线性约束下双对称矩阵的最佳逼近问题

The Optimal Approximation Problems of Bisymmetric Matrix Under the Linear Restriction

将求矩阵方程AX=B双对称解的问题等价地转化为求一类矩阵方程对称解的问题.通过后者容易得出矩阵方程AX=B的双对称解,给出了解集合的表达式.研究了矩阵方程AX=B的双对称解集合的最佳逼近问题.当矩阵方程AX=B有双对称解时,其最佳逼近解存在且惟一,给出了最佳逼近解的表达式.

Based on sufficiently studying properties of bisymmetric matrices, solving the bisymmetric solutions of the matrix equation AX=B is equivalently reformulated into the problem for solving symmetric solutions of a class of matrix equations. By means of common solutions of the simple latter problem, bisymmetric solutions of the matrix equation AX=B are easily obtained and the expression of the solution set is also given. Moreover,the optimal approximation problem of the bisymmetric solution set of the matrix equation AX=B is also studied. When the matrix equation AX=B has an bisymmetric solution,its optimal approximation solution exists uniquely. The expression of the optimal approximation solution is provided.