Commuting solutions of the Yang–Baxter-like matrix equation for a class of rank-two updated matrices

Abstract

Let $A=I-P{Q}^{\mathrm{T}}$, where $P$ and $Q$ are two $n\times 2$ complex matrices of full column rank and $det\left(Q^{\mathrm{T}}P\right)\ne 0$. All the commuting solutions with $A$ of the quadratic matrix equation $AXA=XAX$ have been founded.