A representation for the Drazin inverse of block matrices with a singular generalized Schur complement

Consider a 2 x 2 block complex square matrix M = [(A)(C) (B)(D)], where A and D are square matrices. Suppose that (I - AA(D))B - O and (I - AA(D)) - O, where A(D) is the Drazin inverse of A. The representations of the Drazin inverse M-D have been studied in the case where the generalized Schur complement, S = A - CA(D)B, is either zero or nonsingular. In this paper, we develop a representation, under certain conditions, for 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. (C) 2011 Elsevier Inc. All rights reserved.