Further results on the group inverses and Drazin inverses of anti-triangular block matrices
Introduction
Let Cm×n be the set of complex m × n matrices. The symbols r(A) and A(1) will denote the rank and any {1}-inverse of a given matrix . Further, AD will stand for the Drazin inverse of A, i.e., the unique matrix satisfying the equations [1]where k = Ind(A), the index of A, is the smallest nonnegative integer k such that
If , then the Drazin inverse of A is reduced to the group inverse, denote by . If , then . In addition, we denote (or ), especially, if A is idempotent, then .
The Drazin inverse is very useful, and has various applications in singular differential or difference equations, Markov chains, cryptography, iterative method and numerical analysis. In 1979, Campbell and Meyer first proposed an open problem of finding a formula for the Drazin inverse of in terms of the blocks of the partition over complex fields, where A and D are square matrices but need not to be the same size. To the best of our knowledge, up to now, this problem has not been solved completely. However, some special cases of the Drazin inverse and group inverse of M were studied under certain conditions [3], [7], [10], [11], [15], [17], [18], [19], [21], [23], and the expressions of the Drazin inverse and group inverse in terms of Banachiewicz–Schur forms were also investigated in [12], [13], [14], [16], [20].
Specially, when ,is the so-called anti-triangular block matrix, where and . In recent years, the Drazin inverse and group inverse for M formed by (1.2) have been considered by many authors. For instance, Deng [6] established some expressions of the Drazin inverse of M when and is either invertible or equal to zero. Deng and Wei [9] got explicit expressions for the Drazin inverse of the anti-triangular block matrix M under the following different casesIn [22], Castro–González and Dopazo provided the representations for MD under two different conditions
Bu et al. [2], [4] determined the conditions for the existence of in the case or , and presented the representations of . For the matrixif , in [2], the authors showed that exists if and only if ; for each matrix A, need not to be idempotent, if , in [4], they showed that exists if and only if . In recent paper [8], Bu et al. considered the group inverse of M in the case A can be expressed as the linear combination or product combination of matrices B and C. Cao and Li [25] mainly considered the group inverse of matrix with the form over skew fields. Zhou et al. [26] established the conditions for the existence and the representations for the group inverse of block matrix M with one or two full rank sub-blocks.
In the next section, some improved results on the group inverse of the anti-triangular block matrix M are obtained, which extend the results given by Bu et al. [2], [4]. Moreover, applying our new results, several representations of the Drazin inverse of anti-triangular block matrix M are derived under certain conditions.
Before giving the main results, we first introduce some important lemmas as follows. Lemma 1.1 [24] Let and . Thenwhere the is an arbitrary {1}-inverses of A. Lemma 1.2 Let . If , then and exist. Furthermore, . Proof In view of , we havewhich meansSimilarly, we can deduceHenceAnd the condition r(B) = r(BC) shows that there exists a matrix X ϵ Cm×n such that B = BCX. ThereforeOn the other hand,Combining (1.6), (1.7) gives r(BC) = r(BC)2. Therefore, exists.The existence of can be proved similarly, we omit the details. With the fact , then is evident. □ Lemma 1.3 [2] Let , where and . Then exists if and only if exists and . Lemma 1.4 [5] Let , and , such that . ThenSpecially, (i) If , then (ii) If , then
Section snippets
Main results
In this section, we first establish the necessary and sufficient conditions for the existence of under some weaker conditions than those in [2], [4], where M is given by (1.2), then present the explicit expressions for . Furthermore, we use our new results to derive some representations for the Drazin inverse of M under certain assumptions.
First, we consider the case that A is idempotent. Theorem 2.1 Let M be partitioned as (1.2). If and , then (i) exists if and only if exists and
Examples
In this section, we give two examples to illustrate Theorem 2.3 and Theorem 2.7. Example 3.1 Consider the block matrix , whereIt is easy to verify that and exists. Furthermore, we can calculate thatHence, applying Theorem 2.3, we get Example 3.2 Consider the block matrix , whereWe observe that , and . Furthermore, we can compute that
Acknowledgements
The authors would like to thank the editors and the reviewers for their valuable comments and helpful suggestions, which improved the paper. This work is supported by the National Natural Science Foundation of China (Grant No. 11171361) and Ph.D. Programs Foundation of Ministry of Education of China (Grant No. 20110191110033).
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