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

Abstract

a b s t r a c tIn this paper, a representation of (P + Q ) D such that PQP = 0, PQ 2 = 0 is given, which recovers the case PQ = 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.

A class of Drazin inverses in rings

“…This equality is called Cline's formula. It provides a technique to present the Drazin inverse of the sum of two elements (see [14,23,26,29]). In this section, we also show Cline's formula for the s-Drazin inverse.…”

On additive properties of the Drazin inverse of block matrices and representations

“…This result was generalized in [9], where authors gave the formula for M d under conditions CA p A ¼ 0 and CA p B ¼ 0. Yang and Liu [13] extended this result and derived the representation for M d when BCA p A ¼ 0 and BCA p B ¼ 0 holds. The following theorem is a generalization of this result.…”

Some Results on the Symmetric Representation of the Generalized Drazin Inverse in a Banach Algebra

“…In [14], authors investigated how to express the Drazin inverse of sums, differences, and products of two matrices P and Q, under the conditions P 3 Q = QP and Q 3 P = PQ. The representations of the Drazin inverse for (P + Q), such that PQP = 0 and PQ 2 = 0, is given in [15]. The generalized inverses in C * -algebras has been investigated in [16] and a symmetry of the generalized Drazin inverse in a C * -algebra has been considered in [17].…”