Yang-Baxter-Like Matrix Equation: A Road Less Taken

This chapter represents a comprehensive analysis of the matrix equation AXA = XAX. We revise some of our published results regarding this topic and provide some new original unpublished results. In particular, we revisit our methods for constructing infinitely many nontrivial solutions, for both regular and singular matrix A, and we revisit our characterization of all permutation and doubly stochastic solutions when A is a permutation matrix. Additionally, we prove new results which concern the case when A is invertible: we obtain the closed-form formula for all commuting solutions and characterize the existence of non-commuting solutions (these conditions cannot be weakened). We also provide an alternative way for proving the existence of doubly stochastic solutions when A is a permutation matrix.