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A symmetric matrix has orthonormal eigenvectors and real eigenvalues. However, there are many kinds of matrices that have orthonormal eigenvectors and complex eigenvalues (for instance, circulant matrices). What are the necessary conditions for a matrix to have orthonormal eigenvectors?

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  • $\begingroup$ The product of a matrix with its Hermitian transpose will have orthonormal eigenvectors. SVD ... $\endgroup$ Commented Feb 7, 2014 at 23:00

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The necessary and sufficient condition for a matrix to have A COMPLETE SET of orthonormal eigenvectors is that it be normal. A matrix $M$ is normal if and only if $$ MM^*=M^*M$$

For real valued matrices, symmetric and skew symmetric are some examples. Orthogonal matrices (thank you littleO for the correction) are another example.

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  • $\begingroup$ @littleO Those are normal matrices. Oh you mean they are not symmetric... $\endgroup$
    – adam W
    Commented Feb 8, 2014 at 0:17
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    $\begingroup$ @littleO You are correct, those are also examples. I thought I was covering all the real but was wrong. $\endgroup$
    – adam W
    Commented Feb 8, 2014 at 0:23

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