On Pseudo SCHUR Complements in an EP Matrix

Transcription

1 International Journal of Scientific Innovative Mathematical Research (IJSIMR) Volume, Issue, February 15, PP ISSN 47-7X (Print) & ISSN 47-4 (Online) wwwarcjournalsorg On Pseudo SCHUR Complements in an EP Matrix Dr Bagyalakshmi Karuna Nithi Muthugobal Assistant Professor in Mathematics Bharathidasan University Constituent College Nannilam, Tamilnadu, India Abstract: It is established that under certain conditions a pseudo schur complement in an EP matrix is as well an EP matrix As an application a decomposition of a partitioned matrix into a sum of EP matrices is given Keywords: EP matrix, Pseudo schur complements, partitioned matrix 1 INTRODUCTION All matrices considered here are complex matrices will indicate the forming of the conjugate transpose matrix For an m n matrix A, any matrix X satisfying AXA A is called a generalized inverse of A is denoted by A The distinctive notation A is used for the Moore- Penrose inverse of A [1] A square matrix A is said an EP matrix if N A N A, where N A denotes the null space of A is said on (A) is the rank of A It is proved in [] that A is EP iff concerning generalized inverses is the following Lemma11 ([], p) ' EPr matrix if A is EP (A) r, where AA A A A well known lemma If X Y are generalized inverses of A, then CXB CYB if only if N( A) M( C ) N( A ) N( B ) or, equivalently, if only if C CA A B AA B for every A (1) Throughout this paper we are concerned with n A A A 1 M A A A A A A n matrices M partitioned in the form Where A D are square matrices With respect to this partitioning a Pseduo Schur complement A A A of A in M is a matrix of the form M A A A1 A A A A For properties of Pseduo Schur complements one may refer to [4], [5] [6] On account of Lemma 11 it is obvious that under certain conditions M A is independent of the choice of A However in the sequel we shall always assume that M A is given in terms of specific choice of A In [7] necessary sufficient conditions are derived for a matrix of the form () with A A1 A (or ) to be EP The results are here extended for general matrices of the A ARC Page 79 ()

2 Dr Bagyalakshmi Karuna Nithi Muthugobal form () If a partitioned matrix of the form () is EP, then in general M A is not EP Here we determine necessary sufficient conditions for M A to be EP In particular, when ( M) ( A ) our result include as special cases the result of paper [8] In [6] we have given here a decomposition of a partitioned matrix into a sum of EP matrices Further it is shown that in an EP r matrix every principal every principal submatrix of rank r is EP r The motivation for our research is the following: 1 The paper of AR Meenakshi [6] in which she extended the result of Katz, IJ Pearl, MH, [9] considering the conditions of EP r matrices, normal EP r matrices sums of EP r matrices The paper of DCarlson, EHaynsworth THMarkhasm [1] in which they gave detailed explanation of the concept generalization of the schur complements by means of the Moore- Penrose inverse The paper of Drazin, MP in which he had explained the concept of pseudo-inverse in associate rings semi groups Our purpose is to generalize these result aspects for the result of pseudo schur complement of EP matrix of order x PSEUDO SCHUR COMPLEMENT MATRIX N N N K N N N N N N 1 K / N K / N K / N B K / N K / N K / N K / N K / N K / N 1 1 () / 1 RESULTS Theorem1 Let be matrix of the form () with N N / N 1, then the following are equivalent (i) is an EP matrix (ii) / are EP, N / M ; N 1 (iii) Both the matrices are EP / 1 / International Journal of Scientific Innovative Mathematical Research (IJSIMR) Page 8

3 On Pseudo SCHUR Complements in an EP Matrix Proof (i) (ii) Let us consider the matrices I Q / 1 Q are nonsingular By assumption I L P / N N I I, Clearly P N / N 1 by using lemma 11 it is obvious that can be factorized as = PQL Hence L N N L But is EP Eg, N N N L Therefore by using lemma 11 again L L holds for every L one choice of L is L /, which gives 1 1 / / implies N N, since " " these imply N N Hence is EP From that it follows 1 1 N N N After substituting / 1 using / / in International Journal of Scientific Innovative Mathematical Research (IJSIMR) Page 81

4 Dr Bagyalakshmi Karuna Nithi Muthugobal we get / / / / / / This implies N / N / since / / we get N / N / Thus / is EP, further N N / N / Hence (ii) holds (ii) (i)since N N N N N / N 1 N / N holds according to the assumption, it can be applied (v) of Theorem 1 of the paper [4] so is given by the formula / / 1 (4) / / According to lemma 11 the assumptions N N N N N imply that 1 / is invariant for every choice of Hence / further, using 1 / /, reduced to the form 1 1 is International Journal of Scientific Innovative Mathematical Research (IJSIMR) Page 8

