Solving generalized A*V = B*V*D with A and B being Hermitian gives COMPLEX eigenvalues in Matlab

0
votes

In Matlab, I have two 142 x 142 Hermitian complex matrices A and B, which are verified by

isequal(A,A')
ans =
       1
isequal(B,B')
ans =
       1

and

A-A'=(all exact zeros)
B-B'=(all exact zeros)

I tried to solve the generalized eigenvalue problem AV = BV*D by

[V,D] = eig(A, B);

and I expected for two things:

  1. Real eigenvalues, V

  2. Eigenvectors are orthogonal. That is, V'BV is an identity matrix.

However, I got COMPLEX eigenvalues and V'BV has nonzero value everywhere. Besides, things didn't change if I use the qz algorithm. Where can possibly go wrong? I have got stuck in this bottleneck for weeks. Any help is highly appreciated!

1
matlabeigenvalue
What is the value of the imaginary components? If it's small (1E-15 or lower depending), just get rid of them. – TroyHaskin
@TroyHaskin Yes, it's about that magnitude. However, I need correct orthogonal eigenvectors for further analysis. That's why I can't just throw the imaginary parts. – Jen-Hao Ou
If you can come up with some (random) example of A and B then I can fully test my code and post an answer. Anyway, I agree with @TroyHaskin that the imaginary part is a small number which comes from the computation of eig. Try eps(max(abs(A(:))) to see to what order of magnitude will the result be in exact, aka below which number you can safely declare they are merely errors and just throw them away. – Yvon
Agreed with @Yvon. To proceed further we'll need an MCVE. Although, I'll say again, if the non-zeros from the orthogonality test are "small", there might be nothing you can do. – TroyHaskin

1 Answers

0
votes

from matlab document:

[V,D] = eig(A,B) returns diagonal matrix D of generalized eigenvalues and full matrix V whose columns are the corresponding right eigenvectors, so that AV = BV*D.

use following command for more accurate results : 

 [V,D]=eig(A,B,'qz')

read the following document carefully about situation where eig function does not return accurate results : Matlab Doc