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

TL;DR

This paper characterizes all anti-commuting solutions to a nonlinear matrix equation similar to the Yang-Baxter equation by leveraging Jordan canonical forms and Sylvester equation properties.

Contribution

It provides a complete characterization of anti-commuting solutions to the Yang-Baxter-like matrix equation using novel methods involving Jordan forms and Sylvester equations.

Findings

All anti-commuting solutions are explicitly characterized.

The solutions depend on the Jordan canonical form of A.

The approach introduces new insights into solving nonlinear matrix equations.

Abstract

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.