Some Results for the Drazin Inverses of the Sum of Two Matrices and Some Block Matrices

Abstract

We give a formula of(P+Q)Dunder the conditionsP2Q+QPQ=0,P3Q=0, andPQPQ=0. Then, we apply it to give some expressions for the Drazin inverse of block matrixM=(ABCD)(AandDare square matrices) under some conditions, generalizing some recent results in the literature. Finally, numerical examples are given to illustrate our results.

The Drazin inverse of the sum of two bounded linear operators and it’s applications

“…In this paper, using the technique of the resolvent expansion, we investigate the existence of the Drazin inverse of P + Q for bounded linear operators P and Q and the explicit representations of (P + Q) D in term of P, P D , Q and Q D under the conditions (1) P 2 Q + QPQ = 0, P n Q = 0, (2) PQ 2 + PQP = 0 PQ n = 0 for some integer n, respectively, which extend the relevant results in [7,12,16,19]. Then, we apply these results to establish representations of the Drazin inverse of the operator matrix, which can be regarded as the generalizations of some results given in [10,16]. Actually, the proof of the main results show the efficiency of the method employed to some extent.…”

“…The following corollary is the case when n = 1 of Theorem 2.5. ] and P k P π = 0 (k ≥ ind(P)), X in [16,Theorem 5] can be simplified 2 , where r = ind(P), s = ind(Q). Thus, the representation of (P + Q) D in [16,Theorem 5] is reduced to the formula of (11).…”

“…The case BC = 0, BDC = 0 and BD 2 = 0 has been studied in [10] and the case ABC = 0, BDC = 0, CBC = 0 and D 2 C = 0 in [16] for matrices. We focus our attention in the generalization of the mentioned results.…”

“…In [11], M. P. Drazin proves that (P + Q) D = P D + Q D if PQ = QP = 0 in an associative ring. In the sequel, many authors begin to consider this problem for matrices and operators, and present explicit representations of (P + Q) D under the conditions such as (1) PQ = QP = 0 (see [11]), (2) PQ = 0 (see [9,12]), (3) P 2 Q = PQ 2 = 0 (see [5]), (4) P 2 Q + PQ 2 = 0, P 3 Q = PQ 3 = 0 (see [13]), (5) PQP = 0, Q 2 P = 0 (or QPQ = 0, P 2 Q = 0 ) (see [7,19]), (6) P 2 Q + QPQ = 0, P 3 Q = 0 (see [16]), (7) P 2 QP = P 2 Q 2 = PQ 2 P = PQ 3 = 0 (see [17]), (8) P D Q = PQ D = 0, Q π PQP π = 0 (see [6]). For more general Drazin inverse problems, we refer the reader to [2,4,14] and their references.…”