All solutions of the Yang-Baxter-like matrix equation AXA = XAX with A satisfying A4 = A

Abstract

In this paper, we construct some explicit solutions to the Yang-Baxter-like matrix equation $AXA=XAX$ for matrices A satisfying $A^4=A$, thereby extending previous results in this field. By analyzing the minimal polynomial of A, we classify the problem into 11 distinct cases. Our approach leverages the Jordan decomposition of A to simplify the original equation, reducing it to a system of matrix equations involving block-diagonal matrices with smaller blocks. We then systematically solve these reduced equations to obtain the general solution. Finally, we present three numerical examples to demonstrate the applicability and effectiveness of our theoretical results.