ideals ofI,,_k, each asimple algebra. Being simple, each ofthese subalgebras isalso atwo-sided ideal inI,,_k_1, byLemma 11.74. Itfollows atonce from thisfactandtherelation I,,_,,_1 -—_I,,_,, +Jthat I,,_,,‘1 isalso adirect sum oftwo-sided ideals of I,,_k_1, eachasimple algebra. Wemust stillshow that thealgebra I,,_,,_1 hasaunit. Lete1betheunit ofthealgebra I,,_,, (which exists bytheinduction hypothesis), andlete2be FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS CHAP. ll theunit ofthesimple algebra J.Then, since ab=ba=0forarbitrary aeI,,_,,, b6J,itiseasy toseethat theelement e:e,+e2isaunit inthe whole algebra I,,_,,_,. Thus wehave justiﬁed theinduction onk,thereby proving theitalicized assertion. But, asalready noted, our theorem isaspecial case ofthis assertion (corresponding tok=n). I Itshould benoted thatwehave incidentally proved that every semisimple algebra hasaunit. Thetwo-sided ideals found inthetheorem, whose direct sum isthegiven semisimple algebra A,willhenceforth becalled thesimple components ofthe algebra A. 11.78. Itwasshown inSec. 11.64 that every simple algebra isisomorphic tothealgebra B(X) forsome ﬁnite-dimensional space Xor,equivalently, to thealgebra ofallsquare matrices ofacertain order. Now letX1,...,X0be asetofﬁnite-dimensional spaces, andletB(X1, ...,X,,) bethesetofall rows oftheform a=(a1,... ,an), where akisanoperator from thealgebra B(Xk) (or,ifconvenient, amatrix oftheappropriate order). Obviously B(X1, ...,X0) isanalgebra with respect tothe"component-by-component" operations deﬁned bytheformulas a—l—b=(a1—l—b1,...,a,,—l—b,,), ).a= (7\a1,...,).a,,), ab=(a1b1,...,a,,b,,), wherea,beB(X1,...,X,,),a=(a1,...,a,,),b=(b1,...,b,,),and).isa complex number. Itfollows from these considerations that Theorem 11.77 hasthefollowing equivalent form: Every semisimple algebra isisomorphic tothealgebra B(X,, ...,X0)for some setofspaces X1,...,X0. Wenote further thatthesimple components ofthealgebra B(X,, ...,X") obviously consist ofrows oftheform (0,...,0,ak,0,...,0),where the kthentry ranges over thewhole algebra B(X,,) andtheremaining entries are allzero. Wewill identify each such component with thecorresponding algebra B(X,,). 11.79. Weconclude this section byﬁnding alltwo-sided ideals ofa semisimple algebra: THEOREM. Every two-sided ideal ofasemisimple algebra Aisthedirect sEc. 11.8 REPRESENTATIONS orSIMPLE AND SEMISIMPLE ALGEBRAS 327 sum ofacertain number ofsimple components ofA. Proof. According toSec.11.78, thesemisimple algebra Aisisomorphic tosome algebra oftheform B(X1, ...,X2)with simple components B(X2), 1<k<n.LetIbeatwo-sided ideal inB(X1, ...,X2), andletI2bethe intersection ofIwith B(X2). IfIcontains theelement aZ(a1.~ --,ak—1s akaam-1, --~,an)! then Ialsocontains theelement ae2= (0,... ,0,a2,0,... ,0), where e2istheunit inB(X2). Itfollows that Icanbewritten asthedirect sum I=I1+~---1-I2. Butitiseasily seen that I2isatwo-sided ideal inthesimple algebra B(X2) forevery k(1<k<n).Hence either I2=={0}orI2coincides with thewhole algebra B(X2). I 11.8. Representations ofSimple andSemisimple Algebras From aknowledge ofthestructure ofsimple andsemisimple algebras, wecanwithout particular difﬁculty ﬁnd alltheir representations towithin anequivalence. 11.81. LetAbeasemisimple algebra. Then, bySec. 11.78, wecan identify Awith thealgebra B(X1,...,X2) forsome setofspacm X2 (1<k<n).Therefore, besides thegiven algebra A,weareledinanatural way toconsider nrepresentations T":A ->B(X2), I<k<nofA,deﬁned bytheformula Tl.‘=atEB(X..) forevery a=(al,...,a2,...,an)eA.Since theimage ofthe representation Tl‘isthewhole algebra B(X2), these representations areallirreducible. THEOREM. Every irreducible representation ofasemisimple algebra Ais equivalent tooneoftherepresentations Tl‘(1<k<n). Proof. LetA:B(X,, ...,X2)beasemisimple algebra, with anirre- ducible representation T:A ->B(X), and letZ(T) bethekernel ofthe representation T.Since Z(T) isatwo-sided ideal inA(Sec. 1l.2la), itfollows from Theorem 11.79thatZ(T) isthedirect sumofcertain simple components ofA.LetA,denote thedirect sum oftheremaining simple components of Awhich donotﬁgure inZ(T), andletTi":A,—B(X) betherestriction onto A1oftheoriginal representation T.Thenewrepresentation T11’isnow exact, 328 FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS CHAP. 11 andmoreover irreducible since theimages oftherepresentations T11’andT obviously coincide. The algebra A1,having anexact irreducible represen- tation, must besimple (seeSec. 11.64). Hence A1reduces toasingle simple component, i.e.,A1coincides withB(X2) forsome k(1<k<n).Butthen, asiseasily seen, T,=T21’, a2eB(X2) foreverya= (a1,...,a2,...,a,,)eA. Now, according toSec. 11.51, allexact irreducible representations ofa simple algebra areequivalent. Inparticular, therepresentation TI”:B(X2) -> B(X) and theidentity representation T12’:B(X2) ->B(X2) areequivalent. This means that there exists anisomorphism U:X ->X2such that UTf11k’ = Tfjc’U forevery a2eB(X2). ButT0=Ti}:forevery aeA,asjust shown, while ontheother hand itfollows from thedeﬁnition oftherepresentation T"that Tl;=Tﬁ’. Therefore UT0 =TZU forevery aeA,which proves the equivalence oftherepresentations TandTl’. I 11.82. Next weconsider arbitrary representations ofsimple and semi- simple algebras. Inthis regard, thefollowing general proposition will be found useful: LEMMA. Given anarbitrary algebra A,letT:A-> B(X) beanyrepresen- tation ofA,andletX1,...,X2beminimal invariant subspaces ofTspanning alinear manifold which coincides withX.TThen Xisthedirect sum ofcertain ofthesubspaces X1,...,X2. Proof Anintersection ofinvariant subspaces ofarepresentation is itself aninvariant subspace. Therefore itfollows from theminimality of thegiven subspaces that foranyk,theintersection ofthesubspace X2+1 with thelinear manifold spanned bythesubspaces X1,...,X2iseither empty or X2+1 itself. Hence byconsecutively choosing those ofthesubspaces X1,..., X2which arenotcontained inthelinear manifold spanned bythepreceding subspaces, wegetthesubspaces whose direct sum isthewhole linear manifold spanned byX1,...,X2,namely thewhole space X. I 11.83. According tothesecond structure theorem, every semisimple algebra Aisisomorphic toanalgebra oftheform B(X1, ...,X2). Inwhat follows, wewillﬁnditconvenient toconsider therealization ofB(X1, ...,X2) intheform ofanalgebra ofrows, each made upofnmatrices oftheappropri- ateorders. The number appearing inthe“i]'th” place inthekthmatrix of therow corrmponding totheelement aEA will bedenoted by).§’;’(a). Moreover, wewilluseegfltodenote theelement ofthealgebra Asuch that TBythelinear manifold spanned bythespaces X2,...,X0wemean thesetofalllinear Combinations oftheform ozlxl +:--+<x,,x,, where x26X2(Cf.Sec.2.51). SEC. 11.8 REPRESENTATIONS orSIMPLE AND SEMISIMPLE ALGEBRAS 329 ).g€*(egf’) =1while allother elements inthematrices ofthecorresponding rowequal zero. Itshould benoted that gkeil’=e. (3) where eistheunit ofthealgebra A. LEMMA. LetT:A —>B(X) bearepresentation ofasemisimple algebra A and suppose thevector y=T,;§1x isnonzero forsome xeXandcertain indices iandk.Then ybelongs tosome minimal invariant subspace ofthe representation T. Proof. LetY={T,,y:a 6A}.Then, since y=Tegcix, itfollows from the rule formatrix multiplication that every element z,6Yisoftheform z,=T,,x, where bissome linear combination oftheelements ell?’(with iand kﬁxed). Itissufﬁcient toshow that ifz,qé0,then z,isacyclic vector with respect totherestriction oftherepresentation Tonto Y. Now letz26Y,sothatz2 Tex, where cisanother linear combination ofthesame elements egil. Using therealization ofthealgebra Aasan algebra ofmatrix rows, weﬁndanelement a6Asuch thatc=ab.Butthen z2:Tex=T,,(T,,x) =T021. Hence thevector z,iscyclic, asasserted. I 11.84. THEOREM. Every representation ofasemisimple algebra Aisa direct sumofirreducible representations andthetrivial representation. Proof. Given anyrepresentation T°:A->B(X°), consider theoperator T‘;where eistheunit inA.Then theformula ><=T‘Zx+(x~T‘3x) obviously deﬁnes anexpansion ofX°asadirect sum ofsubspaces XandX0 invariant with respect toT°,where therestriction ofT°onto X0isthe trivial representation. Wemust still show that therepresentation T:A-> B(X), therestriction ofT“onto X,isadirect sumofirreducible representations. Letxl,...,x,,,beabasis inX.Then T,istheidentity operator inX, andhence, because of(3),thelinear manifold spanned bythevectors ofthe type TA;-ix forallpossible indices i,jandkcoincides with thewhole space X. ByLemma 11.83, every nonzero vector ofthistype liesinsome minimal irreducible subspace oftherepresentation T.Thus theconditions ofLemma 11.82 areinforce. Butthen thespace Xisthedirect sum ofcertain minimal invariant subspaces oftherepresentation T,sothat Tisadirect sum of irreducible representations. I 11.85. Theorems 11.81 and 11.84 together describe towithin anequiva- lence allrepresentations ofsemisimple (including simple) algebras. In F1N1TE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS CHAP. ll particular, weseethat theoperators ofagiven representation ofasimple algebra (singling outthis case forgreater clarity) aredescribed insome basis byquasi-diagonal matrices oftheform Mln , (4)E0 where Mranges over thewhole setofmatrices oftheappropriate order and 0denotes thezero matrix. Inthemore general case ofasemisimple algebra, thecorresponding matrices arequasi-diagonal matrices oftheform """""""" \.___ ___________ . (5) lM.% where each ofthematrices M1,...,M2appearing intheindicated larger blocks ranges independently over thewhole setofmatrices oftheappropriate order (ingeneral different fordifferent matrices). 11.86. Incidentally wehave described allsimple andsemisimple matrix algebras (i.e., algebras which themselves consist ofmatrices). Infact, by merely assigning each matrix ofsuch analgebra itsoperator (inanybasis), wegetanexact representation ofthealgebra. This and thepreceding considerations immediately imply thefollowing assertion: SEC. 11.9 SOME FURTHER RESULTS 331 Every simple (orsemisimple) matrix algebra consists ofallmatrices ofthe form P“1LP, where Pisaﬁxed nonsingular matrix andLranges over theset ofallmatrices oftheform (4)(oroftheform (5)). For algebras containing theunit matrix, wegetasomewhat different result: Every simple matrix algebra containing theunit matrix consists of_all matrices oftheform P"LP, where Pisaﬁxed nonsingular matrix, Lranges over thesetofallquasi-diagonal matrices oftheform ZiaJ -0. <6) and Mranges over thesetofallmatrices oftheappropriate order. Every semisimple algebra containing theunitmatrix consists ofallmatrices ofthe form P“‘LP, where Pisaﬁxed nonsingular matrix, Lranges over thesetofall quasi-diagonal matrices oftheformEiiiiiiiiiiiiii .............. ~ 9[E...............- ............ andeach ofthematrices M1,...,M2ranges independently over thewhole set ofmatrices oftheappropriate order. 11.9. Some Further Results Thus wehave completed thedescription ofsimple andsemisimple ﬁnite- dimensional algebras, aswell astheir representations. Further investiga tion ofﬁnite-dimensional algebras liesbeyond thescope ofthis chapter. 332 FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS c1-1A1>. 11 Nevertheless, togive perspective, wenow citesome well-known results along these lines. 11.91. Wet1derburn’s theorem. Every ﬁnite-dimensional algebra isthe direct sum (regarded asalinear space) ofitsradical andsome semisimple algebra.'l' 11.92. The radical ofaﬁnite-dimensional algebra consists only ofnil- potent elements. Moreover, forevery such algebra there exists apositive integer nsuch that theproduct ofanynelements ofitsradical equals zero.3‘, 11.93. Every representation ofaradical algebra isdescribed insome basis bymatrices with zeros onandbelow theprincipal diagona1.§ PROBLEMS 1.Prove thatevery leftideal ofthe algebra B(K2) isthesetofall operators whose nullspaces oontain some subspace K’CK2. 2.Prove thatevery rightidealofthealgebra B(K2) isthesetofalloperators whose ranges arecontained insome subspace K’CK2. 3.Find allmaximal leftandright ideals ofthealgebra B(K2). 4.Given anysemisimple algebra Boflinear operators overaspace C2,introduce ascalar product (x,y)inC2suchthatA6Bimplies A*6B. 5(Converse ofProblem 4).Given anyalgebra Boflinear operators overaspace C2,prove thatifthereexists ascalar product (x,y)inC2suchthatA6Bimplies A*6B,then thealgebra Bissemisimple. 6.Suppose theconditions ofProblem 5aresatisﬁed. Prove thatBisasimple algebra iftheintersection ofthecommutator B(Sec. 11.61) andthealgebra B itself consists only ofoperators which aremultiples oftheidentity operator. 7.LetBbethesimple algebra consisting ofallmatrices oftheform (6)made upofm2blocks: M 0 ... 0 0 M ... 0 O O ... M TSee e.g., N.Jacobson, TheTheory ofRings, American Mathematical Society, New York (I943), p.I16. iSee e.g., N.G.Chebotarev, Introduction totheTheory ofAlgebras (inRussian), Gostekhizdat, Moscow (I949), Sec.8. §Here, ofcourse, itisnotasserted thatthematrices oftheoperators oftherepresentation range over thewhole setofmatrices ofthistype. Seee.g., A.Y.Khelemeski, Onalgebras ofnilpotent operators andrelated categories (inRussian), Vestnik MGU, Ser.Mat. Mekh., no.4(I963), pp.49-55. PROBLEMS 333 Show thatthecommutator ofBcanberepresented (inthesame basis) byall matrices oftheform 7\11E 7\12E '''7*1mE 7.01E THE ---7.2,,,E 9 7‘1rl1E 7‘m2E i'i)‘ntn1E where the1,2(j,k=l,...,m)arearbitrary complex numbers. Inparticular, show that theintersection ofBandBconsists only ofmatrices which are multiples oftheunitmatrix. 