5 On Pseudo SCHUR Complements in an EP Matrix / / Using / / 1 1, a similar way gives B K N B K N / / / / The relations / / / / results, is EP Thus (i) holds (ii) (iii) By corollary 8 in [7] / is EP iff / are EP, further N N N /, 1 / is EP iff / are EP, further N N N / N 1 This proves the equivalence of (ii) (iii) The proof is complete Theorem Let be a matrix of the form () with N N 1 N / N, then the following are equivalent (i) is an EP matrix (ii) / are EP, further N N N / N 1 ; International Journal of Scientific Innovative Mathematical Research (IJSIMR) Page 8

6 Dr Bagyalakshmi Karuna Nithi Muthugobal (iii) Both the matrices / 1 / are EP Proof Theorem follows immediately from theorem 1 from the fact that is EP iff is EP In the special case when 1 we get the following Corollary Let 1 with N N N / N, then the following are equivalent (i) is on EP matrix; (ii) / are EP matrix (iii) The matrix Remark4 / is EP The condition taken on is the previous Theorems are essential This is illustrated in the following example Let N N N 1 K N N N N N N K K N K N K N 1 B K N K N K N K N K N K N International Journal of Scientific Innovative Mathematical Research (IJSIMR) Page 84

7 On Pseudo SCHUR Complements in an EP Matrix B The Rank of is 6Hence is EP / Clearly / are EP N N N / N 1 further / International Journal of Scientific Innovative Mathematical Research (IJSIMR) Page 85 N N But are not EP Thus theorem 1 as well as corollary fail Theorem 5 1 / / /, N B K N N 1 / Let be of the form () with r Then is an EP r matrix if only if is EP r Proof 1 Since r, we have by reason of the corollary of theorem 1 in [5], that N N, / / / / 1 / / N B K N N B K N B K N /

8 Dr Bagyalakshmi Karuna Nithi Muthugobal 1 According to lemma 11 these relations are equivalent to, Let us consider the matrices I P, Q I I I 1 L B K N / / P Q are nonsingular by assumption 1 it holds P Q Therefore can be factorized as EP r Hence PLP Since is EP r, consequently, L is a well N( L) N( L ) so we have according to lemma of paper [] that N N PLP N PL P N This shows that is EP Conversely, let us assume that is EP r since =PLQ one choice of Q P we know that N N, therefore by lemma 11 holds, eg, is 1 1 or equivalently 1 1 Form it follow N N ie is EP r therefore Taking into account International Journal of Scientific Innovative Mathematical Research (IJSIMR) Page 86

9 On Pseudo SCHUR Complements in an EP Matrix 1 We have the theorem is proved In the special case when is nonsingular is automatically EP r theorem 5 reduces to the following Corollary6 (see theorem 9 in [8]) Let of the form () with nonsingular Then is EP if only if 1 Corollary7 Let be nxn matrix of rank r Then is EP r if only if every principal sub matrix of rank r is EP r Proof Suppose is an EP r matrix Let be any principal submatrix of such that r Then there exists permutation matrix P such that 1 PP r According to Lemma in [] p9 s is EP, now we conclude from theorem 5 that is EPr as well Since was arbitrary, it follows that every principal sub matrix of rank r is EP r The converse is obvious 4 APPLICATION We give condition under which a partitioned matrix is decomposed into complementary summs of EP matrices 1 are called complementary summs of if 1 Theorem41 1 Let of the form () with /, where / 1 If / are EP matrices such that International Journal of Scientific Innovative Mathematical Research (IJSIMR) Page 87

10 Dr Bagyalakshmi Karuna Nithi Muthugobal / 1 / / into complementary summs of EP matrices Proof Let us consider the matrices then can be decomposed account that I 1 I / Taking into N N, N N / We obtain by the corollary after theorem 1 in [6], that 1 Since is EP, 1, we have from theorem 5 1 that 1 is EP Since 1 /, theorem 1 of paper [6] gives International Journal of Scientific Innovative Mathematical Research (IJSIMR) Page 88

11 On Pseudo SCHUR Complements in an EP Matrix N / N I 1 N / N I I / I Thus by the corollary of the just applied theorem 1 in [6], we have / Further, using, we obtain I / 1 I / / I / I Thus by theorem 5 is also EP Clearly, where both 1 1 are EP matrix / Hence 1 1 are complementary summs of EP matrices Remark 4 Any matrix that is represented as the sum of complementary summs of EP matrices is itself EP For if k i such that each i i 1 k i i 1 i 1 k N N N N i is EP REFERENCES rk then i [1] Penrose R, A generalized inverse for matrices, