# Orthonormalization of non-hermitian matrix eigenvectors

I have been told that orthonormalization of the eigenvectors of a non-hermitian matrix has to use a different definition of inner product than when the matrix is hermitian. Why is this so, and how do I orthonormalize non-hermitian matrix eigenvectors?

I think it has to do with the fact that there are complex exponentials in my matrix, and that by taking their complex conjugate, I don't satisfy hermitian-ness.

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The eigenvectors of a hermitian matrix for different eigenvalues are orthogonal; for a non-hermitian matrix (actually, for a non-normal matrix) they are in general not orthogonal. Moreover, a non-hermitian $n \times n$ matrix might have fewer than $n$ linearly independent eigenvectors.
If you do have $n$ linearly independent eigenvectors, you can apply the Gram-Schmidt process to get an orthonormal set of vectors (however, they may not be eigenvectors).
Not trivial, but not too bad: on the order of $n^3$ floating-point operations, comparable to a matrix multiplication by the standard algorithm. –  Robert Israel Apr 16 '12 at 21:23