Drazin inverse of partitioned matrices in terms of Banachiewicz–Schur forms

Let be a partitioned matrix, where A and D are square matrices. Denote the Drazin inverse of A by AD. The purpose of this paper is twofold. Firstly, we develop conditions under which the Drazin inverse of M having generalized Schur complement, S=D-CADB, group invertible, can be expressed in terms of a matrix in the Banachiewicz–Schur form and its powers. Secondly, we deal with partitioned matrices satisfying rank(M)=rank(AD)+rank(SD), and give conditions under which the group inverse of M exists and a formula for its computation.