# A Representation for the Drazin Inverse of Block Matrices with a Singular Generalized Schur Complement

## Document Type

Article

## Publication Date

5-15-2011

## Publication Title

Applied Mathematics and Computations

## DOI

10.1016/j.amc.2011.02.061

## ISSN

0096-3003

## Abstract

Consider a 2×2 block complex square matrix *M=[ABCD]*, where *A * and *D * are square matrices. Suppose that* (I-AA ^{D})B=O* and

*C(I-AA*, where A

^{D})=O^{D}is the Drazin inverse of

*A*. The representations of the Drazin inverse

*M*have been studied in the case where the generalized Schur complement,

^{D}*S=A-CA*, is either zero or nonsingular. In this paper, we develop a representation, under certain conditions, for

^{D}B*M*

^{D}when

*S*is singular and group invertible. Moreover, this formula includes the case where

*S=O*or nonsingular. A numerical example is given to illustrate the result.

## Recommended Citation

Li, Xiezhang.
2011.
"A Representation for the Drazin Inverse of Block Matrices with a Singular Generalized Schur Complement."
*Applied Mathematics and Computations*, 217 (18): 7531-7536.
doi: 10.1016/j.amc.2011.02.061

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/88