Why matlab cannot return orthogonal eigenvectors for a normal matrix?

Now I have a normal matrix

A = [0 1 0 1 0 0 0 0 0;
     0 0 1 0 1 0 0 0 0;
     1 0 0 0 0 1 0 0 0;
     0 0 0 0 1 0 1 0 0;
     0 0 0 0 0 1 0 1 0;
     0 0 0 1 0 0 0 0 1;
     1 0 0 0 0 0 0 1 0;
     0 1 0 0 0 0 0 0 1;
     0 0 1 0 0 0 1 0 0];

It should have orthogonal eigenvectors, however, matlab eig function cannot return the orthogonal ones. Is it the problem with the algorithm?

Answer 1

[David Goodmanson]

Hello Ran, If a matrix A is normal it can be diagonalized with a unitary matrix, A = U D U', but no one is saying that the set of eigenvectors of A necessarily provides the matrix U. Lot of times it doesn't.

[Ran Yang]

Hi David, Thanks a lot for your valuable answer. I agree with you since eigenvectors are not unique.