I am looking for the eigenvectors & eigenvalues of a non-symmetric, complex-valued matrix. The matrix is a Hermitian matrix with some diagonal imaginary terms added, which is then non-hermitian. The obtained eigenvalues from the subroutine zgeev are precise, but, the eigenvectors are not, when I compare the results using zgeev with the results obtained from Mathematica. I think the problem is that it is not guaranteed that, the obtained eigenvectors from zgeev are orthorgonal, though, I have no idea why that the results obtained from Mathematica are perfect, ie. The eigenvectors are orthornomal and complete. Should I impose further procedure after using zgeev, to make sure the eigenvectors are orthonormal?
Thanks in advance for any reply&comments.