8.Forwhatsemisimple matrix algebra Bdoesthecommutator Bcoincide with Bitself? 9.Describe every semisimple commutative algebra B(BCB). 10.Describe every semisimple matrix algebra Bforwhich BCB. 11.Prove thatB:Bforevery semisimple algebra B. 12.LetBbethealgebra consisting ofallpolynomials inasingle operator A (hence Biscommutative, sothatBDB).Under what conditions does B=B? 13."Show thatifthealgebra Bgé{0}consists onlyofnilpotent elements (i.e., if Al‘I0forsome k=k(A) forevery A6B),then theequality CB=Bcannot hold forany C6B. 14.Analgebra Bissaidtobenilpotent ifthereexists anumber psuchthatthe product ofanypelements ofBequals zero. Show thatanalgebra Bequal tothe direct sum B,+---+B2,ofitsright ideals isnilpotent ifeach ideal B,» (_/'=1,...,m)isnilpotent. 15.Prove that ifaﬁnite-dimensional algebra Bconsists only ofnilpotent elements, thenBitselfisnilpotent. 16.Given anilpotent algebra Bofoperators inthespace K2,letM1CK2be theintersection ofallnullspaces ofalltheoperators A6B,letM2CK2bethe intersection ofall subspaces carried intoM,bytheoperators A6B,letM3CK2 betheintersection ofallsubspaces carried intoM2bytheoperators A6B,and soon.Show that {O}CM1CM2C--- CM,,:K2, where each setisaproper subset ofthenextandpistheindex ofnilpotency of B,i.e.,thesmallest number psuch thattheproduct ofanypoperators inB equals zero. 334 FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS CHAP. 11 17.Prove thatforevery nilpotent algebra Bofoperators inaspace K2,there exists abasis inwhich every operator A6Bisspeciﬁed byamatrix oftheform 0A12 A13 "' A1.»-1 0 0A23 “‘ A2.p~—1 A=O O O A32_,, 000-“ O where pistheindex ofnilpotency ofB.(A.Y.Khelemski) *appendix CATEGORIES OF FINITE-DIMENSIONAL SPACES A.l. Introduction A.11. Recently theconcept ofacategory andcertain related ideas have begun toplay animportant role invarious branches ofmathematics.T An example ofacategory isacollection ofsetstogether with mappings ofthe setsintooneanother. Acollection oflinear spaces oralgebras together with their morphisms isanother example ofacategory. Theexact deﬁnition ofacategory isasfollows: LetMbeasetofindices oz,andlet.7!’beasetofelements X2(at6szl)called objects ofthecategory 1’.Suppose that forevery pair ofobjects X0andX2there isaset$02 of other elements A0,,called mappings oftheobject X2into theobject X0such that theproduct ofthemappings A22andA02isdeﬁned forarbitrary at,B,Y andbelongs to$22, where multiplication isassociative, i.e., A8Y(AvBAB1) =(AsvA~ta)Ae¢ forarbitrary oz,B,Y,S.Inparticular, theset$22ofmappings oftheobjects X2into themselves isdeﬁned, and (associative) multiplication ofmappings isdeﬁned in-Q22. Finally, itisrequired that theset.922 contain theunit '1’Seee.g., H.Cartan andS.Eilenberg, Homologieal Algebra, Princeton University Press, Princeton, NJ. (I956); Séminaire A.Grothendieck, Algébre Homologique, Secretariat Mathématique, Paris (I958); A.G.Kurosh eta1.,Elements ofthetheory of categories (inRussian), Uspekhi Mat. Nauk, vol.I5,no.6(I960), pp.3C52. 335 336 APPENDIX element 1,,which hastheproperty that l,A,0=A20, A21,=A2, forarbitrary oz,BandY.Instead of$22wewillusually write simply $2. Aset.7!’ofobjects X2andmappings A0,,with theproperties justenumer- ated iscalled acategory. Acategory .7!’iscalled linear ifintheset$0,, of mappings A0,,(with arbitrary ﬁxed atand B)there aredeﬁned operations of addition ofmappings andmultiplication ofmappings bynumbers (from the ﬁeld K).This makes theset$0,,intoalinear space over theﬁeld K.Thus in alinear category theset.Q,becomes analgebra with aunit(over theﬁeld K). A.12. Inthisappendix wewillconsider linear categories whose elements areﬁnite-dimensional linear spaces (ofdimension >1) over theﬁeld Cof complex numbers, while themappings arelinear mappings (morphisms) of onesuch space intoanother. Thus westart with thefollowing deﬁnition: LetX,,(at6M)beasetof ﬁnite-dimensional complex linear spaces, andforevery atlet$2beanalgebra oflinear operators carrying X2intoitself. Moreover, suppose that forevery pair ofindices atandBthere isaset$02 oflinear operators A02carrying X2 intoX0such that1)if380,,contains theoperators A0,,andB02,then330,, contains theoperator sum A02+B02, and 2)if$0,, contains theoperator A02, then $0,, contains theproduct AA02 where 7.isanarbitrary complex number. Afamily oflinear operators with these two properties will be called alinear family. lnparticular, thelinear family .9312, coincides with the algebra Q2. Itisalsoassumed that .QY03'J0, C53,2 (1) forarbitrary 0.,BandY,i.e.,thatevery product Avt:Ae1 (AveEﬁve»At»:EM151) belongs to#12,. Such asetofspaces X2together with algebras £2and linear families £02 will becalled acategory ofﬁnite-dimensional spaces or simply acategory, andwillbedenoted by1’. Ifwechoose abasis inevery space X2,then thealgebras 38,,andlinear families 330, canbeidentiﬁed with thealgebras and linear families ofthe corresponding matrices, afact which will henceforth beexploited system- atically. Inwhat follows, wewillﬁndthecategories oflinear spaces corresponding togiven algebras .932, conﬁning ourselves tothecase where the.9312are semisimple algebras containing theunit matrix. According toSec. 11.86, forsuch analgebra thespace X2canbedecomposed into adirect sum of subspaces X2,invariant under alltheoperators A22, where ineach subspace X2,thealgebra J,isasimple algebra containing theunit matrix, i.e., is APPENDIX 337 described insome basis bythesetofallquasi-diagonal matrices oftheform E A5 ' s E where Cranges over thesetofallmatrices oftheappropriate order. Webegin with ananalysis ofsome special cases forwhich general results canafterwards bestated. Thus inSec.A.2weconsider thecase where every algebra W2iscomplete, i.e.,isthealgebra ofalllinear operators acting in thespace X2.The opposite case where each Q2isanalgebra ofoperators oftheform 7.E(multiples oftheidentity operator E)isconsidered inSec. A.3. The results ofSec.A.4pertain tothecase ofsimple algebras $2,this being anatural generalization ofthecase ofthealgebras {7.E}. InSec. A.5 weconsider thecase where each algebra W2isanalgebra ofalldiagonal matrices ofagiven order, while inSec.A.6thegeneral category withsemi- simple algebras W2isreduced tothecategories considered inthepreceding sections. A.13. We now recall thenotation and rules ofoperation governing matrices oflinear operators mapping alinear space Xinto alinear space Y (seeSecs. 4.41-4.43). LetXbeann-dimensional space with basis el,...,e2, andletYbeanm-dimensional space with basisfl, ...,f,,,. Then with every linear operator Amapping Xinto Yweassociate anm><nmatrix an a12 '''am a a ---a A= 21 22 211 aml am2 ''' amn (with mrows andncolumns), where thenumbers al,-,a2,-,...,a,,,,- inthe jthcolumn arethecoefficients oftheexpansion ofthevector Ae,6Ywith respect tothebasis fl,...,f,,,. Moreover, letZbeak-dimensional space with basis gl,...,g2. Then with every operator Bmapping thespace Y into thespace Zweassociate ak><mmatrix Ibn biz blm B: I721 I722 I72.» bkl bk2 bk". 338 APPENDIX Theoperator C=BAmaps Xinto Yandhasthek><nmatrix C11 C12 '''cut C21 C22 C2C: ”, cm ck‘! '''ckn obtained bymultiplying thematrices BandAinaccordance with theformula m c,,2=Z1b,,,-a,»2 (p=1,...,k;q=1,...,n).,2 A.14. The following fact, slightly generalizing Examples 4.44a—b (and proved inthesame way), willoften befound useful: LEMMA. Given anm><nmatrix A:lla,,,ll, suppose Aismultiplied from theleftbyak><mmatrix BIllbnll with allitselements equal tozero except thesingle element b,.0,o I1.Then theresult isak><nmatrix BA whose r0throwconsists oftheelements ofthes0throwofthematrix Awhile allother elements ofBAvanish. Ontheother hand, thematrix Aismultiplied from theright byann><lmatrix C:llc,.,ll withallitselements equal tozero except thesingle element c,.m, theresult isanm><lmatrix ACwhose s,th column consists oftheelements oftherlthcolumn ofthematrix Awhile all other elements ofACvanish. A.15. Itfollows from thelemma that ifanm><nmatrix Aismultiplied from theleftbyak><mmatrix Bandfrom theright byann><lmatrix C, where Band Chave theindicated properties, then theresult isak><l matrix BAC allofwhose elements vanish with the(possible)-exception ofthe single element, equal toaxon, appearing inther0th row and s,th column (cf.Example 4.44c). A.2. TheCase ofComplete Algebras A.21. Suppose thecategory J1"consists ofﬁnite-dimensional linear spaces X,,where forevery atthealgebra 5%,ofoperators acting inX2is complete, i.e.,isthealgebra ofalllinear operators inX2.Fixing arbitrary bases el,...,e2inthespace X,andfl, ...,f,,,inthespace X2,wecanidentify theoperators inthesets33'“, 33,2, 332,, .9822with thecorresponding matrices. Letnbethedimension ofthespace X,andmthedimension ofthespace X2.Suppose thefamily $2, contains anonzero operator A,sothat the corresponding m><nmatrix A=lla,,,,ll hasatleast onenonzero element, saya202“. Wecanassume without lossofgenerality thata20,0:I.ltfollows APPENDIX 339 from thecondition (1)and theassumption that .931,and W2arecomplete matrix algebras that theproduct ofAfrom theleftbyanm><mmatrix and from theright byann><nmatrix isitself amatrix inthefamily $2,. But, according toSec.A.I5,there isalways anoperation ofthiskind leading toan m><nmatrix with aunique nonzero element equal tolinanypreassigned position. Hence, since anym><nmatrix isalinear combination ofsuch matrices, weseethat W21contains allm><nmatrices, i.e.,W2,isacomplete family ofoperators mapping X,intoX2. A.22. Aswewillseebelow, thecategory .7!’justdescribed canberelated toacertain partially ordered set. Deﬁnition. AsetSissaidtobepartially ordered ifforevery pairof elements A,B6Sthere isarelation, denoted bythesymbol <(and read “1ess than orequal”) satisfying thefollowing axioms: a)IfA<BandB<A,thenA=B; b)IfA<BandB<C,thenA<C; c)A<Afor every A. Asomewhat more general concept isthat ofaprepartially ordered set, bywhich wemean asetSwith arelation <satisfying only axioms b)and c).Inthiscase, ifA<BandB<A,wecallAandBequivalent andwrite A~B.Then A~BandB~Ctogether imply A~C.Infact, byaxiom b),itfollows from A<B,B<Cthat A<Candfrom C<B,B<Athat C<A.ButA<Cand C<Atogether imply A~C. Therefore the relation <allows ustopartition thewhole setSinto (equivalence) classes Mn? ,...,where each class Mcontains allelements equivalent toAas well asagiven element A,while elements Aand Bbelonging todistinct classes arenonequivalent. Next weintroduce therelation <fortheclasses .Mand Wthemselves, writing M<.Qifthere exist elements AEM, B6.Qsuch thatA<B. This deﬁnition isindependent ofthechoice oftheelements A6M,B6.Q. Infact, suppose A,6M,B,6%, sothat A~A2,B~B2.Then A,<A< B<B,and hence A,<B,asrequired. The fact that axioms b)and c) forapartially ordered sethold fortheclasses M,.Q,...now follows from thefactthattheyholdfortheelements A,B,....Toshow thataxiom a) also holds fortheclasses M,.Q,...,letM<Q,.Q<Mand choose arbitrary elements AEM, B6.Q.Then A<BandB<A,sothat Aand Bareequivalent. Butthen Mand .Qcoincide, ie.,M=W,asrequired. Thus byintroducing anequivalence relation inaprepartially ordered set S,intheway indicated, wearrive atapartially ordered setofclasses of equivalent elements. 340 APPENDIX A.23. Wenow resume ourstudy ofthecategory 1’.Itfollows from Sec. A.21 that given any pair ofspaces X1and X2,there arejust four possibilities: a)$12and$21arebothcomplete setsofoperators; b)$12isacomplete setand $21consists ofthezero element alone; c)$21isacomplete setand $12consists ofthezero element alone; d)$12and$21both consist ofthezero element alone. If$12isacomplete setand noassumptions atallaremade about $21, we write X1<X2(therelation X2<X1hastheanalogous meaning). Aswenow show, therelation <makes thecategory .7!’into apre- partially ordered set.Infact, $11isacomplete setofoperators forthegiven space X1,byhypothesis, and hence X1<X1.Moreover, ifX1<X2and X2<X2,then $12and $22arecomplete setsoflinear operators mapping X1into X2andX2intoX2,respectively. Since allourspaces have dimension >1,there isobviously anonzero operator intheset$12. Infact, lete16X1, e26X2, e1,EX3 beﬁxed nonzero vectors. Then such anoperator can be obtained astheproduct AB, where theoperator A6$12carries e2into e1 andtheoperator B6$22carries e1,into e2.BySec.A.21, $11,isacomplete setofoperators carrying X2intoX1,sothatX1<X1,.Thus axioms b)andc) aresatisﬁed, andthecategory .7!’hasbeen made intoaprepartially ordered set. A.24. Inaccordance with Sec. A.22, wenow introduce anequivalence relation inJi’,writing X1~X2ifX2<X1andX1<X2,i.e.,ifboth $12and $21arecomplete setsofthecorresponding linear operators. Then theset ofspaces X2decomposes intoclasses ofequivalent spaces, andthesetofall such classes becomes apartially ordered setwhen equipped with arelation asinSec.A.22. Conversely, every partially ordered setofclasses $2ofﬁnite-dimensional spaces deﬁnes acategory ofthetype under consideration. Infact, forspaces X1andX2belonging tothesame class wespecify $12and $21ascomplete setsofoperators, while forspaces X1andX2belonging toclasses $1and$2 such that $1<$1,(i.e., such that $1<$2but$1eé$2), wespecify $12 asacomplete setand $111asthesetconsisting ofthezero element alone. Moreover, ifX1and X1belong tononcomparable classes $1and$1, we specify that $11and $11both consist ofthezero element alone. Thedescription ofcategories oftheindicated type isnow complete. A.3. TheCase ofOne-Dimensional Algebras A.31. Turning tothecase where thegiven algebras $2areallone- dimensional, weconsider twosimple examples: a.Letthecategory .7£’1consist oftwo spaces X1and X2ofthesame dimension, andlettheset$21consist ofanoperator Amapping X1onto X2 APPENDIX 341 inaone-to-one fashion together with allitsmultiples AA,7.6C,while the set$12consists oftheoperator Bwhich istheinverse ofAtogether with all itsmultiples p.B,p.6C.Then obviously 'QI2‘Q2l ={AB}, '%I‘Ql2 = b.Letthecategory Lconsist oftwoarbitrary spaces X1and X2with ﬁxed subspaces X1CX1and X;C_X2,and lettheset$21 consist ofall operators carrying X1intoX;with X1going into{O},while theset$12consists ofalloperators carrying X2into X1with X;going into {O}.Then obviously ‘QWQ21 ={O}, ‘Q21-Q12 = Itwillnow beshown that thecategories .7£’1and .