Group Inverse for a Class 2 2 Circulant Block Matrices over Skew Fields

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1 International Mathematical Forum Vol no Group Inverse for a Class 2 2 Circulant Block Matrices over Skew Fields Junqing Wang Department of Mathematics Dezhou University Shandong China tjgdwjq@sina.com bstract. In this paper we give the existence and the representation of the group inverse for circulant block matrix M B K n n and 2 B 2 B over skew field. Some relative additive results are also given. Mathematics Subject Classification: 47H09; 47H10 Keywords: skew; circulant block matrix ;group inverse 1. Introduction Let K be a skew field and I be the Unit matrix. K n n and respectively denote the set of all n-order matrices and the conjugate transpose of over K. For K n n the matrix X K n n is said to be the group inverse of if X XX X X X.We then write X. It is well known that if exists it is uniqueand the conclusion is given in[1]. On representations of the group inverse of block matricesthe authors have made efforts in [2]and [3]. ctually Generalized inverses have wide applications in many areas such as special matrix theory singular differential and difference equations and graph theory;see [4] [5] [6] [7] and [8]. Since the problem of finding an explicit representation for the Drazin group inverse of a 2 2 block matrix where and D are required to be C D square matrices was proposed by Campbell and Meyer in 1979 considerable progress has been made. condition for the existence of the group inverse of is given in [9] under the assumption that and I + C C D 2 B are

2 1976 Junqing Wang both invertible over any field;howeverthe representation of the group inverse is not given.nd the representation of the group inverse of the block matrix over skew fields has been given in 2001 in [10]. Though the 0 C representation of the Drazin group inverse of the block matrix C 0 proposed as a problem by Campbell in 1983 in [11] is square 0 is square null matrix has not been given there are some achievements about representations of the Drazin group inverse of the block matrices under special C 0 conditions. Some results are received on matrices over the complex fielde.g. in [12] when B I n and in [13] when B C P P PP P 2 P. Some results are over skew fields e.g. in [14] when I n and rankcb 2 rankb rankc and in [15] when B 2. In addition Group inverse of the product of two matricesas well as some related properties over skew field are given in [16]. In this paper we mainly give necessary and sufficient conditions for the existence and the representation of the group inverse of a block matrix M B K n n and 2 B 2 Bsimilarlywe also reach a few conclusions under certain conditions. 2. Preliminaries Lemma 2.1 Suppose K n n then exists if and only if rank rank 2. Lemma 2.2 Let M K 0 C n n k r r then M exists if and only if C exists and rankmrank+rankcand then we have M X 0 C where X 2 BI CC +I BC 2 BC. Lemma2.3 Let 2 B 2 Band rank B rank[i B + BI B]then B exists. Proof. One part rank[ B 2 ] rank[ B 2 +B] rank[i B+ BI B] rank B;nd another part rank[ B 2 ] rank B apparentlythen one can get rank[ B 2 ]rank Bthen according to the conclusion of Lemma 2.1 B exists. Lemma 2.4 Let 2 B 2 Band rank + B rank[i B + BI B]then + B exists. Proof. One part rank[+b 2 ] rank[+b 2 +B 2I] rank[i B +BI B] rank + B;nd another part rank[ + B 2 ]

3 Group inverse for a class circulant block matrices 1977 rank + B apparentlythen one can get rank[ + B 2 ]rank + Bthen according to the conclusion of Lemma B exists. 3. Conclusions Theorem. suppose M B K n n and 2 B 2 Bthen i M exists if and only ifrank B rank[i B+BI B]. ii if M exists and rank+b rank[i B+BI B] I 0 B then M X I B 1 where X B 2 B[I + B + B ]+[I B B ]B + B 2 B B + B. Proof. iproof of sufficient conditions.it is easy to prove that B 0 rankm rank rank 0 + B rank + B+rank B; rankm 2 rank 2 + B 2 B + B B + B 2 + B 2 + B + B rank B + B + B + B 0 rank 0 + B BB B rank + B+rank[I B+BI B]. For the given condition rank B rank[i B+BI B]we can get rankm rank + B +rank B rank + B +rank[i B+BI B] rankm 2 ;nd with rankm rankm 2 we can easily obtain rankm rankm 2.Then according to the Lemma 2.1M exists. Proof of the necessary conditions.m existsthen rankm rankm 2 and then rank + B+rank B rank + B+rank[I B+BI B]so rank[i B+BI B] rank B can be proved. ii with the condition rank B rank[i B +BI B] and the Lemma 2.3 B exists.similarlywith the condition rank + B rank[i B+BI B] and the Lemma B exists. I 0 B B I 0 For M 0 + B B B 0 and 0 + B 0 + B we know rankm rank + B+rank B.nd with the existence of + B and B according to the Lemma 2.2M has the form of 1.

