The Inverse Problem for Part Symmetric Matrices on a Subspace

Abstract

In this paper, the following two problems are considered:
Problem Ⅰ. Given $S \in R^{n×p}, X, B \in R^{n×m}$, find $A \in SR_{s,n}$ such that $AX=B$, where $SR_{s,n}={A \in R^{n×n} | x^T(A-A^T)=0, \ {\rm for} \ {\rm all} \ x \in R(S)}$.

Problem Ⅱ. Given $A^* \in R^{n×n}$, find $\hat{A} \in S_E$ such that $\|\hat{A} -A^*\|={\rm min}_{A \in S_E} \|A-A*\|$, where $S_E$ is the solution set of Problem Ⅰ.

Then necessary and sufficient conditions for the solvability of and the general from of the solutions of problem Ⅰ are given. For problem Ⅱ, the expression for the solution, a numerical algorithm and a numerical example are provided.