%essentially exhaust allcategories consisting oftwo spaces with $,={7.E} (j=1,2),i.e.,that thefollowing alternative holds foranysuch category 1’:Either $12$21 = {O},inwhich case $21$12 ={0}also andthecategory .7!’iscontained ina category ofthetype L,orthespaces X1andX2have thesame dimension and .7!’isacategory ofthetype .7!’1. A.32. Thus let.7!’beacategory consisting oftwo spaces X1and X2 subject tothecondition $1={7.E}, $2={7.E}. LetN1CX1betheinter- section ofthenull spaces (Sec. 4.62) ofalloperators A216$21, and let N2CX2betheintersection ofthenull spaces ofalloperators A126$12. If$21X1 CN2and $12X2 CN1,wearedealing with asubcategory ofa category ofthetype .7£’2inwhich X1: N1,X1=N2.Therefore weassume that $21X1 isnotcontained inN2,say,andhence thatthere isavector x16X1 andanoperator A216$21such that A21x1 =x2does notbelong toN2. Every operator B216$21carries x1into avector collinear with x2,and every operator C126$12carries x2into avector collinear with x1.Infact, letA21x1 =x2,B21x1 =y2, and consider anoperator Cf,6$12 such that C12x2 eé0.Then, bythebasic condition, Cfzxz =C‘f,A,1x1 =71x1, where 7.#0. Replacing Cf,byamultiple ofC11,, wecan assume that 7.: 1. Moreover Bmqzxz =B21x1 =y2,while atthesame time B,1C‘;,x, =p.x2, and hence y2=11x2. Since, conversely, x1=Cfzxz and x1$5N1bythe deﬁnition ofx1,wehave analogously C12x2 =p.x1forevery C126$12. Moreover, inthegiven case, N1andN2reduce totheset{0}consisting ofthezero vector alone. Infact, if216N1,then A21(x1 +z1)=A21x1 =x2, i.e.,thevector x1intheabove construction canbereplaced byx1+z1.But then Cgzxz isamultiple ofboth x1andx1+z1,sothatx1andz1arecollinear. Therefore z1=0,since x16N1.Itfollows thatN1={O}.Similarly, starting with x2,weﬁndthatN2={O}. Wenow seethat x1canbechosen tobeanynonzero vector ofthespace X1,since there isalways anoperator A216$21carrying x1'into anonzero 342 APPENDIX vector. Hence theoperators oftheset$21 establish aone-to-one corre- spondence between allthestraight lines ofthespace X1and some setof straight linesofthespace X2,infactthesetofallstraight linesofthespace X2bythesymmetry ofourconstruction. Next weprove that thewhole set$21reduces tothesetofmultiples ofa single operator. Letx1960beanarbitrary vector ofthespace X1,andletx2 beanonzero vector determining thestraight lineinthespace X2corresponding tox1.Asweknow, there isanoperator Ag,6$21carrying x1into precisely x2.Every other operator A216$21 carries x1into 71x2forsome 7..First suppose A21x1 =71x2,where 7.eé0.Then theoperator 1 B21=ZA21 carries x1into precisely x2.Moreover, B21coincides with Ag,everywhere. Infact, suppose tothecontrary that Aglyl =y2, B21y1 =z2¢y2. This can happen only ify2 eé0,z2=\p.y2, p.eé1orify2=0,z2eé0.Let21=otx1+ By1beanonzero vector with ateé0,B¢0.Then thevectors A2121 and B2121 arecollinear, asproved above. Butthisisimpossible inourcase, since Ag1(°iX1 -l"13)/1) =“X2‘l'PY2. B21(°iX1 -l"B)/1) =“X2‘l'fall-Y2 ify2 eé0,while Ag1(°iX1 ‘l'I5)/1) =“X2, B21(°iX1 ‘l'13)/1) =“X2 ‘l'I322 ify2=0.This contradiction shows that ifA21x1 =71x2, 7.eé0,then A21= M21. Now suppose A21x1 =0.Then, asjust proved, Ag,+A21=Ag,and hence A21=0.Thus $21reduces tothesetofmultiples ofaﬁxed operator A21, andsimilarly‘ $12reduces tothesetofmultiples ofaﬁxed operator BQ2. Theproducts Ag1Bf2 andB‘{2Ag1 arenonzero and, bythebasic assumption, give operators which aremultiples oftheidentity operator. Hence the operators Ag,and Bf,areinverses ofeach other (apart from anumerical factor). Butthis ispossible only ifthespaces X1and X2have thesame dimension. Thus, ﬁnally, wehave proved that every category .7!’ofthe indicated type which isnotasubcategory ofacategory ofthetype L, isa category ofthetype .7!’1. A.33. Thecategories ofthetypes .%’1and.%arenottheonly possible categories with twospaces X1,X2andalgebras $1={7.E}, $2={7.E}. In fact, suppose thatintheset$21ofacategory .7!’ofthetype .7£’2wechoose a linear subset without increasing N1ordecreasing N2(for example, by imposing asuitable extra linear homogeneous condition ontheelements of thematrices oftheoperators A21). Then wegetacategory 1”which satisﬁes thegiven conditions butdoes notcoincide with 1’.Inthesetofallcategories with $,={7.E} (j=1,2),partially ordered with respect tosetinclusion, APPENDIX 343 thecategories ofthetype Larecharacterized bythefact that they are maximal, inthesense that nocategory ofthetype .7!’2,except forsingular cases where X1={0}orX1={O},canbeenlarged while preserving the properties ofacategory andtheconditions $1={7.E}. Infact, suppose that acategory .7!’ofthetype Lcanbeenlarged byincluding anoperator A11 taking avalue y2¢X1forsome x16X1—-X1,where X1={O}.LetB1116$12 beanoperator carryingy2 intoanonzero vector x16X1.Then B‘11A‘11x1 =x1, contrary tohypothesis. Moreover, suppose that inthecategory .7!’weinclude anoperator A21 carrying avector x16X1intoanonzero vectory1 6X2.Then clearly X1eéX1, since otherwise X1=$11X1 =$11X1 ={O}, and there cannot exist a vector x1mapped intoanonzero vector. Hence there isanoperator B126$12 carrying avector y26X2—-X1into x1.Butthen A12B21y2 =y1, contrary to hypothesis. Similarly, assuming that X1eé{O},weﬁndthat itisimpossible toinclude asingle extra operator intheset$12. Thus ourcategory .7!’ofthetype L isindeed maximal, under theassumption that X1eé{O},X1eé{O}. A.34. The singular cases must beconsidered separately. For example, suppose X1={O},sothat $12consists ofthezero operator alone. Then, if X1eéX2,thecategory isnonmaximal, and wecanenlarge theset$21to include alloperators mapping X1into X2without dropping theconditions $,={7.E} (j=1,2).This gives a“trivial” maximal category, where $12={0}and$21isacomplete setofoperators mapping X1intoX2.There isananalogous maximal category with $21={0}and $12acomplete set. Thus, ﬁnally, weﬁnd that thegeneral category ofthetype Lismaximal under thefollowing conditions: 1)X1eé{O},X1eé{O};2)X1={O},X1=X2; 3)x;=x1,X1={0}. A.35. Wenow turn tothegeneral case ofacategory with anarbitrary number N<00ofspaces X2,oz6M.Here wehave thefollowing analogue ofthealternative proved inSec.A.31: THEOREM. If$1=---=$2={7.E}, theneither theproduct$12.$2_2_1 --- $22$21 vanishes, orthespaces X1,...,X2allhave thesame dimension and $2=={).A1?,} where theA1,areﬁxed invertible operators such that A1)kAg.k—1 '''A‘is2A‘21 =E- Proof. Suppose theproduct $12$2_2_1- --$22$21 contains anonzero operator, which istherefore equal to7.Ewith 7.eé0,and letr,bethe dimension ofthespace X,-(j=1,...,k).Consider thecategory .%’0made upofthetwospaces X1,X2andthefollowing setsofoperators $21, $11: gala =g1k'Qk,k-I '''9321 -Q21 =ﬂu 344 APPENDIX ($21 isthelinear manifold spanned bythecorresponding operator products A12A2.2_1~~~A1,2, each mapping X2into X1). Since clearly $‘1’2$g1 eé{O}, itfollows from Secs. A.31—A.32 that X1and X2have thesame dimension r1=r2,while $11=$21={AA11} where A111isaninvertible operator with inverse (A‘11)" and$‘11=={p.(A‘11)"}. Similarly, applying thesame argument tothecategory .)£’0’made upofthetwospaces X2,X1,andthelinear manifolds $13, $111spanned bytheoperators oftheform (Ag1)_1A1kAk.1.C1 '''Ass andA22, respectively, weﬁndthat r2=r1,and$22={).Ag1} where Ag1isan invertible matrix. Continuing inthisway, wearrive atthedesired conclusion after ksteps. I A.36. Inthissection andthenext, when considering acategory made up ofNspaces, wewill assume that allthecyclic products $,,0$0., -'~$0,, vanish. Otherwise, wewould simply identify thecorresponding spaces which areallofthesame dimension. First consider thefollowing concrete category, which wedenote byL": LetX12,...,X1111 beN—~ 1arbitrary subspaces ofthespace X1,and for distinctj, k,l,...let X112 =X1, F)X12, X11111 =X11 F)X12 F)X11, ..., where wesuccessively form intersections ofthespaces X11two atatime, three atatime, andsoon.IfNisﬁnite, thelastintersection willbeX12___N, theintersection ofallN—-1oftheselected subspaces, while ifNisinﬁnite there willbenolastintersection. Letthesame construction becarried out inalltheremaining spaces X2,X1,,...,where theindex ofthewhole space isalways theﬁrstoftheindices appearing inthesymbol used todenote any ofitssubspaces. Thus toanysetofdistinct indices j,k,...(inthat order) there corresponds aunique subspace ofX,.Asforthesets$02, wedeﬁne $21asthesetofalloperators mapping X1into X2such that every subspace X1,“_2 goes into X21202 ifthesequence j,...,kdoes notcontain theindex 2and into theset{0}otherwise, with theother sets $0,, being deﬁned similarly. Wenow prove that 1'1)‘ isinfactacategory. Given operators A216$21 andB226$22, consider theoperator B22A21 carrying thespace X1into the space X1,.Theoperator A21carries thesubspace X1,-___2 into X21,___2 andthen B22carries X21,___2 into X221,___2 CX21,___2, Hence B22A21 6$21, asrequired. Moreover, ifthere isasequence ofoperators A1,,...,A21 mapping the space X1into itself, then theresulting operator carries X1intoX1,___21 ={O}, inkeeping withtherequirement that$1,---$21=10}. APPENDIX 345 A.37. Next weshow thatevery categoryf made upofN<oospaces X1,X2,...with $1={7.E} iscontained inacategory ofthetype 1’. Let X,2bethetotal image inthespace X,ofthespace X2under theaction ofall operators in$12, andletX,21___,,,,, bethetotal image inthespace X,ofthe space X2,under theaction ofalloperators oftheform A,2A2, ---Am (in that order). Then X,-2,____,,2 iscontained intheintersection ofX12, X,,,. .., X12,. Infact, ifz6X,2,___,,,,, then —- ‘Z ‘Z...‘Z 0‘...‘Z ‘Zz1:2A,2A21 AMA," A,,,2z,,,, where 2;6Xm,orequivalently, _ at at__Iat at Z1:2A:'kAk1 Anya» where ll ll ll __ yq 1 2 AQT .-- CX0’ ButAj‘2A‘1,- ~-A1126$2,, andhence z6X21,asrequired. Note also that A1, carries X,-___,,, intoX,,___,,,. Itisnow clear that ourcategory iscontained inacategory ofthetype L” with deﬁning subspaces X12. Inparticular, allthemaximal categories must beofthetype LN. However, itisnotclear what conditions onthe subspaces X12make acategory ofthetype LNmaximal. (Recall that inthe case oftwo spaces X1and X2,anecessary and sufficient condition for maximality ofacategory ofthetype Listhat thespaces X12and X21 either beboth dilferent from {O},orelsethat oneofthespaces bethewhole space while theother isthespace {0}.) A.4. TheCase ofSimple Algebras Ifthegiven algebras $2areallsimple, then, bySec. 11.86, each $2 consists ofthesetofallquasi-diagonal matrices oftheform F E - (2) C insome basis, where Cranges over thesetofallsquare matrices oforder r2. 1'Ofthespecial type under consideration. 346 APPENDIX A.41. First weconsider acategory 1’with just twospaces, aspace Xl ofdimension nlwith klblocks ofsizeml(sothatnl=klml) andaspace X2 ofdimension n2with k2blocks ofsizem2(sothat n2=kzmz). Then every matrix ‘llzlEﬂzl canbepartitioned into blocks asfollows: ml mzl A11 A12 ''' ‘H21: A21 . ... . k2 Akll ' ''‘Ak¢kl mm /<1 Similarly, every matrix Q3”6ﬁllcanbewritten intheform m2_._ mil Bu B12 3% Bf‘i11i k2 THEOREM. Either ‘lI2lQ3l2 =0(forarbitrary ‘1[2ledzl, 23”eﬂw), or kl=k2andthematrices Amareallmultiples ofan(arbitrary) ﬁxed matrix Aandthematrices BMareallmultiples ofan(arbitrary) ﬁxed matrix M, with theconstants ofproportionality making upapair ofmutually inverse matrices Zand§oforder kl=k2.T Proof. Ifthematrices ‘>1[,land Q3l2belong tothecategory 1’,then so does their product (from theappropriate sides) bymatrioes Cland C2of theform (2).Therefore, along with theequality Q[21Q312 =C2» wealso have ' Q[21c1Q312 =C2 1'Acategory ofthesecond type willbedenoted byJG. APPENDIX 347 foranarbitrary matrix Cloftheform (2). Recalling theruleformultipli- cation ofblock matrices (Sec 4.51), wehave AllCBll +Al2CB2l +---+Al,,‘CBk‘l =A2lCBl2 +Allcslz +---+A2,,‘CB,,‘2 (3) =---=Ak‘lCBlkg +Ak:2CB,,,, +---+A,,2,,‘CB,,‘,,,, AllCBl2 +Allcsll +---+Alk‘CB,,‘2 =0, LetCbethematrix with asingle nonzero element, equal tol,appearing in therthrowandsthcolumn (r<ml,s<ml). Ingeneral, ifAisanym2><ml matrix andBanyml><m2matrix, then ACB isaml><m2matrix ofrank 1, withtheelement a,,,b,q appearing inthepthrowandqthcolumn. With this choice ofC,theformulas (3)become alibi?