# Orthogonal basis of a hermitian matrix

Is there a way to extract an orthogonal basis for a hermitian matrix in Mathematica?

A simple diagnolization returns a non orthogonal basis and later using a Grahm-Schmidt process which isn't aware of the original problem, doesn't guarantee them remaining eigenvectors. Or am I missing something?

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Are you sure that your matrix is Hermitian? What is the result of Max@Abs[M-ConjugateTranspose[M]] where M is your matrix? It should be zero. –  ybeltukov Sep 14 at 21:37
Yea both that and the hermitian matrix q say that it is. –  user9522 Sep 14 at 21:43
Both? Do you solve the generalized eigenvalue problem? –  ybeltukov Sep 14 at 21:45
I see! Could you provide a minimum example? –  ybeltukov Sep 14 at 21:50
Without the code used to find the basis, it's hard to guess how to improve it. It may be a numerical issue. Have you tried Eigensystem and JordanDecomposition? -- they seem to return orthonormal bases when the matrix is Hermitian. You could orthogonalize the bases for each eigenspace separately. –  Michael E2 Sep 15 at 15:08