A Note on the Formulas for the Drazin Inverse of the Sum of Two Matrices and Its Applications

Abstract

The Drazin inverse has applications in a number of areas such as control theory, Markov chains, singular differential and difference equations, and iterative methods in numerical linear algebra. The study on representations for the Drazin inverse of block matrices stems essentially from finding the general expressions for the solutions to singular systems of differential equations, and then stimulated by a problem formulated by Campbell. In 1983, Campbell (Campbell et al. (1976)) established an explicit representation for the Drazin inverse of a 2 × 2 block matrix M in terms of the blocks of the partition, where the blocks A and D are assumed to be square matrices. Special cases of the problems have been studied. In 2009, Chunyuan Deng and Yimin Wei found an explicit representation for the Drazin inverse of an anti-triangular matrix M, where A and BC are generalized Drazin invertible, if A^πAB=0 and BC (I–A^π) =0. Afterwards, several authors have investigated this problem under some limited conditions on the blocks of M. In particular, a representation of the Drazin inverse of M, denoted by M^d. In this paper, we consider the Drazin inverse of a sum of two matrices and we derive additive formulas under the conditions of ABA^π=0 and BA^π=0 respectively. Precisely, for a block matrix M, we give a new representation of M^d under some conditions that AB=0 and DCA^π=0. Moreover, some particular cases of this result related to the Drazin inverse of block matrices are also considered.