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

Authors

Xiezhang Li, Georgia Southern UniversityFollow
Yimin Wei, Fudan University

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