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 June 14Jun 14 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

Update: working on proof of such similarity test for general matrices, it is convening to use Schur decomposition rotating A and B to upper-diagonal (can be complex), can be chosen with same diagonals thanks to tested Tr(A^k)=Tr(B^k). Then seems we should use induction d\to d+1 from d x d matrix A, vector v, scalar a:

From their equality for A and B, and right hand side using powers lower by 1, we should conclude Tr(A^k (A^T)^l) = Tr(B^k (B^T)^l)$ and equality of vectors ...

Edited June 15Jun 15 by Duda Jarek

On 6/14/2026 at 1:19 PM, studiot said:

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

Is that happens when you enter an MRI suite wearing a 10kg metal chain round your neck? https://www.bbc.com/news/articles/cx2n39dvp0po

53 minutes ago, exchemist said:

Is that happens when you enter an MRI suite wearing a 10kg metal chain round your neck? https://www.bbc.com/news/articles/cx2n39dvp0po

Funny you should mention this as I'm taking my wife for an MRI scan tomorrow.

+1

MRI is, of course, a fusion of two very modern technologies as I'm sure you are familiar with NMR in Chemistry and we have the Maths of finite patch models.

Duda's models are not meshes but aggregated solid blobs (bodies).

43 minutes ago, studiot said:

Funny you should mention this as I'm taking my wife for an MRI scan tomorrow.

+1

MRI is, of course, a fusion of two very modern technologies as I'm sure you are familiar with NMR in Chemistry and we have the Maths of finite patch models.

Duda's models are not meshes but aggregated solid blobs (bodies).

I had one a couple of weeks go. I hope your wife has the patience for a long, boring and noisy procedure. The only point of interest for me was the contrast agent, a gadolinium complex.

My basic approach is to represent entire shape as Gaussian times polynomial, then find rotation invariants of this polynomial - as features for chemoinformatics, or vector to compare for shape similarity metric.

Interesting mathematics to speedup MRI: https://en.wikipedia.org/wiki/Compressed_sensing