A note on the representations for the Drazin inverse of 2 × 2 block matrices

By Xiezhang Li, Yimin Wei. 2007

64 citations recorded by the source corpus.

Abstract In 1979, Campbell and Meyer proposed the problem of finding a formula for the Drazin inverse of a 2 × 2 matrix M = A B C D 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 MD, has recently been obtained under the assumptions that C(I − AAD)B = O and A(I − AAD)B = O together with the condition that the generalized Schur complement D − CADB be either nonsingular or zero. We derive an alternative representation for MD under the same assumptions, but with the condition on the Schur complement in the hypothesis replaced by the condition that R ( CAA D ) ⊂ N ( B ) ∩ N ( D ) , where R ( · ) and N ( · ) are the range and null space of a matrix.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/57cd05e71719dc3043d0de37f26af800a715ff24

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

On the Drazin inverse of the sum of two operators and its application to operator matrices (2009, background): “Several authors have considered the problem for 2 × 2 block matrices, with square diagonal blocks, under certain conditions on the individual terms [5,9,13,18].”
A note on the Drazin inverse of an anti-triangular matrix (2009, background): “Several authors have considered the problem for 25 2× 2 blockmatrices, under certain conditions on the individual blocks (see [9,11,15,18,21,23,29]).”
A note on the representation for the Drazin inverse of 2×2 block matrices (2008, background): “Many other papers have considered this open problem and each of them offered a formula for the Drazin inverse and specific conditions for the 2 × 2 block matrix M = [ A B C D ] to satisfy so that the formula is valid (see [4,5,10,13,17,18]).”