Constructing solutions of the Yang–Baxter-like matrix equation for singular matrices

Let A=UVH be an n×n complex singular matrix of rank r, where U, V are two n×r matrices of full column rank. Some sufficient and necessary conditions are derived for X being a nontrivial (commuting) solution of the quadratic matrix equation AXA=XAX such that we can construct infinitely many (commuting) solutions of the matrix equation. When VHU is nonsingular, all the solutions of the matrix equation are characterized and can be found and constructed by solving smaller matrix equations. In particular, all the commuting solutions can be obtained by finding the projections that commute with VHU. The method of constructing the solutions is free of computations of Jordan canonical forms and matrix inverses.