CODEWITHSUNDEEP
numpyscipylinear-algebraleast-squares
Ben
Ben

Reputation: 377

Solve (underdetermined) linear system with commutator form in Python

Consider three (complex) square matrices A, X, and B, that form a generally underdetermined linear system

A @ X - X @ A == B

The system is to be solved for X. I would like to use, e.g., the lstsq functions of numpy or scipy to find a particular solution for X.

Note that in my case, A and B are Hermitian, and thus the solution X is necessarily anti-Hermitian.

It is of course always possible to use a custom least squares minimization, but is there a way to use existing linear algebra solvers to solve this directly?

Upvotes: 1

Views: 77

Answers (1)

Ben
Ben

Reputation: 377

Thanks to @Reinderien in the comments for the solution idea. Scipy has a solve_sylvester function specifically for this problem, namely,

import scipy

X = scipy.linalg.solve_sylvester(A, -A, B)

This could use optimizations for my case, but it will work well enough. Unfortunately, the function does not seem to support batching.

EDIT AND WARNING: As ThomasIsCoding points out in the comments, there might be issues with using the Bartels-Stewart algorithm (on which solve_sylvester is based) in my particular case. I checked that it worked in a very simple case. But then, in a very complicated case, it gave me nonsense. So tread carefully. I don't have time to investigate further at the moment.

Upvotes: 2

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