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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 390592, 10 pages
The Expression of the Drazin Inverse with Rank Constraints
Department of Mathematics, Dezhou University, Dezhou 253023, China
Received 9 October 2012; Accepted 5 November 2012
Academic Editor: C. Conca
Copyright © 2012 Linlin Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By using the matrix decomposition and the reverse order law, we provide some new expressions of the Drazin inverse for any block matrix with rank constraints.
Let be a square complex matrix. The symbols and stand for the rank and the Moore-Penrose inverse of the matrix , respectively. The Drazin inverse of is the unique matrix satisfying where is the index of , the smallest nonnegative integer such that . We write .
The Drazin inverse of a square matrix plays an important role in various fields like singular differential equations and singular difference equations, Markov chains, and iterative methods.
The problem of finding explicit representations for the Drazin inverse of a complex block matrix, in terms of its blocks was posed by Campbell and Meyer [1, 2] in 1979. Many authors have considered this problem and have provided formulas for under some specific conditions [3–6].
In this paper, under rank constraints, we will present some new representations of which have not been discussed before.
Lemma 2.1 (see ). Let and be square matrices of the same order.
If , then where .
If and , then .
Lemma 2.2 (see ). Let where , are square matrices with , . Then where
Lemma 2.3 (see ). Let , . Then if and only if , satisfy where , and with .
Lemma 2.4 (see ). Let . Then
3. Main Results
In this section, with rank equality constraints, we consider the Drazin inverse of block matrices.
Let , where is invertible and is singular. It is easy to verify that can be decomposed as Let where . According to Lemma 2.3, we have the following theorem.
Theorem 3.1. Let , where is invertible and is singular. If where , , , then has the following form:
Proof. From Lemma 2.3 and (3.1), we know that if
where , then
From Lemma 2.4, we have
Note that , . Then we get
Let , . Then can be rewritten as the following three matrix products:
Since is nonsingular, then Thus, we have From the above equality and the condition (3.3), (3.5) is easily verified.
Let , where , . It is easy to verify that the matrix can be decomposed as where is the generalized Schur complement of in .
Let where . Then we have the following theorem.
Theorem 3.2. If , and the matrices , satisfy then, where , .
Proof. From Lemma 2.3 and (3.14), we get that if
Similar to the proof of Theorem 3.1, we derive that the rank condition (3.18) can be simplified as (3.16).
Next, we will give the representation for . Let
Since , , and , then . From Lemma 2.2, we get
From Lemma 2.1 and the fact , it follows that Substituting in (3.19), the conclusion can be obtained.
From Theorem 3.2, we can easily obtain the following corollaries.
Corollary 3.4. If , and the rank equality (3.16) hold, then where .
Next, we will consider another decomposition of involving the generalized Schur complement .
Let , where . Then can be decomposed as where is the generalized Schur complement of in .
Let where . Then we have the following theorem.
Theorem 3.5. Let , where . If , and the matrices , satisfy the following rank equality: then where .
Proof. From Lemma 2.3 and (3.25), we get that if the following rank condition
holds, then . From the same method used in Theorem 3.1, we can verify that the above condition (3.29) can be reduced to (3.27).
Next, we will give the representation for . For , we write where .
From the condition , we get , according to Lemma 2.2, we have Let . By Lemma 2.1 and the fact , we get Therefore, we get
Remark 3.6. In addition to the decompositions of in (3.14) and (3.25), the matrix also can be decomposed as other matrix products involving the generalized Schur complements or . In these cases, new formulas for would be derived by the method used in this paper.
In this paper, we mainly discuss the Drazin inverse of block matrices under rank equality constraints. Comparing with the existing results, it is obvious that our results have more strong restrictions, but the methods used in this paper are different from those in previous relevant paper.
The author wishes to thank Professor Guoliang Chen and Dr. Jing Cai for their help. This research is financed by NSFC Grants 10971070, 11071079, and 10901056.
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