+a}?b??+"""+ai’£‘bl‘;‘ =a?}-bi?+a??b??+"""+a3‘b’§;? =~~~=a’;:‘b:;"+a::*bf;“=+~~~+a’;¥“bl?"’, <4) aiibi?+a}?b??+""‘+all-‘b’§;? =0,.......................... .., where thesuperscripts denote theindices ofthecorresponding matrices. Wecanregard (4)asasingle matrix equation 11 12 11¢ 11 12 1k arr am" am‘ baa baa baa. 21 22 ___ 21¢, 21 22 ___ 21¢; .... am" am" am bra bsa baa Am'B-to _" kzl 76:2 ... kzki kil k12 ... Mk: am‘ awr am bsq baa baa A 0 ... 0 ... 0 Z0A k2_ 00...)\} \__,__/ k2 348 APPENDIX Similarly, wehave 21 22 _ 21¢ 21 22 ___ 2 E1 ___ baa baa baa: arr am aw? sq pr_"11 12 _ 11¢, 11 12 ___ uq bill b5U bill a1)?‘ av?‘ aPT R11 7612 ___ Ink; kgl It-22 ___ kgklbaa bsa baa am am am- a0 0 0 ...= “ 0kl. 0 0 ... H mi /<1 Thus weseethat thematrices 1,,andEsq(with parameters p,r,s,q)form a category connecting thespace Xlofdimension klwith thespace X2of dimension kl,subject totheconditions $1={XE}, W2={HE}- Wecannow apply thealternative proved inSec. 12.32. Namely, ifklqék2, then infactA=0,p.=0,while ifAqé0(orifp.qé0)foratleast onesetof indices p,q,r,s,then kl=k2and thematrices 11,,areallmultiples ofa single invertible matrix Z,while thematrices Emareallmultiples ofthe inverse matrix §=1-1: 1,,==xl,,/T, 11,,=u,,§. Thematrix 11,,consists oftheelements ofthematrices Allappearing inthe pthrowandrthcolumn. Hence ii’;=7\,,,a”‘, where the6”‘aretheelements ofthematrix Zoforder kl=k2:Itfollows that thematrices K,,,areallmultiples with coefficients E”ofaﬁxed matrix A=IIM,-ll, and similarly forthematrices FM. Moreover, thematrices I and§areinverses ofeach other, asalready noted. I A.42. Thus if‘l[2lQ3l2 960,then kl=k2and thecategory 1’isofthe form EHA 512A ‘H21: 511A . . ’ (5) ,;ki1A . 511M 512M| 2312: ~ ". APPENDIX 349 where A==llkmll isanm2><mlmatrix andM=Hp.,qH isanml><m2matrix. Among thematrices Aﬁguring inthegiven category there must beanonzero matrix A0(since ﬂzlﬂlz qé{0}), andhenoe anym2><mlmatrix must bea matrix Asince wecangetanynonzero matrix bymultiplying A0from the right byC2and from theleftbyCl.Hence ifﬂzlﬂlz qé{O},thesetﬁll consists ofallmatrices oftheform (5),where Z=llﬁikll isaﬁxed invertible matrix andAranges over thesetofallm2><mlmatrices. The situation is similar ifﬂlzﬂzl qé{O}.Itisn0w'clear that theinequalities ﬂlzﬂﬁzl qé{0} andﬂzlﬂlz qé{0}either both hold orboth failtohold. A.43. Theabove results canbeformulated interms oftensor products, anapproach which allows ustoexplain some further facts aswell. Thus we begin withthefollowing deﬁnition: Given ak-dimensional linear space Xwith abasis el,...,ekand an m-dimensional linear space Ywith abasis fl,...,fm, bythetensor product XxY=Zofthespaces XandYwemean thesetofallﬁnite formal sums 11 Xx»><yv» v=1 where xv6X,yv6Y.Here itisassumed that [X1XY]+[X2XY]=l(X1 ‘l'X2)Xylr [X><Y1]+[X><Y2]=[X><(yl+y2)], 11 11 11 ZMX» XyvZixv X)\vyv ==E)\v[xv X v==1 v=1 v=1 Itfollows that Zisalinear space ofdimension <km, where allthevectors ofZcanbeexpressed,in terms ofvectors oftheform ei><f, (i=l,...,k; j=1,...,m).Itisfurther assumed that thevectors e,-><f,- arelinearly independent andhence form abasis forthespace Z,sothat thecoefficients c,-,-intheexpansion ,§l\/1*3==_iilei X./jl (6) can beuniquely determined. Wecan write (6)somewhat diﬁerently by summing over theindex i.This gives m k m 8=§1(§1¢t,-ea) Xfa=21)‘: Xfa, H.where the k xi=Zcirei i=1 arearbitrarv vectors ofthespace X(nolonger necessarily basis vectors). 350 APPENDIX A.44. LetAbeanoperator mapping aspace Xlofdimension klinto a space Xlofdimension kl,andletBbeanoperator mapping aspace Ylof dimension mlinto aspace Ylofdimension ml.Then bythetensor product C:AxBoftheoperators Aand Bwemean theoperator mapping the space Zl=XlxYlinto thespace Zl=XlxYlinaccordance with the formula C[e§ ><fill=Aef><Bel (7) (thesuperscript istheindex ofthespace). If k2 M2 1Aei=2l1¢>\@i, Bf?zzbmflfl>\=1 |A=1 then(7)takes theform k2 m2c[e§><f}]=gZa,-lb,,,[e§ ><ff]. >-'5>- Next weﬁndthestructure ofthematrix Coftheoperator Cwith respect tothebases e}><fl‘andei><ffarranged inXlxYlintheorder @l><fl,@é><fl,---mi, ></‘Lei ></‘Le; Xf;9"‘9e;1Xf§!~~'7 ei><f1,.,,eé ><fi,.,,---,el,><fin, and similarly inthespace XlxYl.According toSec. A.l3, thematrix C hastheform 41111711 11121711 '''a1klb11 '''a11b1ml a12b1ml ''a1klb1m, 11211711 11221711 ‘'‘a2klb11 '''a21b1ml a22b1ml ‘‘‘a2k,b1ml akllbll akl2b1l ‘‘‘aklklbll '‘‘akl1b1ml akl2b11nl ‘‘‘aklklblmg a11bml1 a12bml1 a1klbml1 a11bmlml aizbmlml a1klbmlm, akl1bml1 akl2bml1 aklklbmll ‘"akl1bmlml aklzbmlml aklklbmlm, or Ab“l ‘Am, I Abmll Abmlml when written asablock matrix. APPENDIX 35I A.45. Applying Secs. A.4l and A.44, weseethat theoperators ofthe algebra ﬁlconsidered above aretheoperators inthetensor product ofan ml—dimensional space Xland akl-dimensional space Ylwhich aretensor products ofanarbitrary operator C6ﬁ(Xl) andtheunitoperator Esﬁ(Yl). Moreover, theoperators ofthesetﬁllarethetensor products ofanarbitrary operator A6ﬁ(Xl, Xl)andaﬁxed invertible operator A6ﬁ(Yl, Yl), while theoperators ofthesetﬁllarethetensor products ofanarbitrary operator M6ﬁ(Xl, Xl)andtheinverse operator A“. A.46. The following formula obviously holds forproducts oftensor products ofoperators: (AxB)(C XD)=(AC) x(BD). Hence, multiplying theoperators All6ﬁllandAll6ﬁll, weﬁndthat (AXA)(MXA-1)=(AM)X(AA—1) =(AM)XE6al, asmust bethecase foracategory 1’. A.47. Next weﬁndtheinvariant subspaces ofthealgebra ﬁ={CxE} ofoperators acting inthespace Z=XxY.These subspaces aretensor products oftheform XxYo,where Yoisanarbitrary subspace ofY,since (Cx E)(X xYo):CXx EYoeX ><Yo. Toseethat Zhasnoother invariant subspaces, let z=Zx,><y, beanyvector inZ(itcanbeassumed that thevectors xiarelinearly inde- pendent), and suppose Ccarries thevectors xiinto given vectors 1?,-EX. Then (Cx E)Zx1' ><y,-=Z>?i Xyr» andhence anysubspace invariant under alltheoperators CxEwhich contains thevector EX, ><y,also contains every vector Xx,» ><y,-.This proves the italicized assertion. lfweapply every operator AxAofthecategory 1’toaninvariant subspace XlxYloCXlxYl,then, since thematrix Aisarbitrary, the resulting image inthespace Zl=XlxYlisthesubspace AXl xAYlo = XlxYlo. Hence theoperators ofthecategory J!’establish aone-to-one correspondence between theinvariant subspaces ofthespaces ZlandZl,at thesame time establishing aone-to-one correspondence between theordinary subspaces ofthespaces YlandYl. A.48. Everything said above isvalid under thecondition ﬁllﬁll qé{0} (orequivalently ﬁllﬁll qé{0}). Ifﬁllﬁll =ﬁllﬁll ={O}, theabove 352 APPENDIX scheme does notwork, andthematrices ofthecategory 1’donotingeneral consist ofblocks which aremultiples ofaﬁxed matrix A.The situation is then thesame asinSec.A.32, andwecanapply theresult proved there, i.e., ourcategory 1’iscontained insome category ofthetype ifl(just which onetobeexplained below). A.49. Wenow turn tothecase ofacategory made upofanarbitrary number ofspaces Z,(oz6.M)’[ andsimple algebras ﬁl. Two spaces Zland Zlwillbecalled cognate ifﬁllﬁll qé{O},sothat thematrioes ofﬁllareof theform (5).Itisclear that therelation ofbeing cognate istransitive. In fact, ifZliscognate toZlandZliscognate toZl,then Zliscognate toZl,, since, bythearbitrariness ofthematrices A,there arenonzero matrices in theproduct ﬁllﬁll. Hence wecanpartition thewhole setofspaces Zl,into nonintersecting classes ofcognate spaces. IfZland Zlbelong todistinct classes, then ﬁllﬁll =ﬁllﬁll ={O}. Wecannow repeat thescheme ofSec. A.36 with certain modiﬁcations. Suppose oursetofspaces Z,ispartitioned into various classes Gl,..., G,,...ofcognate spaces, where thespaces belonging totheclass G,areof theform X,,xY,,and Y,denotes essentially onespace inwhich invertible operators act. Weﬁrst consider thespaces Y,bythemselves, andconstruct forthem acategory just asinSec. A.36 (satisfying thecondition ﬁ,-sﬁsl ={0})bychoosing arbitrary subspaces Y,“andthen forming their intersections Y,u,, Yum, ....This category consists oftheoperators Al, mapping Y,intoY,andatthesame time carrying thesubspaces Ym, ...CY, into thesubspaces Y,-ll“, ...CY,-.Then forthespaces Z,weconstruct thefollowing category, denoted by1’ftIfZ,and Zl,arecognate, the operators A”,6ﬁ,,,arethose previously constructed, while Z,=X,XY, andZl,=X,xY,belong todistinct classes, theoperators A,,,arearbitrary operators mapping Z,into Z,andatthesame time carrying every invariant subspace X,xY,q,,___ intoaninvariant subspace X,xY,,q,,___. Wenow verify thatevery category 1’with simple algebras ﬁ,iscontained inacategory ofthetype J£’;v. Suppose Z,=X,xY,and Z,=X,xY, belong todistinct classes ofcognate spaces. LetZl,,bethetotal image in thespace Z,ofthespace Z,under theaction ofalloperators inﬁ,,,. Then Zl,,isobviously aninvariant subspace ofZ,,and hence isoftheform X,xYl,,where Yl,,issome subspace ofY,.Similarly, letZ,,,,___,,,, bethetotal image inthespace Z,ofthespace Z,"under theaction ofalloperators ofthe form A,-,,A,,, ~~~Am. Then Z,,,,___,,,, isalso aninvariant subspace, which is easily seen tobecontained intheintersection ofZl,,Zl,,...,Zl,", byan argument likethat given inSec. A.37. Itfollows that ourcategory J!’is contained inacategory ofthetype ff, asasserted. 1'Wetemporarily denote each space byZl,instead ofXl,,reserving Xl,fortheﬁrst factor inthetensor product Zl,=Xl,,><Yl,,(cf.Sec. A.45). APPENDIX 353 A.5. TheCase ofComplete Algebras ofDiagonal Mam<;e5 Suppose thegiven algebras ﬁl,areallcomplete algebras ofdiagonal matrices. Then ineach space X,there isaﬁxed basis inwhich thematrices oftheoperators A,6ﬁ,,,arealldiagonal. Relative tothese bases, the operators All6ﬁll, arealso speciﬁed bycertain (rectangular) matrices, so that ourproblem canbestated asaproblem inmatrix theory. A.51. First consider acategory J!’with two spaces Xland Xl,and let Allbethematrix ofanyoperator inﬁll. Then, bythedeﬁnition ofacategory, theproduct B12=A1-412B2r (8) where Aland Blaresuitable diagonal matrioes, isalso thematrix ofan operator inﬁll. Suppose Alisthe(diagonal) matrix whose only nonzero element, equal tol,appears inthejthrowandjthcolumn, while Blisthe matrix whose only nonzero element, again equal tol,appears inthekth rowandkthcolumn. Then, byLemma A.l4, alltheelements ofthematrix Bllvanish with the(possible) exception ofthesingle element appearing in thejthrowandkthcolumn, andthiselement isjusttheelement a,,,ofthe matrix All.Thus theoperation (8)replaces every element ofthematrix All byzero, except theelement al,,which itleaves unchanged. This leads tothefollowing conclusion about thestructure ofthefamily ﬁll: Thefamily ﬁllconsists ofallmatrices with arbitrary elements ataﬁxed setofpositions andzeros everywhere else. A.52. LetSlldenote theﬁxed setofpositions inthematrices ofthe family ﬁll atwhich arbitrary elements areallowed. Wenow explain the connection between thesetsSlland Sll. LetAll6ﬁllbeamatrix whose only nonzero element, equal to1,appears inthejlth rowandklth column,l' sothat (jl,kl)6Sll, and letBll6ﬁll beany matrix with arbitrary nonzero elements atthepositions ofSll. Then theproducts Cl=AllBll and Dl=BllAll arediagonal matrices, byhypothesis. Ontheother hand, by Lemma A.l4, thejlth rowofClconsists oftheelements oftheklth row of Bllwhile alltheother elements ofClvanish. Sinoe Clmust bediagonal, we seethat alltheelements oftheklth row ofBllvanish with the(possible) exception oftheelement inthejlth column. Similarly, theklth column of Dlconsists oftheelements ofthejlth column ofthematrix Bllwhile all other elements vanish, andsince Dlmust bediagonal, alltheelements of thejlthcolumn ofBllvanish with the(possible) exception oftheelement in 1'Forsimplicity, ifAllisanoperator inﬁllandAllis itsmatrix, wewrite All6ﬁll aswell asAllE ﬁll, andsimilarly forﬁll, etc. 354 APPENDIX theklth row. Thus, (jl,kl)6Sll, alltheelements ofthejlth column andklthrowofanarbitrary matrix ofﬁllvanish with the(possible) exception oftheelement attheintersection ofthisrowandcolumn. Wearenow able todetermine thestructure ofthesetSllfrom aknowl- edge ofthesetSll.Bysuitably interchanging rows and columns ofthe matrices ofﬁll(which isequivalent tointerchanging elements inthebases ofthespaces XlandXl),wecanseetoitthattherows andcolumns appearing ﬁrstinthematrices ofﬁllcontain nopositions inthesetSll,while therows andcolumns with only oneposition each inSllcome next andtherows and columns with atleast two positions each inSllcome last. Thus amatrix All6ﬁllhastheform lY 3 n 0 1 .0. A12: . . ... , (9) 1;0 . 10. Q . .11 where thepositions corresponding tothesetSllareoccupied byones andall other positions areoccupied byzeros. Next weconstruct thegeneral matrix Bll6ﬁll, with nrows and m columns: 1on L3 m 1....()....()... 0 1 .0.. B21: . . ... (10) 30 . 10 ()......