Complete commuting solutions of the Yang-Baxter-like matrix equation for diagonalizable matrices
Complete commuting solutions of the Yang-Baxter-like matrix equation for diagonalizable matrices
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DOI：
10.1016/j.camwa.2016.04.047
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发表时间：
2016-07
期刊：
影响因子：
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通讯作者：
Qixiang Dong;Jiu Ding
中科院分区：
文献类型：
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作者：
Qixiang Dong;Jiu Ding
设A是可对角化的方阵。利用二次矩阵方程A的Jordan形结构，结合关于Sylvester方程解的唯一性的一个著名定理，求出了二次矩阵方程A X A=X A X的所有对易解。进一步研究了给定矩阵A的两类特殊类型，包括循环矩阵和等于其某些幂的循环矩阵。此外，当A是Household矩阵时，基于谱扰动结果，构造了所有的非对易解。
Let A be a square matrix that is diagonalizable. We find all the commuting solutions of the quadratic matrix equation A X A= X A X, by taking advantage of the Jordan form structure of A, together with the help of a well-known theorem on the uniqueness of a solution to Sylvester’s equation. Two special classes of the given matrix A are further investigated, including circular matrices and those that are equal to some of their powers. Moreover, all the non-commuting solutions are constructed when A is a Householder matrix, based on a spectral perturbation result.