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How can I possibly characterize the set of matrices such that $A^T A = A A^T$?

I know that if I have $A^T A = A A^T = I_n$ then it is the set of orthogonal matrices ($A^{-1}=A^T$). But now how can I describe such a set? Thanks in advance.

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    $\begingroup$ Assuming that the matrices in question have real entries, $A^TA=AA^T$ iff $A$ is normal. $\endgroup$ – Math1000 May 25 '18 at 15:03
  • $\begingroup$ Why not just describe them by $A^TA=A^TA$? Do you want to know properties of such matrices, i.e., normal matrices? What do you really mean by "description"? The question is not particularly clear. $\endgroup$ – Dietrich Burde May 25 '18 at 15:04
  • $\begingroup$ They are precisely the real matrix that are unitarily equivalent to a diagonal (possibly complex) matrix. math.stackexchange.com/questions/2843109/… $\endgroup$ – Bob Jul 22 '18 at 18:25
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These are the so-called normal matrices (assuming that we are talking about real matrices here).

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