Abstract
The equation \(XAX=AXA\) is called the Yang–Baxter-like matrix equation. We examine this equation for matrices of order 2, assuming that \(A\) is a nonsingular matrix; moreover, we are only interested in nonsingular solutions. Using a unified rule, one can associate each solution with a matrix that commutes with \(A\) , in other words, with an element of the centralizer \(\mathcal{M}_{A}\) of \(A\) . No obvious reasons exist for two distinct solutions \(X_{1}\) and \(X_{2}\) to generate one and the same element of \(\mathcal{M}_{A}\) . Nonetheless, all the solutions, and there are infinitely many of them, produce one and the same matrix in the centralizer. We give an explanation for this amazing fact.
