Eigenvalues of hermitian matrix

[fanxiu]

1. Let A^H be the hermitian matrix of matrix A, and how the eigenvalues of A^H be related to eigenvalues of A?



3. what I have done is

equation no.1: (A^H-r₁*I) * x₁ = 0,
And equation no.2: (A-r₂*I) * x₂ = 0

time no.1 both sides by x₂^H
((A*x₂)^H-r₁*x₂^H)* x₁ = 0

Then we have (conjugate (r₂) -r₁)*x₂^H*x₁ = 0

only if x₁ and x₂^H are not orthogonal vectors,
conjugate (r₂) = r₁

so i just reach this conditional conclusion. but i am not sure what else can be know about the eigenvalues of hermitian matrix

 

[kurt.physics]

AH. The eigenvalues of a Hermitian matrix AH are related to the eigenvalues of A in the following way: if x1 and x2 are eigenvectors of A corresponding to eigenvalues r1 and r2 respectively, then x2H*x1 = 0 if and only if conjugate(r2) = r1. This means that the eigenvalues of AH are either the same as the eigenvalues of A or their complex conjugates, depending on whether the corresponding eigenvectors are orthogonal or not.