On the commuting solutions to the Yang-Baxter-like matrix equation for a class of special rank-two matrices

Abstract

Let $A=I-PQ^{\T}$, where $P$ and $Q$ are two $n \times 2$ complex matrices of full column rank such that $\det (Q^{\T}P)=0$. We find all the commuting solutions of the quadratic matrix equation $AXA=XAX$.