().. n0 0 0~~~ APPENDIX 355 Since thematrix Allhasaoneinrowat+1andcolumn y-1-1,(hl,mall-ix Bllcanhave aoneincolumn oz+1androwy-1-1but, 1nanyevent the remaining elements ofthisrow andcolumn must vanish The Same is’u,ue ofalltherows from Y+ltoSandcolumns from at-1-1top_Ifthematrix Allhastwoones incolumn S+l,then alltheelements ofthecorresponding rowofthematrix Bllvanish, andthesame istrue ofallcolumns from 8-1-1 ton(which contain atleast twoones). However, ifacolumn ofthematrix Allcontains only asingle one, then there aretwoones insome suitable row with index >13, andthiscauses thecolumn ofthematrix Bllwith thesame index tovanish. Asaresult, thewhole lower right-hand corner ofthematrix Bllisoccupied byzeros. Infact, let(j,k)beanyposition inthiscorner, and consider thecorresponding position (k,j) inthematrix All. Then the kthroworjthcolumn ofAllhasatleast twoones, since otherwise wewould have putthisrow orcolumn inan“earlier” position. This means that the kthcolumn orjth row ofBllconsists entirely ofzeros, sothat inanyevent there must beazero attheposition (j,k).Thelower left-hand corner ofthe matrix Bllalsoconsists entirely ofzeros. Infact, ifaoneappeared anywhere inthelower left-hand corner ofBll, sayattheposition (j,k),then, bythe symmetry oftheconstruction, alltheelements inthejthcolumn ofAll except possibly theelement inthekthrow (i.e., intheupper right-hand corner ofAll) would have tovanish, which isimpossible since thiscolumn must have aone inthelower right-hand corner. Asimilar argument shows that theupper right-hand corner ofBllalso consists entirely of zeros. Asfortheelements intheupper left-hand corner ofBll, they can bearbitrary. Thus itisclear thatourcategory canbeenlarged byincluding allelements ofthelower right-hand corner ofthematrix (9)inthesetSll(provided Sll does notalready contain allthese elements) andincluding allelements ofthe upper left-hand corner ofthematrix (10)inthesetSll.Thecategory then becomes maximal, since itisnolonger possible toenlarge Sllwithout making Sllsmaller. Ingeometric language, themaximal category made up oftwospaces XlandXlisconstructed asfollows: Thespace Xlisthedirect sum ofthree subspaces X3,X},Xfandthespace Xlisthedirect sum ofthree subspaces Xg,Xé,Xg,where X}andX;have thesame dimension. Theeﬂect ofanoperator Allissuch thatX‘;ismapped into{O},X}ismapped intoX;by adiagonal matrix andXfismapped into Xiinanarbitrary way, while the eﬂect ofanoperator Bllissuch thatXgismapped intoX‘;inanarbitrary way, X;ismapped into X}byadiagonal matrix andX§ismapped into {O}.An arbitrary (nonmaximal) category diﬁers from themaximal category inthat theoperators mapping Xfinto Xiarenotarbitrary, butrather correspond tomatrices with zeros incertain ﬁxed positions, while thesame istrue ofthe operators mapping Xginto X‘;(there isnoconnection whatsoever between thepositions occupied bythese zeros). 356 APPENDIX A.53. Next weconsider acategory 1’involving arbitrarily many spaces X,(ot6M). First ofall,itisclear thatevery subcategory ofthecategory 1’ made upofapair ofspaces Xl,Xlandcorresponding families ﬁll, ﬁllis constructed inthewayjust described, i.e., ﬁll,isthefamily ofallmatrices witharbitrary elements atsome prescribed setofpositions Sll,while ﬁllis thefamily ofallmatrioes with arbitrary elements atsome other prescribed setofpositions Sll. Inthisregard, weintroduce thefollowing notation: If Sisanysetofpositions inanm><nmatrix, then ﬁ,,,,,(S) isthesetofall m><nmatrices with arbitrary elements atthepositions Sandzeros every- where else. Now letSlbeasetofpositions inanm><nmatrix andSlasetofposi- tions inann><pmatrix. Suppose Sistheproduct SlSl, deﬁned astheset ofallpositions inanm><pmatrix atwhich onecangetnonzero elements in theproduct ﬁ,,,,,(Sl)ﬁ,,,,(Sl). Inother words, aposition (i,k)belongs to thesetSlSl ifandonly ifthere exists anindexj such that (i,j) belongs toSl and (j,k)belongs toSl.LetSll,...,Sl,,beacollection ofsuch setsof positions foranm><nmatrix, and letSll,...,Sllbeananalogous collection forann><pmatrix. Then thegeneral formula ‘Q Q QC‘H...C“in....inas /'\_.i\-/ US11US21=i=1 1=1 == isaneasy consequence ofthedeﬁnition ofaproduct ofS-sets. Interms ofproducts ofS-sets, wecan write theconditions forour category intheform SSCD SSCS (12) beea 1 beav aw where Disthesetofallpositions along theprincipal diagonal ofthe appropriate square matrix. A.54. Wenow construct afamily ofconcrete categories ofacertain type. Tospecify acategory J!’means tospecify allthefamilies ﬁll, orequivalently inthepresent case, tospecify allthesetsSll. Choosing Sllarbitrarily, we then choose Sllinsuch awaythat SllSll CD,SllSll CD(wehave already described how thisisdone inSec. A.52). Suppose Sllhasbeen constructed foralljandklessthan n,insuch awaythattheconditions (12)foracategory aresatisﬁed. Then S1,,and Sn,»(j<n)areconstructed asfollows: S,lis chosen arbitrarily, and Sl,ischosen tosatisfy theconditions Sl,,S,,l CD, S,,lSl,, CD.Suppose S1,,and SMarechosen forallj <kinsuch away that (12) holds. The required sets S",and S,,,,must satisfy thefollowing conditions implied by(12): 3)Snkskn CD»Sknsnk CD; b)Sjkskn CSimSkrtsrtj CSki’Sinsnk CSmSnkski CSm; C)Skn 3Skisina Sm3SmS1k~ APPENDIX 357 The conditions a)and b)represent “upper bounds” and condition c) “lower bounds” forthesetsS,,,,andS,,,,. Wenow show thatthese conditions arecompatible. Suppose, forexample, that 11-1 11-1 Sim=USkisim S1111=US11:iS:ik' (13) i=1 j=1 Then, byformula (ll)andtheinduction hypothesis, 11-1 11-1 11-1 11-1 Snkskn =US11:iS:ik USkisin =U US11:iS.'lkSkiS1'11 j—1 1'—1 j—1 ‘-1 11-1 11-1 C jsiisin CUsnisin CD1 11 j=1 i=1 11-1 11-1 11-1 Sikskn =S111USkisin =USikskisin CUS1151» CS111-s=1 i=1 1-1 This proves theﬁrst oftherelations a)andtheﬁrst oftherelations b),and itisclear that theremaining relations canbeproved bysimilar arguments. Thus theinduction isjustiﬁed andourconstruction iscorrect. Itispossible, ofcourse, toconstruct acategory byusing arbitrary S,,,, and Sl,satisfying theconditions a)—c), andnotjustsetsofthespecial type (13) used toprove thecompatibility ofthese conditions. Inthis way we obtain alarge family ofconcrete categories, ineach ofwhich only thesets S,larearbitrary, while theremaining setsSllsatisfy theextra conditions a)—c). A.55. Wenow seethat every category J!’such that ﬁllﬁll, Cﬁ(D) belongs tothefamily justconstructed. Infact, thesetsS,,landSl,,aredeﬁned in1’forevery n,while theremaining sets S,,,,and Sllmust satisfy the conditions a)—c). Butthen 1’isacategory ofthefamily described inSec. A.54. Itwould beinteresting todescribe theform ofthemaximal categories ofthisfamily. A.6. Categories andDirect Sums A.61. Given acategory 1’with basic spaces X1}, algebras ﬁf’and families ofoperators ﬁg, wherep =1,...,k,-andq=1,...,kq,wenow show how toconstruct anew category whose basic spaces aredirect sums ofthespaces Xf’andwhose basic algebras arethecorresponding direct sums ofthealgebras ﬁf. 358 APPENDIX Thus letX,»bethedirect sum ofthespaces X},...,X19",andletﬁ,-be thedirect sum ofthecorresponding algebras ﬁ},...,ﬁfi(i.e., inthespace X?anoperator A6ﬁ,actslikeanyoperator inthealgebra ﬁf). Tospecify anoperator Al,»6ﬁ,,, weusetheblock matrix A‘,-1»A2?-~A2" AilA5? ~Ail"Aii=- . (14) kl 1:2 ... Irk-Aii A1i AliL where theblock Ag?corresponds toanarbitrary operator of1’mapping the space Xfinto thespace X‘;(p=1,...,k,;j :1,...,kj).Toshow that thisgives acategory, wenote that if 11 12 ___ 11¢Bil B11 B111 21 22 ___ 21¢B B11 B11 B11‘ 51: "i 1 H k2 ... M- 1111 1221 ___ 111-111A5131: +A1131‘: + +A,'1’Biithen A118’-I Z .. ... where each sum ofproducts again belongs totheappropriate family of operators, bythedeﬁnition ofthecategory 1’.Thus ourruleleads toanew category 17,which wecallanextension ofthecategory 1’. A.62. Itturns outthat theconverse isalso true, i.e.,ifthebasic spaces X,ﬁguring inacategory aredirect sums ofcertain spaces Xf’(p:1,...,k,-) and ifthecorresponding algebras ﬁ,aredirect sums ofalgebras ﬁf (p=1,...,k,»)ofoperators acting inXf’,then thewhole category isan extension 17intheabove sense ofacategory 1’,constructed from the spaces Xf’andalgebras ﬁf. Infact, let1’’beacategory oftheindicated APPENDIX 359 type. Then inanappropriate basis chosen inthesubspaces Xf’,every matrix A,ofanoperator ofthealgebra ﬁ,»hasthequasi-diagonal form AlI1A,1 A1?‘’ where Afisasquare matrix oforder rf’(p:1,...,k,»).Every matrix Al,» ofanoperator ofthealgebra ﬁ,,isablock matrix oftheform (14), where Ag?isarectangular matrix with rj?rows andrfcolumns. With each block AZ?wecanassociate inanatural wayanoperator Ag?mapping thespace X5’ into thespace X‘,?.Using allsuch operators, weconstruct anew category 1’‘Q’with basic spaces Xf’,algebras ﬁf’and families ﬁg? ofoperators Ag? speciﬁed bythematrices Agf’. Wenow show that thiscollection ofobjects does infactdeﬁne acategory. LetAﬂf’beanoperator mapping Xfinto X§,andletAZ?beanoperator mapping X‘;into X’,'.Then theproduct A’,'§’=A’,'§A‘§f belongs tothefamily ﬁ{,?’. Infact, thecategory 1”’contains thematrix with itsqpth block equal toAlfandallother blocks equal tozero, aswell asthematrix with itsrqth block equal to andallother blocks equal tozero. The product ofthese two matrices, which belongs tothecategory 1”’, isamatrix with itslith block equal toA1,?’andallother blocks equal tozero. Therefore A’,'f6ﬁff’, asasserted. Thus alltheconditions foracategory aresatisﬁed. Itistrue that the operators mapping thespace Xfinto thespaces Xfwith thesame subscript have notyetbeen deﬁned. However, allsuch operators canbesetequal to zero without destroying therequirements foracategory. A.63. Since every semisimple algebra ofoperators acting inaspace X,- allows ustodecompose thespace X,intoadirect sum ofspaces Xiinwhich thealgebra now acts asasimple algebra, weseethat thestructure ofa general category with semisimple algebras reduces tothat ofacategory with simple algebras (this problem was considered inSec. A.4). The matrix of every operator All6ﬁl,ofthecategory isofthe form (14)inanappropriate basis, where each block Ajfisthematrix ofanoperator inthefamily ﬁg? ofsome category ,7£’,Z” with basic spaces Xf,Xf’andsimple algebras ﬁg,ﬁf. Some blocks ofthematrix A,,may beidentically zero foralltheAli.Ifwe 360 APPENDIX denote thesetofallvanishing blocks byS”,thequestion arises ofhow the setsS,»,-arerelated forvarious indices iandj.Asimilar problem wascon- sidered inSec.A.5forthecase ofone-dimensional blocks. Themethod used there isalsoapplicable tothepresent case, andleads tothefollowing result: Ifthecategory determined bytheintersection ofthejthblock rowandithblock column ofthematrix All6ﬁllisofthetypeJ!’lorJ£’l(involving invertible matrices),l' then alltheblocks intheithblock rowandjthblock column ofthe matrix Alldetermine zero categories, with the(possible) exception ofthe block attheintersection ofthisrowandcolumn. Ifthecategory inquestion isof thetype1’l,thenmatrices ofacategory ofthetype 1%appear intheindicated blocks andhave zeroproducts with thegiven matrix. Wecannowdetermine thestructure ofthegeneral category, asinSec.A.5. Remark. A.Y.Khelemski (loc. cit.) hasfound thecategories corre- sponding tonilpotent algebras ﬁl. 1‘SeeSec.A.3l andthefootnote onp.353. HINTS AND ANSWERS Chapter I 1.Ans.a)+;b) +. 2-A"S- "11"s2"2:1"44, "41"12"2s"s4, "s1"42"2:1"44~ 3.Ans. (-l)"l"_1)/2. 4.Hint. Consider thedeterminant allofwhose elements equal 1. 5.Ans. A=(mq-np)(ad -bc). 6.Hint. Multiply theﬁrstcolumn by104,thesecond by103,thethird by102, thefourth by101,andaddthem tothelastcolumn. Then useCorollary 1.45. 7.Ans.Al=-29,400,000, Al=394. 8.Hint. P(x) isobviously apolynomial ofdegree 4.Weﬁrstﬁnditsleading coeﬁicient, andthen determine itsroots bymaking rows ofthedeterminant coincide. Ans.P(X)=-—3(X2 -l)(X2-4). 9.Hint. Add allthecolumns totheﬁrst. Ans. A=[x+(n-l)a](x -a)"_1. 10.Hint. Thedeterminant ontheleftisapolynomial ofdegree ninx,,with roots xl,...,x,,_l, andhence canberepresented intheform 11-1 (A +Bxn) 1-I (X11 '_xk)" k=1 36| 362 I-IINTs AND ANSWERS Another representation ofthesame determinant intheform ofapolynomial of degree ninx,,canbeobtained byexpanding itwith respect toitslastcolumn. Equating thecoeﬁicients ofxi,‘andthose ofx;"2, ﬁndAandB. 11.Ans. cl=O,cl =2,cl=—2,cl =0,c5 =3. 12.Ans. =O,where il<il<---<ikand il<ié< I---<ilareﬁxed, andatleast oneoftheilldiffers from thecorresponding ill (Cauchy). 13.Hint. Itissufﬁcient forthecorresponding fourth-order determinant tobe nonzero. 14.Hint. Usetheresults ofSecs. 1.96-1.97. Chapter 2 1.Ans. No,since wecannot multiply by-1andstaywithin theset. 2.Ans. No,since wecannot addtwovectors which aresymmetric with respect tothegiven lineandstillstaywithin theset. 3.Ans. Yes. Inparticular, thenumber 1ePserves asthe“zero vector” ofthe space P. 4.Hint. SeeSec.1.96. 5.Hint. Assuming linear dependence oftheform alt" +alt” +'''+<xkt'b EO, divide byt'1anddifferentiate. Then useinduction ink. 6.Hint. Show thatthezerovector alsohasaunique expansion with respect to thesystem el,el,...,e,,.From thisdeduce thelinear independence ofthe vectors ofthesystem. 7.Ans. Yes, consisting ofasingle vector, i.e.,anyelement x€P different from 1. 8.Ans. 1. 9.Ans. Theintersection isthelineofintersection (intheusual sense) ofthetwo planes, while thesumisthewhole space. 10.Hint. SeeSec.2.34. 11.Ans. No.Itcanbereplaced byanyother vector ofthehyperplane. 12.Ans. With the“point” interpretation, theproperty means that every hyperplane contains thelinepassing through anytwoofitspoints. 