On The Best Approximate Solutions of The Matrix Equation $AXB=C$

TL;DR

This paper investigates the best approximate solutions for the matrix equation $AXB=C$, including least squares solutions over symmetric and skew-symmetric matrices, with numerical examples illustrating the methods.

Contribution

It provides explicit forms of the best approximate solutions for both consistent and inconsistent cases, extending to symmetric and skew-symmetric matrix sets.

Findings

Derived implicit forms of best approximate solutions.

Addressed solutions over symmetric and skew-symmetric matrices.

Included numerical examples demonstrating the solutions.

Abstract

Suppose that the matrix equation $AXB=C$ with unknown matrix $X$ is given, where $A$, $B$, and $C$\ are known matrices of suitable sizes. The matrix nearness problem is considered over the general and least squares solutions of the matrix equation $AXB=C$ when the equation is consistent and inconsistent, respectively. The implicit form of the best approximate solutions of the problems over the set of symmetric and the set of skew-symmetric matrices are established as well. Moreover, some numerical examples are given for the problems considered.