4 1978 Junqing Wang Corollary 3.1 suppose M K n n and 2 then i M exists; I ii if M exists thenm 2 I Proof. We only need to replace B with in Theoremand easily prove that 2 existsthen the conclusion comes true. I Corollary 3.2 suppose M K I n n and 2 then i M exists;iiif rank + I rank Ithen I 0 I M X I I where X I 2 [I + I + I ]+[I I I ] + I 2 I + I. Proof. In Theorem we replace B with I.For rank I rank[i I+ II I and rank + I rank Ithen I and + I existso the Corollary 3.2 can be proved easily. Corollary 3.3 suppose M K n n and 2 then i M exists;ii if M exists then I 0 M X I where X 2 [I + + ]+[I ] Proof. In Theorem we replace B with.for and + are respectively anti-hermitian matrix and the Hermite matrixthey are unitarily similar to diagonal matricesso and + existthen the Corollary 3.3 can be proved easily. Corollary 3.4 suppose M K n n and 2 then i M exists;iiif rank + rank then M I 0 X I where X 2 [ + + ]+[I ] Proof. In Theorem we replace B with.for rank ran[i + I ] and rank + rank so and + exists.therefore it then the Corollary 3.4 can be proved easily.

5 Group inverse for a class circulant block matrices 1979 References [1] Wa-Jin ZhuangInvolutory functions and generalized inverses of matrices over an arbitrary skew fieldsnortheastmath[j] Chinese 1: [2] G. Wang e.tgeneralized Inverse Theory and ComputationsScience PressBeijing [3] K.P.S. Bhaskara RaoThe Theory of Generalized Inverses over Commutative RingsTaylor-Francis New York [4] S.L. Campbell and C.D. Meyer Generalized Inverses of Linear TransformationsDover New York [5] M. Catral D.D. Olesky P. van den DriesscheGroup inverses of matrices with path graphselectron[j] Linear lgebra 17: [6] G.H. Golub C. GreifOn solving block-structured indefinite linear systemssim [J] Sci.Comput. 24: [7] I.C.F. Ipsen note on preconditioning nonsymmetric matricessim [J]Sci. Comput.23: [8] Y. Wei H. DiaoOn group inverse of singular Toeplitz matriceslinear lgebra ppl.399: [9] X. Chen R.E. HartwigGroup inverse of a triangular matrixlinear lgebra ppl.237/238: [10] C. CaoSome results of group inverses for partitioned matrices over skew fieldsheilongjiang University Natural Science Journal Chinese 18: [11] S.L. CampbellThe Drazin inverse and systems of second order linear differential equationslinear Multi-linear lgebra 14: [12] N. Castro-Gonzales Representations of the Drazin inverse for a class of block matriceslinear lgebra ppl. 400: [13] C. Cao X. TangRepresentations of the group inverse of some 2 2 block matricesint.math. Forum 31: [14] C. BuOn group inverses of block matrices over skew fields[j] Math. 35:49C [15] C. Bu J. ZhaoJ. ZhengGroup inverse for a class 2 2 block matrices over skew fieldsppl. Math. Comput. 204: [16] C. Bu C. CaoGroup inverse of product of two matrices over skew field[j] Math Study35: [17] Wajin Zhuang Guidance of matrix theory on skew field[m] Science pressbeijing2006. Received: February 2012