Group inverse for a class block matrices over skew fields
Introduction
Suppose K is a skew field. Let denote the set of all matrices over K. For , the matrix is said to be the group inverse of A, if it holds thatDenote by . By [1], if exists and it is unique.
The research on the representations of the group inverse for block matrices is an important problem in generalized inverse of matrices (see [2], [3]). The generalized inverse of block matrices have important applications in automatic, probability statistical, mathematical programming, numerical analysis, game theory, econometrics, control theory and so on (see [4], [5], [6], [7]).
In 1979, Campbell and Meyer proposed the problem of finding a formula for the Drazin inverse of a matrix in terms of its various blocks, where the blocks A and D are required to be square matrices (see [6]). At the present time, there is not a known representation for this problem. Ref. [8] gave the existence and the representation of group inverse for block matrix over skew fields. At present, the representation of the Drazin (group) inverse for block matrice of the form (A is square, 0 is square null matrix) have not been given. Some people only give the existence and the representation for the Drazin (group) inverse of such block matrix in some especial cases. For example, [9] gave the existence of the group inverse for block matrix ( is the identity matrix) if and only if existed over skew fields, if it also satisfied , then paper [9] gave the representation of the group inverse of it; [10] gave the existence of the group inverse for block matrix is the identity matrix) if and only if over the field of complex numbers, and gave the representation of the group inverse of it; [11] gave the existences and the representations of some group inverse for block matrices such as the form over the field of complex numbers, where is the transpose conjugate of P; [12] gave the representation of the group inverse for block matrix is the transpose conjugate of A) over the field of complex numbers.
In this paper, we give the existence and the representation of the group inverse for block matrix , in this case, some relative results in paper [10], [11] are the corollaries of this paper.
Section snippets
Some Lemmas
Lemma 2.1 Suppose , then exists if and only if . Lemma 2.2 Suppose , then exists if and only if exists and . If exists, then . Lemma 2.3 Suppose , then and exist. Lemma 2.4 Let , if , then there is an invertible matrix such that where is the identity matrix, . Proof By , then there are invertible matrices such that[3]
[8]
[13]
Conclusions
Theorem 3.1 Suppose , where , then exists if and only if ; If exists, then
Proof
- (i)
exists if and only if , that is .
- (ii)
By Lemma 2.3, we can know and exist. Let , where
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