Group inverse for the block matrices with an invertible subblock☆
Introduction
The group inverses of block matrices have numerous applications in many areas, such as singular differential and difference equations, Markov chains, iterative methods, cryptography and so on (see [1], [2], [3], [4], [5], [6]). In 1979, Campbell and Meyer proposed an open problem to find an explicit representation for the Drazin inverse of a block matrix , where the blocks A and D are supposed to be square matrices but their sizes need not be the same (see [1]). Until now, this problem has not been solved completely, and there is even no known expression for the Drazin (group) inverse of which was posed by Campbell in 1983 in [3]. However, there are many literatures about the existence and the representation of the Drazin (group) inverse for the block matrix under some conditions (see [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]). For example, in [7], the existence of the group inverse of M over a field F was investigated under the condition: A and are invertible. The representations of the Drazin inverse and the group inverse of M over the complex number field were given in [12], when and . In [13], [14], [16], the authors studied the existence and representations of the group inverse of M over skew fields under different conditions: in [13], A is square, and . In [14], and . And in [16], A is square and .
Let be the set of all matrices over a skew field K, the identity matrix over K and the rank of . For a square matrix A, the Drazin inverse of A is the matrix satisfyingwhere is the index of A, the smallest nonnegative integer such that . It is known that is existent and unique (see [20]). For , if there exists a matrix satisfying the matrix equationthen we say that X is a {1}-inverse of A and it is denoted by . Denote the set of all {1}-inverses of A by (see [20]). Furthermore, if the matrix X also satisfies the matrix equationsthen we say that X is the group inverse of A and it is denoted by . It is well known that if exists, it is unique (see [20]). Notice that the group inverse of a matrix exists iff .
In this paper, we study the necessary and sufficient conditions for the existence of the group inverse for the block matrix (A is square) when and D satisfy one of the conditions listed below:
- (1)
A is invertible, exists, where ;
- (2)
D is invertible, exists, where ;
- (3)
B or C is invertible.
We also give the expressions of the group inverse for M under these conditions respectively.
Section snippets
Lemmas
Lemma 2.1 Let , then exists if and only if . Lemma 2.2 Let , then there are invertible matrices such thatandwhere , and are arbitrary. Lemma 2.3 Let , where and are invertible, . Thenwhere , , and are arbitrary. Lemma 2.4 Let , where is invertible, . If exists, and is[20]
[21]
Main results
Theorem 3.1 Let , where is invertible, . If exists, then exists if and only if R is invertible, where and ; If exists, then
where
Proof
- (i)
Since A is invertible, then we havewhere , so .
Moreover, as exists, we get
Acknowledgement
The authors would like to thank two anonymous reviewers for their helpful comments.
References (21)
- et al.
On group inverse of singular Toeplitz matrices
Linear Algebra Appl.
(2005) - et al.
On group inverses of M-matrices with uniform diagonal entries
Linear Algebra Appl.
(1999) The group inverse of the transformation
Linear Algebra Appl.
(1997)- et al.
The group inverse of a triangular matrix
Linear Algebra Appl.
(1996) - et al.
A note on the representations for the Drazin inverse of block matrices
Linear Algebra Appl.
(2007) A note on the representation for the Drazin inverse of block matrices
Linear Algebra Appl.
(2008)- et al.
Representations of the Drazin inverse of a class of block matrices
Linear Algebra Appl.
(2005) - et al.
Group inverse for a class block matrices over skew fields
Appl. Math. Comput.
(2008) - et al.
Formulas for the Drazin inverse of special block matrices
Appl. Math. Comput.
(2006) - S.L. Campbell, C.D. Meyer, Generalized inverses of linear transformations, Pitman, London, 1979, Dover, New York,...
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2011, Applied Mathematics and Computation
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Supported by Natural Science Foundation of the Heilongjiang Province, No. 159110120002.