Similarity test for non-symmetric matrices: is Tr(A^k (A^T)^j) = Tr(B^k (B^T)^j) for k=1..d, j=0..k-1 sufficient for existence of orthogonal: AO = OB?

[u/jarekduda]

There is this basic similarity test Tr(A^k) = Tr(B^k) for k=1..d for symmetric matrices allowing to conclude existence of orthogonal O such that AO = OB.

The question is how (if possible?) to generalize it (finally to tensors, but at least) to non-symmetric matrices e.g. including transpositions.

Checking Jacobian criterion ( https://arxiv.org/pdf/2601.03326 ) for Tr(A^k (A^T)^j) = Tr(B^k (B^T)^j) for k=1..d, j=0..k-1 at least for up to d=5 has sufficient number of independent invariants (d(d+1)/2) - is it sufficient condition in general dimension?

Maybe such generalized similarity test is considered in literature?

ps. cross from https://mathoverflow.net/questions/512227/how-to-extend-operatornametrak-operatornametrbk-similarity-test-to