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

In this paper, a representation of ( P + Q ) D such that P Q P = 0 , P Q 2 = 0 is given, which recovers the case P Q = 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 × 2 partitioned matrix M = ( A B C D ) , where A and D are square matrices. Finally, two numerical examples are given to illustrate our results.