The Drazin inverse of the sum of two matrices and its applications

In this paper, a representation of (P+Q)D(P+Q)D such that PQP=0PQP=0, PQ2=0PQ2=0 is given, which recovers the case PQ=0PQ=0 studied by Hartwig et al. [R.E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl., 322 (2001) 207–217]. Furthermore, we apply our results to establish the representations for the Drazin inverses of a 2×22×2 partitioned matrix M=(ABCD), where AA and DD are square matrices. Finally, two numerical examples are given to illustrate our results.