Anti-commuting Solutions of the Yang-Baxter-like Matrix Equation
Mohammed Ahmed Adam Abdalrahman, Huijian Zhu, Jiu Ding, Qianglian Huang
TL;DR
This work addresses finding all solutions to the nonlinear matrix equation $AXA = XAX$ for a given matrix $A$, with a focus on anti-commuting solutions where $AB = -BA$. The authors develop a Jordan-form based framework, reducing the problem to a homogeneous Sylvester-type system and showing that the essential information is contained in the zero-eigenvalue block of $A$, via a key result that relates the original problem to a simplified equation on the nilpotent part. The main contributions include a complete characterization: if $A$ is nonsingular, the only anti-commuting solution is $B = 0$, while if $A$ is singular, nontrivial anti-commuting solutions exist only within the zero-eigenvalue Jordan blocks and admit a concrete block-structured form determined by the sizes of the Jordan blocks. The paper also provides constructive procedures and numerical examples to illustrate how to obtain these solutions from the zero-block, offering a pathway to generalize the approach to other non-commuting solution families.
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.