Explicit solutions of the Yang-Baxter-like matrix equation for a singular diagonalizable matrix with three distinct eigenvalues

Abstract

Let $A$ be a singular diagonalizable complex matrix with three distinct eigenvalues. We derive all explicit solutions $X$ of the Yang-Baxter-like matrix equation $AXA = XAX$, by taking advantage of the Jordan form structure
of $A$. The result generates the formula obtained in Chen et al. (2019) and M. Saeed Ibrahim Adam et al. (2019). We give examples to illustrate the validity of the results obtained in this paper.