一类矩阵方程组 AX =B，XD =E的对称解与反对称解

摘要

The symmetric solution and the anti-symmetric solution of matrix equations AX = B , XD =E were considered.Based on the special properties of symmetric (anti-symmetric)matrices and the canonical correlation decomposition (CCD)of a pair of matrices,the necessary and sufficient conditions for the solvability and the general expression of the symmetric solution (anti-symmetric solution)were obtained.Moreover,the related optimal approximation problem to a given matrix over the solution set was solved.%考虑矩阵方程组 AX =B ,XD =E 的对称解与反对称解,利用对称(反对称)矩阵的性质和矩阵对的标准相关分解(CCD),给出了矩阵方程组对称解(反对称解)存在的充分必要条件及解的一般表达式,并讨论了对任意给定矩阵的最佳逼近问题。