Determinantal representations of the Drazin inverse for Hermitian matrix over the quaternion skew field with applications

TL;DR

This paper develops determinantal formulas for the Drazin inverse of Hermitian matrices over quaternions, enabling explicit solutions for certain quaternion matrix equations.

Contribution

It introduces new determinantal representations of the Drazin inverse for quaternion Hermitian matrices and derives explicit solution formulas for related matrix equations.

Findings

Determinantal representations of the Drazin inverse over quaternions.

Explicit formulas for quaternion matrix equations solutions.

Applications to solving quaternion matrix equations.

Abstract

Within the framework of the theory of the column and row determinants, we obtain determinantal representations of the Drazin inverse for Hermitian matrix over the quaternion skew field. Using the obtained determinantal representations of the Drazin inverse we get explicit representation formulas (analogs of Cramer's rule) for the Drazin inverse solutions of quaternion matrix equations $ {\bf A}{\bf X} = {\bf D}$, $ {\bf X}{\bf B} = {\bf D} $ and ${\bf A} {\bf X} {\bf B} = {\bf D} $, where $ {\bf A}$, ${\bf B}$ are Hermitian.