Representations for the Drazin inverse of the generalized Schur complement

TL;DR

This paper derives new formulas for the Drazin inverse of the generalized Schur complement, broadening previous results by relaxing restrictions and extending Sherman-Morrison-Woodbury type formulas.

Contribution

It provides generalized expressions for the Drazin inverse of the generalized Schur complement with weaker assumptions than prior work.

Findings

Derived formulas for the Drazin inverse of the generalized Schur complement.

Extended Sherman-Morrison-Woodbury type formulas.

Relaxed restrictions compared to existing literature.

Abstract

In this paper we present expressions for the Drazin inverse of the generalized Schur complement $A-CD^{d}B$ in terms of the Drazin inverses of $A$ and the generalized Schur complement $D-BA^{d}C$ under less and weaker restrictions, which generalize several results in the literature and the formula of Sherman-Morrison-Woodbury type.