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-CADB, 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.