13.Ans. Ingeneral p+q+1,ifthisnumber does notexceed thedimension ofthewhole space. n1N'rs AND ANSWERS 363 14.Ans. p+q+r+2ifthisnumber does notexceed thedimension ofthe whole space. 15.Ans. With each positive number associate itslogarithm. Chapter 3 1.Hint. Inamatrix ofrank 1thecolumns areproportional. 2.Hint. Wehave towrite theconditions foravector ytobelong tothesubspace Linsuch awaythattheyinvolve only minors ofAoforder k.ButyeLifand only ifthematrix Bobtained byadding toAthecolumn consisting ofthe components ofthevector yhasrank k,orequivalently, ifandonly ifevery minor ofBoforder k+1vanishes. Expanding every minor ofBoforder k+1withrespect toelements ofthelastcolumn, weobtain asystem ofequa- tions inthecomponents ofy,withcoefﬁcients which areminors ofAoforder k. 3.Hint. SeeSecs. 1.51-1.52. 4.Ans. x=(cl,cl,cl,cl),where cl=-16 +cl+cl+5c,,cl =23—203— 204—605. 5.Ans.If(1-1)(1+2)ab0,then 1+1 1 (1+1)2 x=_i zi’ Z=ii1+2’ Y1+2 1+2‘ IfA=1,thesystem hassolutions depending ontwoparameters. IfA=-2, thesystem isincompatible. 6.Ans. Thematrices albl alblcl albl and alblcl "3ba "3baCs must have thesame rank. 7.Ans. Thematrices albl al-blcll "2 b2 "2 b2 4'2 .. and ... a,,b,, a,,b,,c,, must have thesame rank. s.Ans.~X<1>=(1,-2,1,0,0), Xi”=(1,-2,0,1,0), xi”=(5,~6,0,0,1). 364 I-IIN'rs AND ANSWERS 9.Ans. -16 1 1 5 23 -2 -2 -6 x= 0+al 1+al 0+0:3 0, 0 0 1 0 0 0 0 1 forexample. Here theﬁrstcolumn consists ofthecomponents ofavector xo which isaparticular solution ofthenonhomogeneous system, while theother columns consist ofthecomponents ofthevectors ym,j/‘2),yli“forming anormal fundamental system ofsolutions ofthecorresponding homogeneous system. 10.Ans. Therank ofAlis3,andthere isabasis minor intheupper left-hand corner (forexample). Therank ofAlis5,andthebasis minor isthesame asthe determinant ofthematrix. 11.Hint. Move theminor Mintotheupper left-hand corner andthen, byusing theprocedure ofSec.3.62, show thatallthecolumns ofAstarting with the (r+l)stcanbemade intozerocolumns. 12.Hint. IfP75O,look forAintheform POx O1y 13.Hint. Therank ofthematrix llalkll iseither equal tonorlessthan n. 14.Hint. UsetheKronecker-Capelli theorem. 15.Hint. Usetheresult ofProb. 14. Chapter 4 1.Ans. Also n. 2.Ans. c)and g). 3.Ans. Yes. 4.Ans. —l —l 2 2 O-2 bl)/41¢): 1-3 3; b)/41¢): 1-1 1- —l -5 5 2 1 O 5.Ans. ABAB 75A2132. 6.Ans. AB—BA=E. 7.Hint. (A+B)?=A2+AB+BA+B2, (A+B)3=A3+AZB+ABA+AB2+BA’+BAB+BZA+B3. I-IIN'rs AND ANSWERS 365 8.Hint. Useinduction. 9.Ans. Thedimension ofthespace isnm.Forbasis operators wecantakethose corresponding tothematrices A,~,-(i=1,...,n;j=1,...,m),where A,-,~is anymatrix whose elements areallzeroexcept fortheelement intheithrowand jthcolumn. 10.Ans. '0 00 AB=OOO. O00 ll.Ans. 1n cosno —sin noA"= ,B"= . O1 sinno cosno a b 12.Ans. A= ,where bc=-112. c-a 13.Ans. —9 -2 -10 000 a) 6 14 8; b) 000. -7 5 -5 000 15.Hint. UseProb. 14. 16.Hint. Thethree equations fortheunknown elements ofthematrices Aand Blead toequations forthree minors ofanunknown 2><3matrix. Now see Chap. 3,Prob. 12. 17.Hint. SeeSec.4.54. 18.Hint. Express theelements oftheminor Minterms oftheelements appearing intheﬁrstrrows, andthen useTheorem 4.75. 19.Hint. Usethesolution toProb. 18. 20.Hint. SeeSec.4.54. 1—2 7 5—2 1 3)/111 = 1 2; b)B11=O 1-2 ; c)C“=C.21.Ans. ll 0 0 1 23.Ans. IfAisthezeromatrix, thenXisarbitrary. IfdetA75O,thenXisthe zero matrix. IfdetA =0andAisnotthezero matrix, then itsrows are 366 I-I1NTs AND ANSWERS proportional. Leton/[5betheratio ofthecorresponding elements oftheﬁrst andsecond rows ofthematrix A.Then -bl]aq foranyp andq. 24.Hint. SeeSecs. 1.51-1.52, 25.Ans. N0. 26.Hint. Consider theoperator Alsuch that Al[a0 +alt+---+a,,t"] =M0+al+alt+---+ a,,t"“1. 27.Hint. Theoperator Acarries linearly independent vectors intovectors that areagain linearly independent. 28.Hint. Apply theequality AB=BAtoaneigenvector oftheoperator A. 31.Hint. Usetheresult ofProb. 30. 32.Hint. Suitably choosing anoperator Bandusing Prob. 28,reduce the solution toProb. 31. 34.Hint. Usethefactorization oftheoperator A2-112E. 35.Ans. a)Al=2,fl=(1,O,O);7l=1,jl=(1,O,1);A3=-1,f3=(0,1,-1);b)1.=-1.1.=<1.0.<>>; 1.=1.-1.1".-(1.0.1>.f. =<<>.1. -1); c)Al=2,fl =(l,0,0);d)Al =l,fl =(l,0,0, —l);Al =O,ﬂ= (O,1,0,0). 36.Hint. The relation T(A") CN(A’") isnecessary and sufﬁcient forthe equality A’°+’" =0tohold. 37.Hint. Letfl, ...,f,beabasis fortherange oftheoperator A,sothat Ax=2a,-(x)f, forevery x6K,,.Now let C1 A,»x -a,-(x)f,- (i=1,...,r). Chapter 5 1.Hint. Theﬁrstvector ofthenewbasis isx. 2.Hint. Choose anewbasis fl,j$, ...,f,,whose lastn-kvectors form a basis forthespace K’.Write thecondition x€K' intheform ofasystem of equations involving thecomponents ofxinthenewbasis, Usethetransfor- mation formulas toconstruct thecorresponding system ofequations involving thecomponents ofxintheoriginal basis. HINTS AND ANSWERS 367 3.Hint. UseProb. 2andthedeﬁnition ofahyperplane. 4.Ans. Thematrix ofthedesired transformation isC=BA"1. 5.Hint. Letel,el,...,e,,beanarbitrary basis inK,,,andlet Tl L(x) IEllcilcs Ir1 where Zl,El,...,5,,arethecomponents ofthevector x.Begin theformulas forthecoordinate transformation with theequation 11 7'11=2111511- =1>1- 6.Hint. Use Sec. 4.83 andtheinvariance ofthecharacteristic polynomial (Sec. 5.53). 7.Hint. Choose abasis whose ﬁrstmvectors lieinthesubspace R0"). Show that forthisbasis thepolynomial det||A(,, -7.Ellhasthefactor (A-7.1,)“. Now usetheinvariance ofthecharacteristic polynomial (Sec. 5.53). Chapter 6 1.Ans. Inthebasis e,,,e,,_l, ...,el. 2.Hint. SeeSec. 6.44. —l 1 O O O O—l O O O O O3.Ans. 0 0 2 1 0. 0 0 0 2 0 0 0 2 4.Ans. No. El(A) =(7.-2)(7. -1)2,El(B) =(7.-1)(7.2 -57.-2). 5-ArmEn_1(/41) =E.._1(/42) =(1—1)".E..-2(/41) =En-2(A2) =1; En-1(A3) =('7C1)",En-2(A3) 31; E.._.<A.> =fl<1-Io.E.._.<A.> =1- lf=1 6.Hint.E,,_l(A) =(ct-1)",E,,_l(A) -1. 7.Ans. Adiagonal matrix with some oftheroots ofthepolynomial P(A)along itsprincipal diagonal. 368 HINTS AND ANSWERS 8.Ans. Some oftheroots ofthepolynomial P(A)liealong theprincipal diagonals oftheJordan blocks, andthesizes oftheblocks donotexceed themultiplicities ofthecorresponding roots. 9.Hint. Thevectors x,AxandAx’arelinearly dependent. 10.Ans. Polynomials inA,,,(u). 11.Ans. Matrices oftheform blblb, b,,, b,, 01».1». 1»... 1».-.B...= <n>m> OOO bl b,l_,,,+l blbl b,, Ob1 bn-1 B,l,,,= OO bl (n<m).OI‘ O0---O 00 0 12.Ans. Matrices oftheform Bmlml Bmlml II-Bmlmk Bmkml Bmkml '''Bmkmk withtheblocks B,,,__,,,,_ given intheanswer toProb. 11. B,,,,,,,1 0 -0 0B,,,,,,,, 013.Ans. Matrices oftheform 0 0 B,,,,,,,, 14.Ans. Toevery group ofJordan blocks withthesame rootofthecharacteristic polynomial, there corresponds ablock ofthekind given intheanswer toProb. 12.Theremaining elements areallzero. 15.Ans. Ifthemultiplicity ofeach rootofthecharacteristic polynomial equals thesizeofthecorresponding Jordan block, orifthecharacteristic polynomial coincides with theminimal annihilating polynomial, orifalltheelementary divisors except theonewith highest index equal 1. I-IINTs AND ANSWERS 369 Chapter 7 1.Ans. Atensor oforder two, with twocovariant indices. 2.Ans. Forexample, vii—11%—115. where "11=2&1+252+is, -"12=la.‘i1—252, Tia=ia- 3.Hint. SeeSec.7.93. 4.Hint. SeeSecs. 4.54and7.15. 5.Ans. Forexample, A(Xs )1)=‘F171 +‘I272 +°aTas where o',-and1,(i=1,2,3)arethenewcomponents ofthevectors xandy. Thetransformation formulas tothenewbasis are °1=E1+52» °2=‘Z2+2&3, °a=as- 6.Hint. First renumber thevariables insuch away that thematrix ofthe bilinear form A(x,y)istransformed intoaform towhich Jacobi’s method is applicable. 7.Hint. ||-a,»,,|| must bethematrix ofapositive deﬁnite form. Ans. "11 "12all--:O, >O,..., (-1)" detl|a,~l.|l >0. "21 "22 8.Hint. Seetheremark toSec.7.96. 9.Hint. Consider theform onthebasis vectors. 10.Hint. Thelastrowofthedeterminant consists oftheelements al,") =(—l)"“1A(el, ...,el._l, e,,_,.l, ...,e,l) (k=1,2,...,n). 11.Hint. Usetheequation A(el, el)=1toﬁndtheﬁrstpairofbasis vectors. Then construct thesubspace Ldeﬁned bytheequations A(@1.X) =0. A(@2. X)=0- Iftheform A(x,y)does notvanish identically inthissubspace, ﬁndvectors el,cleLsuch thatA(el, el)=1,andsoon. 12.Hint. Consider theform 7! A(x, X)+e2if (e>O), 1-1 andapply thecriterion ofSec.7.96. 13.Hint. Let x(1)=(5_1t>,___,gllt)) (j=1,__,,r) 370 I-IINTs AND ANSWERS beabasis ofthesubspace K’.Then K”consists ofthe vectorsy =(v1l,...,11,) satisfying thesystem A(X"’»y) =2(ib.-.&.<")n. =0<1=1.....1). k=1 i=1 Thematrix ofthecoefﬁcients ofthesystem istheproduct ofthenonsingular matrix lla,-kll oftheform A(x,y) andthematrix ||El1'>l| ofrank r.Now use Corollary 4.67. 14.Ans. A’=A. 15.Hint. Ify-('t1l,...,r,,l)isasolution ofthesystem (44), then (b.y)=(AX.y)=(X.A’y)Z0- Conversely, thesystem (44) isthecondition forthevectors yand a,»= (a,-l, ...,a,-,,)tobeconjugate. If(b,y)=0forallsuch y,then xliesinthe linear manifold spanned bythevectors al,...,a,l. 16.Hint. SeeChap. 4,Prob. 37. 17.Hint. SeeChap. 3,Prob. 1. 18.Hint. First consider thecaseofnonnegative forms ofrank 1,using Prob. 17andthen Prob. 16. Chapter 8 1.Ans. No,since axiom b)fails, andsodoes axiom c)(forA=-1). 2.Ans. No,since axiom b)fails. 3.Ans. Yes. Thenewdeﬁnition ofthescalar product merely corresponds toa change ofunits along thecoordinate axes. 4.Hint. Letel,el,eldenote thevectors directed along three edges ofthe tetrahedron drawn from acommon vertex, andexpress theother edges ofthe tetrahedron asvectors. Ans. 90°. 5.Ans. 90°, 60°, 30°. 6.Ans. j< x2(t)dt +,/jig)/2(t)dt, l>ll/j“:x2(t)dt -/j:y2(t)dtJ1':(x(t)+norat 1 7.Ans. cos<11=W 3-Arm=1)g=(3.1. *1.1'2). /1=(2.1. —1.4); b)g=(1.7.3.3). /1= (-4, -2,6,0). HINTS AND ANSWERS 37| 9.Hint. Usethedeﬁnition ofangle (Sec. 8.33) andtheorthogonality ofthe vector /1toallvectors ofthesubspace R’. 10.Hint. Take thescalar product ofequation (18), p.223with thevector go. 11.Hint. SeeSec.8.52. 12.Ans. yl=i,y2 =y3 =0,y4=~2j,y5 =0,ye=5k, 13.Ans. (1,2,1, 3),(10, ~1,1, —3), (19, ~87, A61, 72). 14.Hint. Assuming thatthedimension ofR”isgreater than thedimension of R’,consider thevector e”eR”which isorthogonal totheprojection ofR’onto R".Then useProb. 10. 15 A_(2n)! .Ans. ,,—T——n(n!)2. 16.Ans. P,,(~1) =(~1)". 17.Hint. Express thecoefﬁcients asscalar products. 18.Hint. Usetheresults ofPr0bs. 15and16. 19.Hint. Expand Q(t)inLegendre polynomials. 1 A"S-Q0‘)=IPW)- Tl 20.Ans. ‘IF(t)||2 =i— .in 2n+1 21.Ans. k(A) =]detA]. 22.Hint. SeeSec.4.75. 23.Hint. This isaquestion ofcomparing thealtitudes oftwohyperparallel- epipeds. 24.Hint. Theinequalities V ,,..., V ,,..., [X1X2 Xm] < [X1X2 xk] (k=1,2,__ _,m) I/[X1, ---9xk—-1!xk+1! ---9x1!l] V[x1,x2, ---axk-1] areeasily obtained from theinequality (37). Multiply them alltogether for k=1,2,...,m,make appropriate cancellations, andthen takethe(m—l)th root. Thegeometric meaning oftheinequality isthefollowing: Thevolume ofan m-dimensional hyperparallelepiped does notexceed theproduct ofthe(m—l)th roots ofthevolumes ofits(m—1)-dimensional “faces.” 25.Hint. Write theinequality (38) forxsl,xsg,...,x,’,andthen multiply these inequalities together forallpermissible values ofs1,s2,...,s,. 26.Hint. Wemust construct ahyperparallelepiped ina2"‘-dimensional space such thattheprojections ofitsedges onto each axishave absolute values no greater than Mandsuch thatitsvolume isexactly M"n"/2. ForM=1,the 372 HINTS AND ANSWERS matrix Amofthecomponents ofthe2'"-dimensional vectors determining this hyperparallelepiped aregiven bythefollowing recurrence formula: /1,,,_, /1,,,_, 11A...= ,A,= ./1,,,_, -A,,,_, 1-1 Comment. Fornvi2"‘,theestimate canbeimproved. 27.Hint. Given anysubspace GCR,letGldenote theorthogonal complement ofG.Forevery xeN(A) andevery zeR, (A'AX)=(LAX) =0, and hence A’z€NJ-(A), i.e., T(A') CNi(A), Ti(A’) 3N(A). Forevery xeTi(A) andeveryy eR, (A'X,y) =(X.Ay)=0, and hence A'x =O,i.e., x€N(A'), sothat T1-(A) CN(A'), T1-(A’) CN(A). Itfollows thatN(A) =T1-(A’), Ni(A)=T(A’). Theother assertion isproved similarly. 28.Hint. SeeSec.4.77. 29.Hint. SeeSec.4.54. 30.Hint. Theangles ofatriangleareuniquelydetermined byitssides. Alterna- tively, thesymmetric bilinear form (Qx, Qy)isuniquely determined bythe quadratic form (Qx, Qx). 31.Hint. Agiven isogonal operator Atransforms theorthonormal basis e,,e2,...,eninto anorthogonal basis fl’=<z,f1, f2’=ilzﬁ, ...,f,;=<z,,f,., where fbﬁ, ...,f,,areunitvectors. LetQbetheisometric operator carrying thevectors f,,}§, ...,f,,intoe1,e2,...,e,,.Then thematrix oftheisogonal operator QAisdiagonal. Show thatthecondition <1,»vi11,-allows onetocon- struct apairoforthogonal vectors which arecarried intononorthogonal vectors bytheoperator QA. 32.Hint. Itissuiﬁcient toshow thatQisanisogonal operator (seeProb. 31). Assuming thatthere isaright angle which isnottransformed intoaright angle, construct aparallelogram whose area changes asaresult ofapplying the operator Q. 33.Hint. Generalize theconstruction ofProb. 32. 