On the projection-based commuting solutions of the Yang-Baxter matrix equation

We study the commuting solutions of the Yang-Baxter matrix equation AXA=XAX when A is an arbitrary square matrix. By characterizing its commuting solutions based on projection matrices, we show that projections can be determined by using the generalized eigenspaces corresponding to the eigenvalues of A. Therefore, commuting solutions can be constructed explicitly. Our results are more general than those obtained recently by Dong (2017), Ding and Zhang (2014), and Ding and Rhee (2013).