关于$\top$-Stein矩阵方程的注记
A note on the $\top$-Stein matrix equation
AI总结：
本文研究了线性矩阵方程$X = AX^\top B + C$的解的存在性条件，探讨了广义本征值问题与线性矩阵方程理论的关系，并提出数值方法及扰动界。
摘要：
本文研究了线性矩阵方程$X = AX^\top B + C$的解的存在性条件，探讨了广义本征值问题与线性矩阵方程理论的关系，并提出数值方法及扰动界。
英文摘要：
This note is concerned with the linear matrix equation $X = AX^\top B + C$, where the operator $(\cdot)^\top$ denotes the transpose ($\top$) of a matrix. The first part of this paper set forth the necessary and sufficient conditions for the unique solvability of the solution $X$. The second part of this paper aims to provide a comprehensive treatment of the relationship between the theory of the generalized eigenvalue problem and the theory of the linear matrix equation. In the finally part of this paper starts with a briefly review of numerical methods for solving the linear matrix equation. Related to the computed methods, knowledge of the residual is discussed. An expression related to the backward error of an approximate solution is obtained; it shows that a small backward error implies a small residual. Just like for the discussion of linear matrix equations, perturbation bounds for solving the linear matrix equation are also proposed in this work.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
