chapter6

Chapter 6 Normal Matrices Normal matrices are matrices that include Hermitian matrices and enjoy several of the same properties as Hermitian matrices. Indeed, while we proved that Hermitian matrices are uni tarily diagonalizable, we did not establish any converse. That is, if a matrix is unitarily diagonalizable, thendoes it have any special property involving for example its spectrum or itsadjoint? As we shall see normal matrices are unitarily diagonalizable. 6.1 Introduction to Normal matrices Definition 6.1.1. Am a t r i x A∈Mnis called normal ifA∗A=AA∗. Proposition 6.1.1. A∈Mnis normal if and only if every matrix unitarily equivalent to Ais normal. Proof. Suppose Ais normal and B=U∗AU,w h e r e Uis unitary. Then B∗B=U∗A∗AU=U∗AA∗U=U∗AUU∗A∗U=BB∗.I fU∗AUis normal then it is easy to see that U∗AA∗U=U∗A∗AU. Multiply this equation on the right by U∗a n do nt h el e f tb y Uto obtain AA∗=A∗A. Examples. (1) Unitary matrices are normal ( U∗U=I=UU∗). (2) Hermitian matrices are normal ( AA∗=A2=A∗A). (3) If A∗=−A,w eh a v e A∗A=AA∗=−A2. Hence matrices for which A∗=−A,c a l l e d skew-Hermitian , are normal. 197 198 CHAPTER 6. NORMAL MATRICES Example 6.1.1. Consider the arbitrary matrix N∈M2(R),written as N=·ab cd¸ . If we suppose that Nis normal then N∗N=·ab cd¸T·ab cd¸ =·a2+c2ab+cd ab+cd b2+d2¸ NN∗=·ab cd¸·ab cd¸T =·a2+b2ac+bd ac+bd c2+d2¸ From this we conclude that b2=c2,o rb=±c. Consider the cases in turn. (i) If c=b,t h e n Nis Hermitian and thus normal. (ii) If c=−b6=0,then ( N∗N)12=ab+cd=b(a−d). On the other hand ( NN∗)12=ac+bd=(d−a)b.F o r b(a−d)=( d−a)b,w em u s t have a=d.This gives that real 2 ×2 normal matrices are either symmetric or have the form N=·ab −ba¸ Note this form includes both rotations and skew-symmetric matrices. Recall the de finition of a unitarily diagonalizable matrix: A matrix A∈Mn is called unitarily diagonalizable if there is a unitary matrix Ufor which U∗AU is diagonal. A simple consequence of this is that if U∗AU =D (where D= diagonal and U= unitary), then AU=UD and hence Ahasnorthonormal eigenvectors. This is just a part of the spectral theorem for normal matrices . Theorem 6.1.1 (Spectral theorem for normal matrices). IfA∈Mn has eigenvalues λ1...λn, counted according to multiplicity, the following statements are equivalent. (a)Ais normal. (b)Ais unitarily diagonalizable. (c)Pn i=1Pn j=1|aij|2=Pn j=1|λj|2. (d) There is an orthonormal set of neigenvectors of A. 6.1. INTRODUCTION TO NORMAL MATRICES 199 Proof. (a)⇒(b). If Ais normal, then AA∗is Hermitian and therefore unitarily diagonalizable. Thus U∗A∗AU=D=U∗AA∗U.A l s o , A,A∗, A∗A=AA∗form a commuting family. This implies that eigenvectors of A∗Aare also eigenvectors of A.S i n c e A∗Ahas a complete orthonormal set we know that U∗AUis also diagonal. It is easy to see also that (b) ⇒(a) We also note that (a) ⇒(d). (b)⇒(c). Suppose U∗AU=D.T h e n U∗A∗U=D∗andU∗A∗AU=D∗D. By Corollary 3.5.2 similarly preserves the trace. We know trace of A∗A is tr ( A∗A)=Pn j=1Pn k=1a∗ jkakj=Pn j=1Pn k=1¯akjakj=Pn j=1Pn k=1|akj|2. Since the trace of D∗DisΣ|λj|2, the result follows. (c)⇒(b). We know that Ais unitarily equivalent to a upper triangular matrix T. We also know that if A∼Bare unitarily equivalentP ij|aij|2= P ij|bij|2. Application of this equality to the upper triangular matrix T yields X i,j|aij|2=X |λj|2+X j>i|tij|2=X |λj|2. Thus tij=0f o r j>i .T h u s Ais uniformly diagonalizable. (d)⇒(b). Trivial. Corollary 6.1.1. LetA∈MnandAis normal. If Uis unitary and if U∗AU is upper triangular then U∗AU is diagonal. Theorem 6.1.2. LetN∈Mn(R).T h e n Nis normal if and only if there is a real orthogonal matrix Q∈Mn(R)such that QTNQ= A 1 A2 ° ... ° An (1) where A iis1×1(real) or Aiis2×2(real) of the form Ai=·αiβj −βjαi¸ . Proof. First of all, any matrix Aof the form given by (1) is normal, and therefore so also is any matrix unitarily similar (real orthogonally similar in this case) to it. 200 CHAPTER 6. NORMAL MATRICES To prove the converse we assume that N∈Mn(R)i sn o r m a l .W ek n o w thatNis unitarily diagonalizable. That is, there is a unitary matrix Usuch thatU∗NU=D, the diagonal matrix of its eigenvalues. Because Nis real, all complex eigenvalues occur in compl ex conjugate pairs. Arrange them as successive diagonal entries in D.I fλis a real eigenvalue, we can assume without loss of generality that the corresponding eigenvector is real. For complex eigenvalues, the corresponding eigenvectors also occur in conjugatepairs. Thus if α+iβis an eigenvector of Nwith corresponding eigenvector written in real and complex parts u=u r+ius.S i n c e Nis real we have thatα−iβis also an eigenvector of Nwith corresponding eigenvector ¯ u= ur−ius.B y t h e f a c t t h a t Nis unitarily diagonalible, these vectors are orthogonal. This means hur,usi=0. Replace the eigenvectors ur±iusby the real an imaginary parts in U. This gives the matrix Q. Now compute QTNQ. It is easy to see that com- puteNur=αur−βvsandNus=αus+βvr. When the first of these vectors (αur−βvs) is multiplied by QTwe obtain the vector [0 ,...,α,−β,0,...0]T. Multiplication by the second gives the vector [0 ,...,β,α,0,...0]T.I n t h i s way the components·αβ −βα¸ arise. Corollary 6.1.2. (a)A∈Mnis symmetric if and only if (1) holds with all blocks 1×1(and real). (b)AAT=Iif and only if (1) has the form  λ1 ... λp ∗ A1 ∗... Ak  whereλ j=±1andAj=hcosθj−sinθj sinθjcosθji θj∈R. 6.2 Exercises 1. If AandBcommute and if Ais normal, then A∗andBcommute.