# Equivalence classes of unitary matrices under “transpose conjugation”

My question seems like a basic linear algebra problem but I wasn't able to find a solution online nor to come up with one in a short period of time, so here it goes:

Q: What are the equivalence classes of $$\mathrm{SU}(n)$$ under the equivalence relation $$A \sim M A M^T$$ for some $$M \in \mathrm{SU}(n)$$?

This feels like an easy question, yet I am happy to receive some sub-complete answers: What are the general procedures to find all equivalence classes? Are there finitely many, or a "nice" parametrization (similar to eigenvalues/maximal tori for the usual conjugation), etc.

Specifically, $$A,B \in \mathrm{SU}(n)$$ are unitary congruent (i.e. there is an $$M \in \mathrm{SU}(n)$$ such that $$A = M B M^T$$) if and only if $$\mathrm{Tr}( (A \overline{A})^k) = \mathrm{Tr}( (B \overline{B})^k)$$ for all $$k=1,\dots, n^2 -1$$ (apparently a lower bound there also works but this one seems more intuitive).
• As it is stated it seems that you are considering matrices up to base change given by the unitary matrix formed of the rows or columns of $M$. Thus, $A\sim B$ if and only if they represent the same linear map. Or am I missing something here? – user526015 Feb 26 at 11:10
• @James I guess that is correct, in a way, but I was looking for a more specific criterion. For example, the identity matrix is equivalent to all matrices of the form $M M^T$, i.e. all symmetric matrices and in particular all diagonal matrices, but I personally wouldn't say that all diagonal matrices represent the same linear map (that I would usually refer to through the relation $A \sim M A M^{-1}$). Hope that makes sense. – Sebastian Schulz Feb 26 at 13:30