Sum of squared minors theorem REVIEWED

Sum of squared minors theorem. 5/16/16. This is all Appendix G now F.6 Theorem: det(RTR) is the sum of the squares of the full-width minors of R We saw and verified an example of this theorem in Section 10.10 for a 3 x 2 R matrix, K2 = det(RTR) (10.10.22) K2 =   det2 + det2 + det2 . (10.10.18)' Before proving the theorem, we have Maple test it for a messy case, just to make sure it is true. Enter a generic 6x4 R matrix as follows: Compute and accumulate the squares of all (6,4) = 15 full-width minors, If we take the resulting "acc" and expand it, we get a series of 4,230 terms each of which contains a product of eight matrix elements of R, Here are four of these terms . We next compute det(RTR), note that it also has 4230 terms, and then we show that det(RTR) = acc. (F.6.1) Fortified with the knowledge that the theorem seems to be true, we proceed: Theorem: Let R be an m x n matrix with m ≥ n. There are (m,n) full-width minors. The claim is that det(RTR) = sum of the squares of the full-width minors. Comment: For m = n there is only one minor for square R which is det(R), the sum of the squares of the minors is then just det2(R), and indeed det(RTR) = det2(R). Proof for n < m : Define a multiindex I as follows I = i1, i2, .... in (F.6.2) where the ir indicate which n rows of the R matrix are included in a certain minor. Each ir takes values in the range 1 to m since R has m rows. Denote a full-width minor of R by minorI . (F.6.3) We shall need the following Lemma: Σ'I [minorI]2 = (1/n!) ΣI [minorI]2 (F.6.4) where Σ'I = Σ1≤i<i<....<i≤m = ordered sum ΣI = Σi,i,...i=1m = symmetric sum (F.6.5) Proof of Lemma: ΣI [minorI]2 = Σi,i,...i=1m [minorI]2 = Σi≠i≠..≠i [minorI]2 // minorI = 0 if two rows are the same = (Σi<i<....<i + n! - 1 other orderings) [minorI]2 = (ΣP [ΣP(i)<P(i)<...<P(i)]) [minorI]2 = Σi<i<...<i [ΣP fP(i)P(i)...P(i)] . // by (A.9.1) with fii...i = [minorI]2 But fii...i = [minorI]2 is a totally symmetric function of the indices since row swaps don't affect a squared determinant. Thus we continue the above to get = Σi<i<...<i [ΣP fii...i] = Σi<i<...<i fii...i [ΣP 1 ] = Σi<i<...<i fii...i [n! ] = Σi<i<...<i [minorI]2 [n! ] = n! Σ'I [minorI]2 QED Lemma Proof of Theorem: We can write minorI as the determinant of a matrix using (A.1.19) minorI = ΣP(-1)S(P) RiP(1) RiP(2) ....RiP(n) = Ri1 Ri2 ....Rin + all signed permutations ≡ ΣP(-1)S(P)RIP(Z) . // in multiindex notation, Z = 1,2....n (F.6.6) Then we first claim that Sum ≡ sum of all full-width squared minors = Σ'I [minorI]2 . (F.6.7) In this ordered sum, each full-width minor of R is included exactly once. For example, for a 3x2 R matrix we had above K2 =   det2 + det2 + det2 . (10.10.18)' i1,i2 = 1,2 i1,i2 = 1,3 i1,i2 = 2,3 Then using the above Lemma we write Sum = Σ'I [minorI]2 = (1/n!) ΣI [minorI]2 = (1/n!) ΣI [minorI] [minorI] = (1/n!) ΣI [ ΣP(-1)S(P) RiP(1) RiP(2 ...RiP(n)] [ ΣP'(-1)S(P') RiP'(1) RiP'(2)..RiP'(n)] = (1/n!) ΣI [ ΣP(-1)S(P)RIP(Z)] [ ΣP'(-1)S(P')RIP'(Z)] // multiindex notation = (1/n!) ΣP(-1)S(P) ΣP'(-1)S(P')ΣI RIP(Z)RIP'(Z) // reorder = (1/n!) ΣP(-1)S(P) ΣP'(-1)S(P')ΣI (RT)P(Z)IRIP'(Z) // matrix transposes = (1/n!) ΣP(-1)S(P) ΣP'(-1)S(P')(RTR)P(Z)P'(Z) // n matrix multiplications = (1/n!) ΣP(-1)S(P) ΣP'(-1)S(QP')(RTR)P(Z)QP'(Z) // ΣP' rearrangement theorem (A.1.3) = (1/n!) ΣP(-1)S(P) ΣP'(-1)S(PP')(RTR)P(Z)PP'(Z) // select Q = P = (1/n!) ΣP(-1)S(P) ΣP'(-1)S(PP')(RTR)ZP'(Z) // (A.8.31) since factored form = (1/n!) ΣP ΣP'(-1)S(P')(RTR)ZP'(Z) // (A.1.11) = (1/n!) ΣP'(-1)S(P')(RTR)ZP'(Z)[ΣP 1] = (1/n!) ΣP'(-1)S(P')(RTR)ZP'(Z)[n!] = ΣP'(-1)S(P')(RTR)ZP'(Z) = ΣP(-1)S(P)(RTR)ZP(Z) = det(RTR) // (A.1.19) QED Theorem (F.6.8) F.7 The Connection between Theorem F.6 and Differential Forms Recall from (10.11.7) that the integral of a k-form for F: Rn → Rm is given by αx' = Σ'I fI(x') dx'^I (F.7.1) ∫S' αx' = ∫S Σ'J Σ'I fI(F(x)) det(RIJ) dxj ^ dxj ^ ... ^ dxj = ∫S Σ'J Σ'I fI(F(x)) det(RIJ) dx^J . // R = (DF) (F.7.2) The context here is that x' = F(x) is a point on a manifold M created by the mapping F: Rn→ Rm as illustrated for example in (10.7.26) which we replicate here (10.7.26) A case of frequent interest is k = n, and in this case there is only one term in the Σ'J sum, so ∫S' αx' = ∫S Σ'I fI(F(x)) det(RIZ) dx1 ^ dx2 ^ ... ^ dxn = ∫S Σ'I fI(F(x)) det(RIZ) dx^Z . Z ≡ 1,2...n (F.7.3) Here the letter Z is our multiindex stand-in so that J = Z means j1 = 1, j2 = 2, ... jn = n. The object det(RIZ) is then in fact a full-width minor (a number) of the m x n R matrix, and in (F.6.3) we called this minorI so det(RIZ) = minorI ≡ mI . (F.7.4) where mI is a compact notation for minorI. Since Rij(x) is generally a function of x (or x' = F(x)), we may regard mI = mI(x'), so this minor value is a function of x' on manifold M. Then, suppressing the arguments fI and mI, we have ∫S' αx' = ∫S [ Σ'I fI mI ] dx^Z . (F.7.5) We may consider fI and mI to be vectors with (m,n) components which we can dot together to get, ∫S' αx' = ∫S [ f m ] dx^Z . // f m ≡ Σ'I fI mI . (F.7.6) For example, for n = 2, m = 3 we would write, showing components of each vector in "standard order", f m = (f12, f13, f23) (m12, m13, m23) = f12m12+ f13m13+ f23m23 = Σ'I fI mI (F.7.7) One could create a minor unit vector in this manner ≡ (F.7.8) where |m|2 = Σ'I (mI)2 = Σ'I (minorI)2 = det(RTR) // (F.6.8) ! (F.7.9) and then we have ∫S' αx' = ∫S [ f ] |m| dx^Z = ∫S [ f ] [ dx^Z] . (F.7.10) The objects in red above are all functionals in the space Λn(Rn). According to the "second definition" described in (10.1.3) we are always allowed to replace the functional dx^Z by dxZ to obtain a normal calculus integral that can be evaluated by normal methods, dx^Z ≡ dx1 ^ dx2 ^ ... ^ dxn → dxZ ≡ dx1dx2.....dxn . (F.7.11) Now consider the special case where the function f = . Then, ∫S' αx' = ∫S [ ] [ dx^Z] = ∫S dx^Z = ∫SdxZ . (F.7.12) which displays the tangent space volume measure dV' = dx1dx2.... dxn shown in (F.5.2). This last integral gives the volume (area) of the surface S' (example below) and for this reason we refer to the n-form α'x with this value of f as the volume measure form μ' . From (F.7.1) then, μ' = Σ'I fI dx'^I = f dx^' = dx^' . (F.7.13) Here we use the same dot product idea as in (F.7.5), where fI and dx'^I are each treated as vectors having (m,n) components. The pullback of this measure form is the integrand of (F.7.10), F*(μ') = dx^Z = dx1 ^ dx2 ^ ... ^ dxn // Sjamaar p 105 above item 8.12 (F.7.14) Consider now a different situation where f(x') = | f(x') | (F.7.15) so that our new function f, considered as a vector with those (m,n) components, points in the direction. In this case we get ∫S' αx' = ∫S [ f ] [ dx^Z] = ∫S | f((F(x))| dx^Z (F.7.16) and this is how one treats the integration of a scalar function over the surface S' such as the temperature function in (10.10.20). In this case we can write the corresponding n-form in x'-space as αx' = Σ'I fI dx'^I = f dx^' = | f | dx^' = (f ) dx^' = (f ) μ' (F.7.17) from which we extract this equation f dx^' = (f ) μ' . (F.7.18) Special case: Hypersurface where n = m - 1 Suppose now that n = m-1, so that the manifold M embedded in Rm is a hypersurface, meaning it has one dimension less than Rm. In this case, (m,n) = (m,m-1) = m and each of our "vectors" above has exactly m components. One can define the following Hodge star objects as discussed at the start of Section 10.3, *dx'i = (-1)i+1 dx'1 ^ dx'2 .. [dx'i]... ^ dx'm where dx'i is missing (F.7.19) so that each *dx'i object is an (m-1)-form (that is, an n-form). These m Hodge dual objects can be combined to form a vector *dx' ≡ (*dx'1, *dx'2, ....