When the Tracy-Singh product of matrices represents a certain operation on linear operators

Given two linear transformations, with representing matrices A and B with respect to some bases, it is not clear, in general, whether the Tracy-Singh product of the matrices A and B corresponds to a particular operation on the linear transformations. Nevertheless, it is not hard to show that in the particular case that each matrix is a square matrix of order of the form n², n > 1, and is partitioned into n² square blocks of order n, then their Tracy-Singh product, (Formula presented.), is similar to (Formula presented.), and the change of basis matrix is a permutation matrix. In this note, we prove that in the special case of linear operators induced from set-theoretic solutions of the Yang-Baxter equation, the Tracy-Singh product of their representing matrices is the representing matrix of the linear operator obtained from the direct product of the set-theoretic solutions.