Explicit formulas for determinantal representations of the Drazin inverse solutions of some matrix and differential matrix equations

TL;DR

This paper derives explicit determinantal formulas for Drazin inverse solutions of certain matrix and differential matrix equations, extending Cramer's rule to singular matrices.

Contribution

It provides new determinantal representations and analogs of Cramer's rule for Drazin inverse solutions of matrix and differential equations involving singular matrices.

Findings

Derived explicit formulas for Drazin inverse solutions.

Extended Cramer's rule to singular matrix equations.

Applied formulas to differential matrix equations.

Abstract

The Drazin inverse solutions 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 both the determinantal representations of the Drazin inverse obtained earlier by the author and in the paper. We get analogs of the Cramer rule for the Drazin inverse solutions of these matrix equations and using their for determinantal representations of solutions of some differential matrix equations, ${\bf X}'+ {\bf A}{\bf X}={\bf B}$ and ${\bf X}'+{\bf X}{\bf A}={\bf B}$, where the matrix ${\bf A}$ is singular.