Orthogonal transformation between commuting matrices

[blegat]

In SumOfSquares, we have pairs of commuting matrices A, B and need to compute an orthogonal transformation X such that X * A = B * X. This is implemented in orthogonal_transformation_to. It basically calls schur(A) and schur(B) but then it's a bit subtle since the schur decomposition is not unique so we need to canonicalize in a similar way for both. It looks a bit like some of the equations of this package so I was wondering if it could fit into one of them.

[andreasvarga]

Could you provide a link to orthogonal_transformation_to? What mathematical condition guarantees the existence of an orthogonal solution?

[blegat]

It's there
https://github.com/jump-dev/SumOfSquares.jl/blob/6ad333b54524aa15ca47be20572bf9d82bfa56c4/src/Certificate/Symmetry/block_diag.jl#L3-L179
I'm not sure what is the condition actually. I just assume that there must exist one

[andreasvarga]

The fact that A and B are commuting (i.e., AB = BA) does guarantee the existence of an orthogonal solution? I was not able to find such a result in the literature.

[blegat]

Thanks for the literature search. Indeed, I don't think the commuting property is the right one. Actually there are pairs of matrices that do not commute for which I can find such orthogonal matrix.