Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACESWe also showed that both Hermitian and unitary matrices (or symmetric andorthogonal matrices in the real case) are examples of normal matrices. We nowdiscuss the properties of the eigenvectors and eigenvalues of a normal matrix.If x is an eigenvector of a normal matrix A with corresponding eigenvalue λthen Ax = λx, or equivalently,(A − λI)x = 0. (8.69)Denoting B = A−λI, (8.69) becomes Bx = 0 and, taking the Hermitian conjugate,we also haveFrom (8.69) and (8.70) we then haveHowever, the product B † B is given by(Bx) † = x † B † = 0. (8.70)x † B † Bx = 0. (8.71)B † B =(A − λI) † (A − λI) =(A † − λ ∗ I)(A − λI) =A † A − λ ∗ A − λA † + λλ ∗ .Now since A is normal, AA † = A † A and soB † B = AA † − λ ∗ A − λA † + λλ ∗ =(A − λI)(A − λI) † = BB † ,and hence B is also normal. From (8.71) we then findx † B † Bx = x † BB † x =(B † x) † B † x = 0,from which we obtainB † x =(A † − λ ∗ I)x = 0.Therefore, for a normal matrix A, the eigenvalues of A † are the complex conjugatesof the eigenvalues of A.Let us now consider two eigenvectors x i and x j of a normal matrix A correspondingto two different eigenvalues λ i and λ j . We then haveMultiplying (8.73) on the left by (x i ) † we obtainHowever, on the LHS of (8.74) we haveAx i = λ i x i , (8.72)Ax j = λ j x j . (8.73)(x i ) † Ax j = λ j (x i ) † x j . (8.74)(x i ) † A =(A † x i ) † =(λ ∗ i x i ) † = λ i (x i ) † , (8.75)where we have used (8.40) and the property just proved for a normal matrix to274