34.Hint. Applying theorthogonalization process tothegiven systems, obtain orthonormal systems e,,e2,...andf,,}Q, ....Using Sec.8.53, show thatthe formulas expressing thevectors x1,x2,...,xkinterms ofel,e2,...arethesame asthose expressing thevectors y,,y2, ...,y,,interms offhﬂ, ....Then deﬁne Qastheoperator which maps thesystem el,e2,...intothesystem f1»fz,- --- HINTS AND ANSWERS 373 35.Hint. Consider theﬁnite systems ei,ef,eé,efz’,...,e,',,eifandf1,f;',f,_;, f;,...,fk, obtained indetermining theangles between thesubspaces R’, R”andthesubspaces S’,S".Byconstruction, (e§.¢’;-I)=(f§,f:-')=¢°5 ‘Pt (I'=1,2,---J‘), (ELF;-) =(f§,f}) =O, (ELF?) =(fZ,f§) =O (5*1)- Show further that(e;,eg)=(f;,f;) =O(using Prob. 9).Then usetheresult ofProb. 34. 36.Hint. Use Prob. ll. 37.Hint. Inthesubspaces L,andL2,lete,,e2,...,emandfbﬁ, ...,f,,,be thebases obtained inconstructing theangles 0:1,<12,...,am.Inthespace R construct abasis el,e2,...,e,,,,em“, ...,e,,which begins with thevectors obtained byorthogonalizing thevectors el,e2,...,e,,,,f1,ﬁ, ...,fm. Expand thevectors xl,x2,...,x,,,,yl,y2,...,y,,,with respect tothisbasis. Show that thematrices ofthese expansions each have only oneminor oforder m,ifwe disregard minors which areknown tovanish. Then usetheexpression forthe volume ofahyperparallelepiped interms oftheminors ofthecorresponding matriX. 38.Hint. SeeChap. 3,Prob. 2andChap. 4,Prob. 17. 39.Hint. Verify theassertion inthespecial basis whose ﬁrstkvectors belong tothesubspace L(x1, x2,...,xk).Togoover tothegeneral case, useChap. 4, Prob. 17,showing that det|ia§’)|| =1. 40.Hint. First consider thecasek=2. 41.Hint. Choose abasis inthespace Rlikethatchosen inProb. 37,andverify thattheformula isvalid inthisbasis. Then goover tothegeneral caseinthe same wayasinProb. 39. 42.Hint. SeeSec.4.54. 43.Hint. Consider theorthogonal complement Zoftheinvariant (with respect toA)subspace Hofallvectors xsuch thatP(A)x =O.Thesubspace Zisalso invariant with respect totheoperator A,andhence with respect to[P(A)]"'1. Butif2EZ, then [P(A)]*"'1z€ H,sothat [P(A)]"'1z =O.From this, deduce that[P(t)]"*1 isanannihilating polynomial oftheoperator A. Chapter 9 1.Hint. UseSec.9.45. 2.Hint. Theoperator Bhasabasis consisting ofeigenvectors el,...,e,,with positive eigenvalues ul,...,u,,.Hence Bze,» =ufei, andanecessary condition forB2=Aisthatthee,»beeigenvectors oftheoperator Aandthatthenumbers (ifcoincide withthe1,».Butthisisalsosufﬁcient forB2=A. 374 HINTS AND ANSWERS 3.Hint. First transform thebasis insuch awayastodiagonalize thematrix ofthegiven operator. Ans. 320 ,/1:242. 025 4.Hint. The operator A’A issymmetric, and theexpression (A’Ax, x)= (Ax,Ax)isnonnegative forarbitrary x6R,,.IfAisnonsingular, thisexpression ispositive forarbitrary xeR". 5.Hint. Q’=QC‘. 6.Hint. Theoperator A’Aissymmetric andpositive (Prob. 4),andhence we canﬁndasymmetric positive operator Ssuch thatS2=AA’. Then construct anoperator Qsuch thatQ=S'1A andshow thatQisisometric. 7.Hint. UseProbs. 2and5. 8.Hint. LetR’CR,,bethesubspace spanned bytheeigenvectors ofthe operator A’Awithnonzero eigenvalues, andletR”betheorthogonal complement ofR’. OnR’letVequal theisometric component ofA(sothat(/E Vx=Ax), and onR”letVx=0. 9.Hint. UseChap. 4,Probs. 28—29. 10.Hint. Apply theorthogonalization process tothevectors oftheJordan basis ofA(Sec. 6.37). Chapter I0 2 2 1 1.Ans. a)41)f+1)§—2v;§; 1)1=-§E1—§E.2+§E3, 2 1 2 l'lz=§E~1+'§E>2_§E~2, 1 2 2 7)3=‘j'E,1+‘j'E,g+'j'E,3§ 1 2 2b)l0v;f+1)§+1)§; W1=-53-1+3-E2—-55-3, -2a--La 7i2_\/51 \/52» ,=i Lvi.33x/’§€1‘i‘3\/g€g‘i"_:T€3» 1-nN'rs AND ANSWERS 375 2 2 2 2 1' 1' 1 1C)m~'/t.+3n,+5n,; 111=5¢1+§¢2+5é3+5&,, 1 1, 1 1 "i2=§E~1+§42“§E~aC5E.4, 1, 1 1 1, l'la=i41“EE~2+5E~a“i44, 1 1 1 1 1043591 “'§5~2“5E~a+§E~4§ E, ,/Ed)11f+11§+11§~311§; 111:-\%s1+—?€2. (/5 (/5"tC7i3+7‘-2» 1 1 1 1, 103:5 _iE~2+§E~a“i~4,.1“>- 1 1 1 1 l'l4:§E»1“§€~2“5E~a+iE~4- 2.Ans. Amaximum forx=(;i;l, O,O)where A(x,x) =l.Aminimum for x=(O,O,;i;l) where A(x,x) =§.Aminimax forx=(O,;l;l,0) where A(x, x)=§,i.e.,thefunction A(x, x)increases ifwegoalong theunitsphere inonedirection from thepoint xanddecreases ifwegointheother direction. 3.Hint. Namely, onthesubspace spanned bythecorresponding canonical basis vectors. 4.Hint. The coefﬁcient 1,,equals thesmallest ofthemaxima oftheform A(x, x)onasystem ofsubspaces, andthecoefﬁcient uhequals thesmallest of themaxima oftheform B(x, x)onthesame system ofsubspaces. 5.Ans. y/x=55%. 6-AMA(x,x) =Y1§+11$+11§,B(x,x) =t1§+211%+3Y1§,E1=111—W2+223, E2=W2—Yla,is=Wa- 7.Hint. Theproblem reduces totheuniqueness ofthecanonical basis ofa symmetric operator with distinct eigenvalues. 8.Hint. Generalize Sec.7.44. 9.Ans. a)Ahyperboloid ofonesheet with itsaxisalong they-axis; b)A hyperboloid ofonesheet with itsaxisalong thex-axis; c)Acircular paraboloid with itsaxisalong thex-axis; d)Acircular paraboloid with itsaxisalong the y-axis, displaced oneunitalong thisaxis; e)Ahyperbolic paraboloid. 10.Ans. a)xi+2);:+32:=6; 3(x—1)=—x1 +2)/1+221, 3y=2x1—y1 +221, 3(z+1)=2x1+2y,—21; 376 HINTS AND ANSWERS b)xi+2y§—32$=6;3(x+1)=—X1+2_y1+22,, 3(y+1)-= 2x1—y1 +221, 32=2x1+2y1—21; c)yf=2x1; 3(x—m)=2x1+2y1+21, 3(y+2m) =2x1—y1 —221, 3(2+2m)=—x1+2y1—221 (marbitrary). 11.Hint. The semiaxes oftheellipsoid aredetermined from thecanonical coefﬁcients ofthecorresponding quadratic form. Usetheresults ofSec.10.25. Chapter 1| 1.Hint. LetK’betheintersection ofthenullspaces ofalloperators belonging toaleftideal JCB(K,,), andletrbethedimension ofK’. Choose abasis inKn whose ﬁrstrbasis vectors lieir.K’.Then theﬁrstrcolumns ofthe matrix ofevery operator AeJconsists entirely ofzeros. Letmbethedimension ofJ,andlet A1,...,A,,,belinearly independent operators inJ.Consider thematrix with n—rcolumns andmnrows obtained bywriting allthematrices A1,...,A,,,on topofeach other andomitting theﬁrstr(zero) columns. Therank ofthismatrix isn—r,andhence ithasn—rbasis rows. Thelinear combinations ofthese rows giveallpossible rows consisting ofn—relements. Now useSec.4.44. 2.Hint. Introducing anonsingular bilinear form (x,y), consider thesetJ*of alloperators A*conjugate totheoperators AeJ.This setisaleftideal. Now useProb. 1. 3.Ans. Amaximal leftideal ofthealgebra B(K,,) isthesetofalloperators carrying aﬁxed vector ofthespace Knintozero. Aminimal leftideal istheset ofalloperators carrying aﬁxed (n—l)-dimensional subspace ofKnintozero. Amaximal right ideal isthesetofalloperators carrying thewhole space K,, intoaﬁxed (n—l)-dimensional subspace. Aminimal right ideal isthesetofall operators carrying thewhole space Knintoaﬁxed straight line. 4.Hint. Let (Xiy) =iéﬁi (X=i5191'»)? =iWei") 7-=1 = =1 ~.>- ~. inthebasis e1,...,eninwhich thematrix oftheoperator AeBtakes theform indicated inSec.11.85. 5.Hint. Ifasubspace C’CC,,isinvariant (with respect tothealgebra B),then soisitsorthogonal complement. Expand C,,asanorthogonal direct sum of irreducible invariant subspaces. Every operator Avi0(ofthealgebra B)acts asanonzero operator inatleast oneofthese subspaces. 1-11NTs AND ANSWERS 377 6.Hint. Deduce from therepresentation ofSec.11.85 thatthecommutator of asemisimple butnonsimple matrix algebra Bintersects Binmatrices other than multiples ofthematrix corresponding totheidentity operator. 7.Hint. Write thedesired matrices asblock matrices consisting ofm2blocks. Then write thecommutativity condition anduseSchur’s lemma. 8.Ans. Forthealgebra Bofalldiagonal matrices 1.10A 0' 07,2 0 s 00 7," where 11,12, ...,2,,arearbitrary complex numbers. Every matrix algebra B=Breduces tothisform insome basis. 9.Ans. LetBbethealgebra ofalloperators under which agiven system of subspaces, whose direct sum isthewhole space C,,,remain characteristic subspaces. Then BCB.Every algebra with BCBreduces tothisform. 10.Ans. Thespace C,,isadirect sum ofsubspaces C“), ...,C“), andthe algebra Bconsists ofalloperators invariant ineach C“)(j=1,...,k).The commutator Bconsists ofalloperators which aremultiples oftheidentity operator ineach CU)(j=1,...,k). 11.Hint. IfBisadirect sumBl"+---+13"“,thenis=iii"+ +13"". 12.Ans. Ifthemultiplicity ofeach root ofthecharacteristic equation ofthe operator equals thesizeofthecorresponding Jordan block (seeChap. 6,Prob. 15). 13.Hint. IfCB=B,then CA=Cforsome AeB.Itfollows thatC=CA= C(CA) =C2A=C3A=---. 15.Hint. LetA1,...,A,,, beabasis ofthealgebra B.Then, ifBisnotnil- potent, oneoftheright ideals A1B, ...,A,,,B, sayA1B, isnotnilpotent (Prob. 14).Moreover A1B vi0(Prob. 13),andtheproblem reduces totheanalogous problem foranalgebra ofsmaller dimension. 16.Hint. IfM,»=M,-+1, then forevery vector x€M,~ there isanoperator A1eB such that A1x¢M,» =M,-+1. Moreover, there isanoperator A2613 such thatA2A1x€ Mi,andsoon.IfM1,viK,,,then forevery x6M,,+1 —M1, there isanoperator Ape Bsuch that A,,x€ M1,—M,,_1, then anoperator A,,_1€ Bsuch thatA,,_1A,, 6MDA1 —M,,_2, andsoon, sothatA1A2 ---Apx viO. 17.Hint. Usethesubspaces M1,...,M1,ofProb. l6. BIBLIOGRAPHY Bellman, R.,Introduction toMatrix Algebra, McGraw—Hill Book Co., Inc., New York(1960). Gantmakher, F.R.,TheTheory ofMatrices, 2vols., translated byK.A.Hirsch, Chelsea Publishing Co., New York (1959). Gelfand, I.M.,Lectures onLinear Algebra, translated byA.Shenitzer, Interscience Publishers, Inc., New York (1961). Halmos, P.R.,Finite-Dimensional Vector Spaces, second edition, D.VanNostrand Co., Inc., Princeton, NJ. (1958). Hamburger, H.L.and M.E.Grimshaw, Linear Transformations, Cambridge University Press, New York (1951). Hoffman, K.and R.Kunze, Linear Algebra, Prentice—Hall, Inc., Englewood Cliﬁ‘s,NJ.(1961). Jacobson, N.,Lectures inAbstract Algebra, Vol.2,Linear Algebra, D.VanNostrand Co., Inc., Princeton, N.J. (1953). Mirsky, L.,AnIntroduction toLinear Algebra, Oxford University Press, New York (1955). Noble, B.,Applied Linear Algebra, Prentice—Hall, Inc., Englewood Cliffs, N.J. (1969). Perlis, S.,TheTheory ofMatrices, Addison—Wesley Publishing Co., Reading, Mass. (1952). Shilov, G.E.,AnIntroduction totheTheory ofLinear Spaces, translated byR.A. Silverman, Prentice—Hall, Inc., Englewood Cliffs, N.J. (1961). Thrall, R.M.andL.Tornheim, Vector Spaces andMatrices, John Wiley andSons, Inc., New York (1957). 379 INDEX Adjoint matrix, 258 Adjoint operator, 198, 238, 254, 258 Adjugate matrix, 116 Alﬁne space, 31,215 A-isomorphism, 199,254Algebra(s), 13611,3121’: ofanalytic functions, 176 commutator of,320 second, 320 commutative, 137 complete, 337,338-340 ofdiagonal matrices, 353-357 composition series of,323 ofdimension n,137 factor, 138 ﬁnite-dimensional, 31513 ideals in.138 ofjets,161 morphism of,138-139, 313 nilpotent, 333 normal series of,323 one-dimensional, 340-345 ofoperators, 169 ofpolynomials, 137ﬁ radical, 317 radical of,317 ofrational functions, 175 representations of,31313 semisimple, 316 structure of,323-327 simple, 315,345-352 structure of,320-322 simple components of,326 subalgebra of,138Algebra(s) (cont.): trivial, 137, 312unitin,137,312 left, 137 right, 137 two-sided, 137 Angle(s): between k-vectors, 245 between subspaces, 244 between vectors, 217 Annihilating polynomial, 143 minimal, 143 Antiself-adjoint operator, 262 Antisymmetric operator, 238 real, structure of,269 Antisymmetry property ofdeterminants, 9 Associativity, 1,2,83,86,136 Basis, 38ﬂ components ofavector withrespect to,39 orthogonal, 222,257 orthonormal, 222,258 Basis columns, 25,59 Basis minor, 25,59 Basis minor theorem, 25,59 Basis rows, 59 Bessel's inequality, 224 Bicontinuous mapping, 294 Bilinear form(s), 179ﬁ canonical basis of,190 canonical coelﬁcients of,192 canonical form of,191 general representation of,180 Hermitian, 247 382 INDEX Bilinear form(s) (cont.): matrix of,181 transformation of,181-182 nonsingular, 182,208 positive deﬁnite, 208 rank of,182 symmetric, 181 inaEuclidean space, 273 Bilinear function (seeBilinear form) Bilinear functional, 180 Bordered minors, 302 Bounded set,217 Buniakovsky, V.Y.,218 Canonicai basis: ofabilinear form, 190 construction of,byJacobi’s method, 192- 196 ofaHermitian form, 252 ofaquadratic form, 185 Canonical coelﬁcients, 185,192 Canonical equation: ofacentral surface, 288 ofanoncentral surface, 289 Canonical form, 13313 ofabilinear form, 191 ofaHermitian form, 252 Jordan, 146 ofthematrix ofanarbitrary operator, 146 ofthematrix ofanilpotent operator, 136 ofaquadratic form, 185 Canonical mapping, 54,139Cartan, 1-1.,335 Category, 33513 extension of,358 ofﬁnite-dimensional spaces, 336 linear, 336 maximal, 343 objects of,335 mappings of,335 Cauchy, A.L.,218, 362 Center (ofasurface), 289, 299 Central surface, 288, 290 canonical equation of,288 proper, 288 inndimensions, 293-294 semiaxes of,291 Characteristic equation, 110 Characteristic polynomial: ofamatrix, 110 ofanoperator. 