Cramer's Rule for Generalized Inverse Solutions of Some Matrices Equations

TL;DR

This paper introduces new determinantal representations and Cramer's rules for generalized inverses, including Moore-Penrose and Drazin, enabling solutions for singular matrix equations and differential matrix equations.

Contribution

It develops novel determinantal formulas and analogues of the classical adjoint for generalized inverses, facilitating explicit solutions for singular and differential matrix equations.

Findings

Derived determinantal representations for Moore-Penrose and Drazin inverses.

Established Cramer's rules for least squares and Drazin inverse solutions.

Provided solutions for differential matrix equations with singular matrices.

Abstract

By a generalized inverse of a given matrix, we mean a matrix that exists for a larger class of matrices than the nonsingular matrices, that has some of the properties of the usual inverse, and that agrees with inverse when given matrix happens to be nonsingular. In theory, there are many different generalized inverses that exist. We shall consider the Moore Penrose, weighted Moore-Penrose, Drazin and weighted Drazin inverses. New determinantal representations of these generalized inverse based on their limit representations are introduced in this paper. Application of this new method allows us to obtain analogues classical adjoint matrix. Using the obtained analogues of the adjoint matrix, we get Cramer's rules for the least squares solution with the minimum norm and for the Drazin inverse solution of singular linear systems. Cramer's rules for the minimum norm least squares solutions and the Drazin inverse solutions of the matrix equations ${\rm {\bf A}}{\rm {\bf X}} = {\rm {\bf D}}$, ${\rm {\bf X}}{\rm {\bf B}} = {\rm {\bf D}}$ and ${\rm {\bf A}}{\rm {\bf X}}{\rm {\bf B}} ={\rm {\bf D}} $ are also obtained, where ${\rm {\bf A}}$, ${\rm {\bf B}}$ can be singular matrices of appropriate size. Finally, we derive determinantal representations of solutions of the 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.