Anti-commuting Solutions of the Yang-Baxter-like Matrix Equation

We solve the Yang-Baxter-like matrix equation $AXA=XAX$ for a general given matrix $A$ to get all anti-commuting solutions, by using the Jordan canonical form of $A$ and applying some new facts on a general homogeneous Sylvester equation. Our main result provides all the anti-commuting solutions of the nonlinear matrix equation.