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

摘要

Consider a 2×2 block complex square matrix M=[ABCD], where A and D are square matrices. Suppose that (I-~(AAD))B=O and C(I- ~(AAD))=O, where ~(AD) is the Drazin inverse of A. The representations of the Drazin inverse MD have been studied in the case where the generalized Schur complement, S=A-~(CAD)B, is either zero or nonsingular. In this paper, we develop a representation, under certain conditions, for ~(MD) 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.