The inverse problem of bisymmetric matrices with a submatrix constraint

摘要

An n * n real matrix A is called a bisymmetric matrix if A=A~T and A=S_nAS_n, where Sn is an n * n reverse unit matrix. This paper is mainly concerned with solving the following two problems: Problem I Given n * m real matrices X and B, and an r * r real symmetric matrix A0, find an n * n bisymmetric matrix A such that AX = B, A_0 = A[(1:r)] where A([1: r]) is a r * r leading principal submatrix of the matrix A. Problem II Given an n * n real matrix A*, find an n * n matrix A in S_E such that ‖A~* - A‖ = min/A∈S_E ‖A~* - A‖ The necessary and sufficient conditions for the existence of and the expressions for the general solutions of Problem I are given. The explicit solution, a numerical algorithm and a numerical example to Problem II are provided.