二阶Yang-Baxter型矩阵方程的解

Solutions to the Yang-Baxter-Like Matrix Equation of Order Two

研究了Yang-Baxter型矩阵方程AXA=XAX的解。当A是二阶矩阵时,通过考虑A可对角化和不可对角化两种情形得到了该矩阵方程有解的充分必要条件,并详细给出了解的具体表示。此外,当A是正定矩阵时,说明了A是矩阵方程AXA=XAX的唯一正定解。

Solutions to the Yang-Baxter matrix equation AXA=XAX are studied. When A is of order 2, by considering whether A is diagonalizable or not, sufficient and necessary conditions for the matrix equation having a solution are given, and explicit solutions are also presented. In addition, if A is positive definite, then it is shown that A is the unique positive definite solution to the matrix equa-tion AXA=XAX.