The least-squares solutions of inverse problems of a class of skew-symmetric matrices (Q2743656)

A skew-symmetric matrix \(A=(a_{ij})\) is an \(n\times n\) real square matrix which is symmetric with respect to its sub-diagonal, i.e., \(a_{ij}=a_{n-j+1,n-i+1}\) for \(i,j=1,2,\dots,n\). The inverse problem discussed in this paper is as follows: Given \(X,Z\in \mathbb{R}^{n\times m}\), \(Y,W\in\mathbb{R}^{n\times t}\), find a skew-symmetric matrix \(A\), such that \(AX=Z\) and \(Y^TA=W^T\). Necessary and sufficient conditions for the solvability and the expression of the general solution of this problem are given. Then the least-squares solution of the problem is discussed: Given \(X,Z\in \mathbb{R}^{n\times m}\), \(Y,W\in\mathbb{R}^{n\times t}\), \(\widetilde A\in\mathbb{R}^{n\times n}\), let NEWLINE\[NEWLINES_E=\{A\text{ skew-symmetric}\mid \|AX-Z\|^2+\|Y^TA-W^T\|^2=\min\}NEWLINE\]NEWLINE then it is shown that \(S_E\neq\emptyset\) and that there is a unique \(A^*\in S_E\) such that \(\|\widetilde A-A^*\|=\inf_{A\in S_E}\|\widetilde A-A\|\). The expression of the solution is given as well.