Generalized eigenvectors not orthogonal

I use eig to solve a generalized eigenvalues problem from two symmetric real matrices and resulting eigenvalues are not orthogonal even though there is no degeneration in the eigenvalues. Minimal code to reproduce this:

A=randn(10); B=randn(10);
A=A+A'; B=B+B';
[V,D]=eig(A,B);
diag(D)
V(:,1:6)'*V(:,1:6)

What do I miss?

[Matt J]

I'm not aware of any result saying they should be orthogonal. The material here

http://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix#Generalized_eigenvalue_problem

mentions they will be B-orthogonal, but only if B is positive definite.

Answer 1

[MA]

They are orthogonal, what is the problem?

clear all
close all
clc;
A=randn(10);
B=randn(10);
AA=A+A';
BB=B+B';
[V,D]=eig(AA);
[VV,DD]=eig(BB);
diag(D);
diag(DD);
V(:,1:10)'*V(:,1:10)
VV(:,1:10)'*VV(:,1:10)

[MA]

in your case must be x=y:

clear all
clc;
A=randn(10);
B=randn(10);
AA=A+A';
BB=B+B';
[V,D]=eig(AA,BB);
%x=y
x=AA*V
y=BB*V*D

[Uri Cohen]

The eigenvectors are orthogonal, while the generalized eigenvectors are not, also in your example...

    A=randn(10); AA=A+A';
    B=randn(10); BB=B+B';    
    [V,D]=eig(AA);
    V*V' % eye(10)
    [V,D]=eig(AA, BB);
    V*V' % not eye(10)