126 Characteristic space (seeEigenspace) Characteristic value (seeEigenvalue) Chebotarev, N.G.,332 Circular paraboloid, 297 Class (ofcomparable elements), 48 Cofactor: ofanelement, 12 ofaminor, 22 Cognate spaces, 352 Columns ofnumbers: linear combination of,10,24 coelﬁcients of,24 linearly dependent, 27 product of,with anumber, 24Columns ofnumbers (cont.): sumof,24 Commutativity, 1,137 Commutator, 320 second, 320 Comparable elements (ofasubspace), 48 Complementary minor, 21 Complex numbers, ﬁeldof,3 Components, 34 simple, ofanalgebra, 326 ofavector, withrespect toabasis, 39 Composition series, 323 Conical surface, 288,294-296 Conjugate operator (seeAdjoint operator) Conjugate subspace, 190,252 Conjugate surface, 299 Conjugate vector: toasubspace, 190,252 toanother vector, 190,252 Coordinate transformation(s) ,118ﬁ consecutive, 120 matrix of,119 operator of,119 orthogonal, 239 unitary, 259 Courant, R.,276 Cramer’s rule, 20,35 Derivatives ofapoiyuomiai, 163 Descending principal minors, 193 Determinant(s), 6ff antisymmetry property of,9 column operations on,11 elements of,6 evaluation of,16-17 expansion of2 withrespect toacolumn, 12 withrespect toarow, 12 Gram, 230 linear property of,10 ofamatrix, 6 order of,6 product of,103 ofaproduct ofmatrices, 103 quasi-triangular, 23 terms of,6 transpose of,9 triangular, 14 Vandermonde, 15 Diagonal matrix, 100 Diagonal operator, 100 Diagonalizable operator, 100 Dimension: ofahyperplane, 53 ofalinear manifold, 51 ofalinear space, 40 over asubspace, 45 ofanalgebra, 137 ofthenullspace ofanoperator, 94 oftherange ofanoperator, 93 ofasumofsubspaces, 47 Direct sum, 45,314 orthogonal, 223 Directed linesegment, 31 Distortion coelﬁcient, 242 Distributivity, 2,83,86,136 Eigenray (seeInvariant direction) Eigenspace, 110 Eigenvalue, l08ﬁ Eigenvector, l08ﬁ Eilenberg, S.,335 Elementary divisor, 151 Elementary operations, 67Ellipsoid, 292 Elliptic paraboloid, 297 Embedding, 54,139 Epimorphism, 53 ofanalgebra, 139 Equivalence classes, 339 Equivalent elements, 339 Euclidean isomorphism, 221 Euclidean space(s), 215ﬁ embedding of,inaunitary space, 263tI Euclidean—isomorphic, 221 Factor aigebra, 138 Factor space, 49 Faguet, M.K.,243 Field(s), lﬁ axioms, 1 ofcomplex numbers, 3 isomorphic, 2 ofrational numbers, 2 ofrealnumbers, 2 First structure theorem, 322 Fourier coelﬁcients, 222 Fredholm’s alternative, 73 Fredho1m’s theorem, 212 Fundamental system ofsolutions, 65 normal, 66 Fundamental theorem ofalgebra, 3 General sointion, 63,66 Gram determinant, 230 Hadamard inequality, 234 Hamilton—Cayley theorem, 155 Hardy, G.H.,2,3 Hermitian (bilinear) form, 247 canonical basis of,252 canonical form of,252 (Hermitian—) symmetric, 248 nonsingular, 249 positive deﬁnite, 253 rank of,249 Hermitian conjugate matrix, 258 Hermitian conjugate operator (see Adjoint operator) Hermitian matrix, 248 Hermitian quadratic form(s), 249,308-310 symmetric, 249 canonical form of,309 simultaneous reduction of-two, 310 stationary values of,309 Hermitian-symmetric matrix (see Hermitian matrix) Hermitian-symmetric operator (seeSelf-ad- joint operator) Homeomorphic ﬁgures, 294INDEX 383 Homogeneous linear system, 43 Homotopic ﬁgures, 294 Hyperbolic paraboloid, 298 Hyperboloid ofonesheet, 292 Hyperboloid oftwosheets, 292 Hyperparallelepiped, volume of,232 Hyperplane, 52 dimension of,53 Hypotenuse (inaEuclidean space), 220 Ideai: left, 138 proper, 138right,13s two-sided, 138 Identity matrix (seeUnit matrix) Identity operator, 78,99 Identity transformation, 120 Imaginary numbers, 3 Inclusion relations, 31 Incompatible system oflinear equations, 4, 234 Index ofinertia, 206 negative, 206,251 positive, 206,251 Index ofnilpotency, 333 Integers (inaﬁeld), 2 Interpolation with least mean square error, 237 Invariant, 131 Invariant direction, 108 Invariant matrix, 202 Invariant operator, 201 Invariant subspace, 106,313 Inverse element, 32 uniqueness of,33 Inverse matrix, 105 Inverse operator, 105 Inversion, 5 Invertible element (ofanalgebra), 137 Isogonal operator, 244 Isometric operator, 239 real,structure of,270 Isomorphism: ofalgebras, 139 ofﬁelds, 2 oflinear spaces, 53 Jacobi’s method, 192-196, 252 Jacobson, N.,332 Jet(s): addition of,161 algebra of,161 invertibility of,167 multiplication of,161 product of: withanother jet,161 with anumber, 161 sumof,161 symmetric, 168 Jordan basis, 146 Jordan block, 147 Jordan canonical form, 146 real, 159 384 INDEX Jordan normal form (see Jordan canonical Matrices (cont.): form) Kernei, 56,313 Khelemski, A.Y.,332, 334, 360 k-linear form, 203 Krasnosyelski, M.A.,237,242,244 Krein, M.G.,242,280 Kronecker—Capelli theorem, 62 Kurosh, A.G.,335 k-vectors, 245 angles between. 245 equal, 245 scalar product of,246 Lagrange’s method, 276,280,285,309 Laplace’s theorem, 23 Lawofinertia, 205,207,251Leftideal,13sLeftinverse, 97,9s,104,137 Left unit, 137 Legendre, A.M.,228 Legendre polynomials, 228-230 Length ofavector, 217,255 Linear combination: ofcolumns, 10,24 ofvectors, 36 Linear dependence: ofcolumns, 27 ofvectors, 36 Linear family (ofoperators), 336 Linear form, 75 coelﬁcients of,76 transformation of,123-124 oftheﬁrstkind, 77 ofthesecond kind, 77 Linear functional, 76 Linear independence ofvectors, 36 over asubspace, 44 Linear manifold (spanned byspaces), 328 Linear manifold (spanned byvectors), 50 Linear operator (seeOperator) Linear space(s), 31ﬁ A-isomorphic, 199,254 basis for,38 cognate, 352 complex, 34 concrete, 34 dimension of,40 over asubspace, 45 direct sumof,45 inﬁnite-dimensional, 40 (K-)isomorphic, 53 n-dimensional, 40 real, 34 subspace of,42 tensor product of,349 Linear subspace (seeSubspace) Linear vector function (seeLinear operator) Matrices: block, 89 multiplication of,89 determinant ofaproduct of,103minors ofaproduct of,91 multiplication of,85 noncommutativity of,85-86 quasi-diagonal, 90 multiplication of,90 rank ofaproduct of,95 sum of,84 transposed, 90 multiplication of,90 Matrix, 5ﬁ adjoint of,258 adjugate of,116 augmented, 62 ofabilinear form, 181 block, 89 characteristic polynomial of,110 coelﬁcient, 18,62 determinant of,6 diagonal, 100 elements of,5 Hermitian conjugate of,258 Hermitian (—symmetric) ,248 identity, 81,99 invariant, 202 inverse, 105 leftinverse of,98 minor of,13,21,59 ofanilpotent operator, 136 nonsingular, 104,119 ofanoperator, 79,98 order of,5 orthogonal, 239 principal diagonal of,5 product of: with anumber, 84 with another matrix, 85 ofaquadratic form, 185 quasi-diagonal, 90rankof,25,59,60,67-71 right inverse of,98 singular, 104 symmetric, 181 trace of,115 transpose of,60,90 transposed, 60unit,s1,99 unitary, 259 Matrix algebra, 330 semisimple, 330 simple, 330McShane, E.J.,276 Mean square deviation, 235 Method ofleast squares, 235-236 Metric geometry, 214 Minor, 13ﬁbasis,25,59 bordered, 302 complementary, 21 oforder k,21 ofaproduct ofmatrices, 91 principal, 126 descending, 193 Monomorphism, 53 ofanalgebra, 139 INDEX Morphism, 53 Operator(s) (cont.): ofanalgebra, 138 kernel of,56 nullspace of,56 range of,55 Multilinear form, 203 antisymmetric, 203 symmetric, 203 Natural numbers, 2 Negative element, 1 Nemirovski, A.S.,316 Nilpotent operator, 133 matrix of,136 Noncentral surface, 289 canonical equation of,289 nondegenerate, 296-299 Nonnegative operator, 271 Norm, 217,255 Normal operator, 238,259 geometric meaning of,268-269 real,structure of,265-269 Normal series, 323 Null space, 56,94 Number ﬁeld (seeField) Operator(s), 53,75,77ﬂ acting inaspace, 98 addition of,82,84 adjoint of,198,238,254,258 annihilating polynomial of,143 minimal, 143 antiself-adjoint, 262 antisymmetric, 238 characteristic polynomial of,126 characteristic space of,110 conjugate of,198 determinant of,125 diagonal, 100 diagonalizable, 100 eigenspace of,110 eigenvalue of,108 eigenvector of,108 elementary divisor of,151 equality of,82 equivalent, 133,153 extension of,from areal toacomplex space, 264 Hermitian conjugate of,254 Hermitian-symmetric, 262 identity, 78,99 invariant, 201 inverse of,105 matrix of,105 invertible, 105 isogonal, 244 isometric, 239 Jordan canonical form of,146 leftinverse of,97,104 mapping aspace K"intoitself, 9811 matrix of,79,98 transformation of,124 multiplication of,82-83 negative of,78 nilpotent, 133nonexpanding, 272 nonnegative, 271 nonsingular, 105 normal, 238,259 real, 265 nullspace of,94 positive, 271 powers of,101 product of: with anumber, 82 withanother operator, 83 projection, 100 range of,93 rank of,93 right inverse of,97,104 rotation, 99 self-adjoint, 262 similarity, 99 spectrum of,169 sumof,82 symmetric, 238 tensor product of,350 trace of,126 unit, 78,99 unitary, 259, 263 zero, 78,98 Operator functions, 169-176 matrices of,171-176 Order: ofadeterminant, 6 ofamatrix, 5 Orthogonal basis, 222,257 Orthogonal complement, 220,257 Orthogonal direct sum, 223 Orthogonal matrix, 239 Orthogonal transformation, 239 Orthogonal vectors, 219,257 Orthogonality ofavector: toaset,220 toasubspace, 220 Orthogonalization theorem, 226 Orthonormal basis, 222, 258 Paraboioid, 296-299 circular, 297 elliptic, 297 hyperbolic, 298 Partially ordered set,339 Particular solution, 63 Perpendicular (dropped onto asub 223 footof,224 Planes (inalinear space), 53 Polynomial algebras, 137ﬁ Positive deﬁnite bilinear form, 208 Positive deﬁnite Hermitian form, 253 Positive deﬁnite quadratic form, 206 Positive operator, 271 Prepartially ordered set,339 Principal minor, 126 descending, 193 Producti ofjets,161 ofmatrices, 85385 space) 386 INDEX Product (c0nt.): ofnumbers, 1 ofoperators, 82-83 ofvectors withnumbers, 32 Projection (ofavector), 223 Projection operator, 100 Pythagorean theorem, 221, 257 Quadratic form(s), 179, 18311 canonical basis of,185 canonical coelﬁcients of,185 canonical form of,185 comparable, 310 inaEuclidean space, 27311 extremal properties of,276-283 Hermitian, 249 inaunitary space, 308-310 matrix of,185 nonsingular, 185 positive deﬁnite, 206 rank of,185,189 reduction of,tocanonical form, 185-189 simultaneous reduction oftwo,283-287 Quadric surface(s), 287-308 analysis of,from general equation, 300- 308 canonical equation of,287 central, 288,290 degenerate, 288,299 noncentral, 289 nondegenerate, 288 Quotient: ofelements ofanalgebra, 137 ofnumbers, 2 Radical (ofaualgebra), 317 Radical algebra, 317 Radius vector, 35 Range, 55,93 Rank: ofabilinear form, 182 ofaHermitian form, 249 ofamatrix, 25,59,60,67-71 ofanoperator, 93 ofaproduct ofmatrices, 95 ofaquadratic form, 185,189 Ratio ofsimilitude, 99 Rational numbers: inaﬁeld, 2 ﬁeldof,2 Real numbers, ﬁeldof,2 Reciprocal element, 2 Representation(s), 313ﬁ direct sumof,314 equivalent, 313 exact, 313 faithful, 313 invariant subspace of,313 minimal, 314 proper, 314 irreducible, 314 kernel of,313 leftregular, 314,318-320 restriction of,313 standard, 318Representation(s) (cont.): trivial, 313 Right ideal, 138 Right inverse, 97,98,104,137Rightunit,137 Rodrigues, J.M.,228 Rotation operator, 99 Scalar product, 214,215ﬂ complex, 25411 ofk-vectors, 246 Scalar quantity, 131 Schur’s lemma, 315 Schwarz inequality, 218, 256 Second-degree curve, 287 Second-degree surface (seeQuadric surface) Second structure theorem, 325 Self-adjoint operator, 262 Semiaxes, 291 Semisimple algebra, 316 representations of,327-330 structure of,323-327Shostak, R.Y.,210 Silverman, R.A.,282 Similarity operator, 99 Simple algebra, 315 representations of,327-330 structure of,320-322 Slope (ofsegment joining matrix elements): negative, 7 positive, 7 Solution space ofalinear system, 43 Space: C(a, b),35c,.,34K,,,34R(a,b),35 R,,,34 V1,34 V1,34 V,,,34 Spectrum, 160 multiplicity of,160 symmetric, 168 Spread ofsubspaces, 242 S-sets, 353 product of,356 Stationary value: ofafunction, 276 ofaquadratic form, 276 Straight lines (inalinear space), 53 Subalgebra, 138 Subspace(s), 4211 angles between, 244 comparable_elements of,48 conjugate, 190,252 direct sumof,45 orthogonal, 223 intersection of,42 invariant, 106,313 nontrivial, 42 orthogonal complement of,220,257 spread of,242 sumof,42 trivial, 42 Sum: ofjets,161 ofmatrices, 84 ofnumbers, 1 ofoperators, 82 ofvectors, 31 Summation convention, 126 Sylvester’s conditions, 253 Symmetric operator, 238 real, structure of,269 System oflinear equations, 3ﬁ augmented matrix of,62 coelﬁcient matrix of,18,62 coelﬁcients of,3 compatible, 4 nontrivially, 61 compatibility of,4,61 condition for,62 nontrivial, 61 constant terms of,4 determinate, 4 homogeneous, 43 incompatible, 4,234 indeterminate, 4 index of,212 solution(s) of,4 distinct, 4 fundamental system of,65 normal, 66 general, 63,66 product of,withanumber, 43 sumof,43 trivial, 43 solution space of,43 Taylor’s formula (forapolynomial), 163 Tensor(s), 126-131 addition of,130 contraction of,130 contravariant, 129 covariant, 129 invariants of,131 mixed, 130 multiplication of,130 order of,129 Tensor product: oflinear spaces, 349 ofoperators, 350 Trace, 115, 126, 131 Transpose: ofadeterminant, 9 ofamatrix, 60,90INDEX 387 Triangle inequalities, 221,257 Trivial solution, 43 Two-sided ideal, 138 Unit (two-sided), 137Unitball,217,256 Unit matrix, 81,99 Unit operator (seeIdentity operator) Unit sphere, 217,256 Unit vector, 217,256 Unitary matrix, 259 Unitary operator, 259,263 Unitary space, 254ﬂ Unitary transformation, 259 Vaudermoude determinant, 15 Vector(s), 31ﬁ angle between, 217 complex conjugate of,263 components of,39 transformation of,121 conjugate: toasubspace, 190,252 toanother vector, 190,252 cyclic, 314 diﬁerence of,34 height of,134 length of,217,255 linear combination of,36 coelﬁcients of,36 linearly dependent, 36 linearly independent, 36 norm of,217,255 normalization of,217,256 orthogonal: toasubspace, 220,257 toanother vector, 219,257 perpendicular dropped from theendof, 223 product of,with anumber, 32 projection of,onto asubspace, 223 purely imaginary, 263 real,263 sumof,31 unit, 217, 256 Wedderbum’s theorem, 332 Zero, 1 Zero column, 27 Zero operator, 78,98 Zero vector, 32 uniqueness of,32