Cramer's rules for the solution to the two-sided restricted quaternion matrix equation

TL;DR

This paper develops explicit determinantal formulas for solving restricted quaternion matrix equations using weighted singular value decomposition and column-row determinants, extending classical Cramer's rule to quaternion matrices.

Contribution

It introduces new determinantal representations for solutions of quaternion matrix equations, generalizing Cramer's rule within the framework of weighted SVD and column-row determinants.

Findings

Derived explicit formulas for quaternion matrix solutions

Extended Cramer's rule to quaternion matrices with restrictions

Unified approach for various cases based on weighted matrices

Abstract

Weighted singular value decomposition (WSVD) of a quaternion matrix and with its help determinantal representations of the quaternion weighted Moore-Penrose inverse have been derived recently by the author. In this paper, using these determinantal representations, explicit determinantal representation formulas for the solution of the restricted quaternion matrix equations, ${\bf A}{\bf X}{\bf B}={\bf D}$, and consequently, ${\bf A}{\bf X}={\bf D}$ and ${\bf X}{\bf B}={\bf D}$ are obtained within the framework of the theory of column-row determinants. We consider all possible cases depending on weighted matrices.