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.

Practical 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 for Tr(A^k (A^T)^l) = Tr(B^k (B^T)^l) for k=1..d, l=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? If not, how to extend it?

Used Mathematica code using Jacobian criterion to find the number of independent invariants, assuming upper-diagonal as in Schur decomposition, getting d(d+1)/2 as required up to d=5:

d = 5; M = Table[If[i > j, 0, Subscript[a, Row[{i, j}]]], {i, d}, {j, d}];
inv = Table[Tr[MatrixPower[M, k].MatrixPower[Transpose[M], l] , {k, d}, {l, 0, k - 1}];
MatrixRank[jac = Table[D[Catenate[inv], v], {v, Variables[inv]}]]

Motivations ( https://arxiv.org/pdf/2601.03326 ), especially if reaching also for tensors, is complete shape description up to rotation e.g. for chemoinformatics, medical imaging:

Edited 10 hours ago10 hr by Duda Jarek

I reckon this must be worth +1 just to introduced to Gaussian splatting.

Yes, one direction here is considering more sophisticated primitives than in gaussian splitting, e.g. multiplied by polynomial.

But direct question is about better rotation invariants than e.g. spherical harmonics - offering only rough description modulo rotation, while here should be complete.

Studying your article will need some heavy lifting, but here is an extract.

Keywords: machine learning, feature extraction, rotation in-

variants, shape descriptors, multivariate polynomials, tensors,

shape similarity metric, medical imaging, image recognition,

chemoinformatics, 3D scene understanding, Gaussian splatting

This one is quite far future work, but maybe will move forward with interns from https://www.qaif.org/events/aintern/aintern-2026