Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations

TL;DR

This paper derives explicit formulas, similar to Cramer's rule, for the minimum norm least squares solutions of certain quaternion matrix equations using the theory of column and row determinants.

Contribution

It introduces explicit representation formulas for quaternion matrix equations, extending classical solutions to the quaternion setting.

Findings

Derived formulas for ${f A} {f X} = {f B}$, ${f X} {f A} = {f B}$, and ${f A} {f X} {f B} = {f D}$.

Extended classical Cramer's rule to quaternion matrices.

Provided explicit solutions in the quaternion matrix equations context.

Abstract

Within the framework of the theory of the column and row determinants, we obtain explicit representation formulas (analogs of Cramer's rule) for the minimum norm least squares solutions of quaternion matrix equations ${\bf A} {\bf X} = {\bf B}$, $ {\bf X} {\bf A} = {\bf B}$ and ${\bf A} {\bf X} {\bf B} = {\bf D} $.