Analogs of Cramer's rule for the least squares solutions of some matrix equations

TL;DR

This paper develops analogs of Cramer's rule for least squares solutions of certain matrix equations, utilizing determinantal representations of the Moore-Penrose inverse to find minimum norm solutions.

Contribution

It introduces new determinantal formulas for least squares solutions of matrix equations, extending Cramer's rule to these cases.

Findings

Derived determinantal representations for least squares solutions.

Extended Cramer's rule to matrix equations with minimum norm solutions.

Provided explicit formulas for solutions of specific matrix equations.

Abstract

The least squares solutions with the minimum norm of the matrix equations ${\rm {\bf A}}{\rm {\bf X}} = {\rm {\bf B}}$, ${\rm {\bf X}}{\rm {\bf A}} = {\rm {\bf B}}$ and ${\rm {\bf A}}{\rm {\bf X}}{\rm {\bf B}} ={\rm {\bf D}} $ are considered in this paper. We use the determinantal representations of the Moore - Penrose inverse obtained earlier by the author and get analogs of the Cramer rule for the least squares solutions of these matrix equations.