Explicit solutions of the Yang-Baxter-like matrix equation for diagonalizable coefficient matrix with two distinct nonzero eigenvalues

Abstract

Let $A$ be a complex diagonalizable matrix with two distinct nonzero eigenvalues $\lambda$ and $\mu$, the Yang-Baxter-like matrix equation $AXA = XAX$ is reconsidered. We correct and improve the results in Shen et al. (2020) when $\lambda^2-\lambda\mu+\mu^2= 0$. We also improve the results in Shen et al. (2020) when $\lambda^2-\lambda\mu+\mu^2\neq 0$. We obtain the explicit structure of the solutions $X$ for the Yang-Baxter-like matrix equation $AXA = XAX$, which are diagonalizable. Finally, we improve other existing relevant conclusions.