The Solvability Conditions for the Inverse Problem of Bisymmetric Nonnegative Definite Matrices

Abstract

$A = (a_{ij}) \in R^{n×n}$ is termed bisymmetric matrix if $$a_{ij} = a_{ji} = a_{n-j+1,n-i+1}, i,j=1,2, ..., n.$$ We denote the set of all $n \times n$ bisymmetric matrices by $BSR^{n×n}$.
This paper is mainly concerned with solving the following two problems:
Problem I. Given $X, B \in R^{n×m}$, find $A \in P_n$ such that $AX=B$, where $P_n = \{ A \in BSR^{n×n}| x^TAx \ge 0, \forall x \in R^n \}$.
Problem Ⅱ. Given $A^* \in R^{n×n}$, find $\hat{A} \in S_E$ such that $$\| A^* -\hat{A}\|_F =\mathop{min}\limits_{A \in S_E} \| A^* - A \|_F,$$ where $\|\cdot\|_F$ is Frobenius norm, and $S_E$ denotes the solution set of problem I.

The necessary and sufficient conditions for the solvability of problem I have been studied. The general form of $S_E$ has been given. For problem II the expression of the solution has been provided.