中心主子阵约束下矩阵方程AX=B的双对称解

Solutions to AX=B for Bisymmetric Matrices under a Central Principal Submatrix Constraint

中心主子阵是指划去周边相同的行和列所得的主子阵。从中心主子阵扩充到双对称矩阵是有效和自然的一种矩阵扩充。通过分析双对称矩阵以及中心主子阵的结构,不仅给出了方程AX=B在中心主子阵约束下有双对称解的充分必要条件,而且给出了通解的表达式。在此基础上,也给出了最佳逼近问题的解的表达式。

A central principal submatfix is a submatrix obtained by deleting the same number of rows and columns in edges of a matrix. This is a feasible and reasoned way to expand the submatrix to bisymmetric matrices. By means of the special structure of bisymmetric matrices and its central principal submatrix, this paper obtains not only necessary and sufficient conditions for the solvability of AX=B in bisymmetric matrices set under a central principal submatrix, but also the general representation of the solutions. Based on this, the expression of the solution for the corresponding optimal approximation problem is given.