#### Title

A Note on the Representations for the Drazin Inverse of 2x2 Block Matrices

#### Document Type

Article

#### Publication Date

6-1-2007

#### Publication Title

Linear Algebra and Its Applications

#### DOI

10.1016/j.laa.2007.01.005

#### ISSN

0024-3795

#### Abstract

In 1979, Campbell and Meyer proposed the problem of finding a formula for the Drazin inverse of a 2 × 2 matrix *M= [ABCD]* in terms of its various blocks, where the blocks *A* and *D* are required to be square matrices. Special cases of the problems have been studied. In particular, a representation of the Drazin inverse of *M*, denoted by *M*^{D}, has recently been obtained under the assumptions that *C*(*I* − *AA*^{D})*B* = *O* and *A*(*I* − *AA*^{D})*B* = *O* together with the condition that the generalized Schur complement *D* − *CA*^{D}*B* be either nonsingular or zero. We derive an alternative representation for *M*^{D} under the same assumptions, but with the condition on the Schur complement in the hypothesis replaced by the condition that *R(CAAD)*⊂*N(B)*∩*N(D)*, where *R*(.) and *N*(.) are the range and null space of a matrix.

#### Recommended Citation

Li, Xiezhang, Yimin Wei.
2007.
"A Note on the Representations for the Drazin Inverse of 2x2 Block Matrices."
*Linear Algebra and Its Applications*, 423 (2-3): 332-338.
doi: 10.1016/j.laa.2007.01.005 source: https://www.sciencedirect.com/science/article/pii/S0024379507000286?via%3Dihub

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/562