当\(A^n = tA^2\)时基于换位子的\(AXA = XAX\)的解
Commutator-based Solutions to $AXA = XAX$ when $A^n = tA^2$
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中文总结 AI 辅助
研究当系数矩阵\(A\)满足\(A^n = tA^2\)时，为类杨 - 巴克斯特矩阵方程\(AXA = XAX\)获取新非交换解的方法，证明一致性标准并举例展示结果，还推广了相关特殊情况，解决了特定非齐次方程。
中文摘要 AI 辅助
本文提供了一种新颖的分析方法，用于在系数矩阵\(A\)满足平方循环条件\(A^n=tA^2\)时，为类杨 - 巴克斯特矩阵方程\(AXA = XAX\)获得新的非交换解。证明了所有一致性标准，并通过实例展示了所得结果。本文的发现同时推广了该主题的几个现有特殊情况。因此，利用所得结果，解决了\(A\)可逆或满足\(A^2 = 0\)时以前未研究的非齐次杨 - 巴克斯特矩阵方程\(AXA = XAX + B\)。
英文摘要
This paper offers a novel analytical method for obtaining new non-commuting solutions for the Yang-Baxter-like matrix equation $AXA=XAX$, when the coefficient matrix $A$ satisfies the square-cyclic condition $A^n=tA^2$. All the consistency criteria are proved and the obtained results are demonstrated on worked examples. The findings provided by this article simultaneously generalize several existing special cases regarding this topic. Consequently, by employing the obtained results, the previously unstudied inhomogeneous Yang-Baxter matrix equation $AXA=XAX+B$ is solved when $A$ is either invertible, or satisfies $A^2=0$.
