MATRICES AND VECTOR SPACES
We also showed that both Hermitian and unitary matrices (or symmetric and
orthogonal matrices in the real case) are examples of normal matrices. We now
discuss the properties of the eigenvectors and eigenvalues of a normal matrix.
Ifxis an eigenvector of a normal matrixAwith corresponding eigenvalueλthenAx=λx, or equivalently,
(A−λI)x= 0. (8.69)DenotingB=A−λI, (8.69) becomesBx= 0 and, taking the Hermitian conjugate,
we also have
(Bx)†=x†B†= 0. (8.70)From (8.69) and (8.70) we then have
x†B†Bx= 0. (8.71)However, the productB†Bis given by
B†B=(A−λI)†(A−λI)=(A†−λ∗I)(A−λI)=A†A−λ∗A−λA†+λλ∗.Now sinceAis normal,AA†=A†Aand so
B†B=AA†−λ∗A−λA†+λλ∗=(A−λI)(A−λI)†=BB†,and henceBis also normal. From (8.71) we then find
x†B†Bx=x†BB†x=(B†x)†B†x= 0 ,from which we obtain
B†x=(A†−λ∗I)x= 0.Therefore, for a normal matrixA,the eigenvalues ofA†are the complex conjugates
of the eigenvalues ofA.
Let us now consider two eigenvectorsxiandxjof a normal matrixAcorre-sponding to twodifferenteigenvaluesλiandλj. We then have
Axi=λixi, (8.72)Axj=λjxj. (8.73)Multiplying (8.73) on the left by (xi)†we obtain
(xi)†Axj=λj(xi)†xj. (8.74)However, on the LHS of (8.74) we have
(xi)†A=(A†xi)†=(λ∗ixi)†=λi(xi)†, (8.75)where we have used (8.40) and the property just proved for a normal matrix to