Yang-Baxter型矩阵方程的反交换解

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

当A为给定复矩阵,X为未知矩阵时,非线性矩阵方程AXA=XAX被称为Yang-Baxter型矩阵方程.对于一些特殊的系数矩阵A,如A是一个幂零矩阵,可对角化矩阵等,部分学者已经给出了Yang-Baxter型矩阵方程解的结构.近年来对于方程交换解的研究取得一定的研究结果,但对于方程的反交换解的研究还处于初始阶段.当系数矩阵A为指数为3的幂零矩阵,本文给出了Yang-Baxter型矩阵方程的求解方法以及反交换解的结构.

When A is a given complex matrix and X is an unknown matrix,the nonlinear matrix equation AXA=XAX is called the Yang-Baxter like matrix equation.Some scholars have given the structure of Yang-Baxter like matrix equation solution for some special coefficient matrix A,for example A is a nilpotent matrix and a diagonalizable matrix.Some results have been made in the study of the commuting solution of Yang-Baxter like matrix equation in recent years,but the study of the anti-commuting solution of the equation is still in the initial stage.When the coefficient matrix A is a nilpotent matrix with index 3,the solution method of Yang-Baxter like matrix equation and the structure of anti-commuting solution are given.