Determinantal representations of W-weighted Drazin inverse solutions of some quaternion matrix equations

TL;DR

This paper derives explicit determinantal formulas for W-weighted Drazin inverse solutions of certain quaternion matrix equations, extending Cramer's rule analogs within quaternion matrix theory.

Contribution

It provides new explicit determinantal representations for W-weighted Drazin inverse solutions of quaternion matrix equations, based on the column-row determinant framework.

Findings

Derived formulas for solutions of quaternion matrix equations.

Extended Cramer's rule to quaternion matrices with W-weighted Drazin inverse.

Provided explicit determinantal representations for specific matrix equations.

Abstract

By using determinantal representations of the W-weighted Drazin inverse previously obtained by the author within the framework of the theory of the column-row determinants, we get explicit formulas for determinantal representations of the W-weighted Drazin inverse solutions (analogs of Cramer's rule) of the quaternion matrix equations $ {\bf W}{\bf A}{\bf W}{\bf X}={\bf D}$, $ {\bf X}{\bf W}{\bf A}{\bf W}={\bf D} $, and ${\bf W}_{1}{\bf A}{\bf W}_{1}{\bf X}{\bf W}_{2}{\bf B}{\bf W}_{2}={\bf D} $.