*dx'm) . (F.7.20) We now define new vectors F and n' as follows, Fi ≡ (-1)i+1f12.[i]..m n'i ≡ (-1)i+1m12.[i]..m (F.7.21) where the notation [i] means that index i is missing from 12...m. It follows that F *dx' = f dx^' (F.7.22) n' *dx' = m dx^' (F.7.23) n' F = m f . (F.7.24) The proofs of the above three lines are basically the same, so we prove just the first line : F *dx = Σi=1m Fi (*dx'i ) = Σi=1m (-1)i+1f12.[i]..m (-1)i+1 dx'1 ^ dx'2 .. [dx'i]... ^ dx'm = Σi=1m f12.[i]..m dx'1 ^ dx'2 .. [dx'i]... ^ dx'm = [ f234..m dx'2 ^ dx'3 ^ dx'4 ..... ^ dx'm + f134..m dx'1 ^ dx'3 ^ dx'4 ..... ^ dx'm + .... ] = Σ'IfI dx'^I with series terms reordered from standard order = f dx^' . (F.7.25) Because vector n is a reordering of vector m where certain terms have minus signs, the sum of the squares of the components of the two vectors is the same, so |n'| = |m| . (F.7.26) Then dividing (F.7.23) and (F.7.24) by |n| one finds ' *dx = dx^' . (F.7.27) ' F = f (F.7.28) Recall from above that μ' = dx^' (F.7.13) f dx^' = (f ) μ' . (F.7.18) Using (F.7.27), (F.7.22) and (F.7.28) these may be written μ' = ' *dx' // Sjamaar p 109 item 8.17 (F.7.29) F *dx' = (F ') μ' // Sjamaar p 107 item 8.16 (F.7.30) Sjamaar's equations are written in the x = φ(t) context rather than x' = F(x) so have *dx, and μ. Example: F : R2→ R3 , k = n = 2 , m = 3. (F.7.31) This example was first treated in Section 10.10 as a no-differential-forms problem, and was then reconsidered as a 2-form problem in Section 10.13 (but there in the x = φ(t) context). Here we write out various objects defined above and show how equations come out with equation number references in italics. αx' = Σ'I fI(x') dx'^I = f12 dx'1 ^ dx'2 + f13 dx'1 ^ dx'3 + f23 dx'2 ^ dx'3 // a 2-form = f dx^' = F *dx' = F dA' = (F ') dA' // see below (F.7.1) f = (f12, f13, f23) (F.7.7) m12 = minor12 = det(R1212) = det // R is a 3 x 2 matrix with 3 full-width minors m13 = minor13 = det(R1312) = det m23 = minor23 = det(R2312) = det (F.7.4) m = (m12, m13, m23) = ( det ), det , det ) (F.7.7) F = (f23, -f13,f12) // agrees with (10.13.16) (F.7.21) n' = (m23, -m13,m12) = ( det ), - det , det ) (F.7.21) = ( det ), + det , det ) = agrees with (10.10.19) where n' is shown as normal to the surface | m |2 = | n' |2 = det2 + det2 + det (F.7.26) ≡ K2 = det(RTR) // agrees with (10.10.18) and (10.10.22) (F.7.9) dx^' = (dx'1 ^ dx'2, dx'1 ^ dx'3, dx'2 ^ dx'3 ) // standard order *dx' ≡ (*dx'1, *dx'2,*dx'3 ) = (dx'2 ^ dx'3 , - dx'1 ^ dx'3, dx'1 ^ dx'2 ) ≡ dA' (10.13.23) f dx^' = (f12, f13, f23) (dx'1 ^ dx'2, dx'1 ^ dx'3 , dx'2 ^ dx'3 ) = f12 dx'1 ^ dx'2 + f13 dx'1 ^ dx'3 + f23 dx'2 ^ dx'3 = Σ'I fI dx'^I F *dx = (f23, -f13,f12) (dx'2 ^ dx'3 , - dx'1 ^ dx'3, dx'1 ^ dx'2) = f23 dx'2 ^ dx'3 + f13 dx'1 ^ dx'3 + f12 dx'1 ^ dx'2 = Σ'I fI dx'^I (F.7.18) n F = (m23, -m13,m12) (f23, -f13,f12) = m23f23 + m13f13 + m12f12 = m12f12 + m13f13 + m23f23 = (m12, m13, m23) (f12, f13, f23) = m f (F.7.24) μ' = ' *dx' = ' dA' = dA' // area measure on surface (F.7.29) F dA' = (F ') dA' = (F ') μ' // this is αx (F.7.30)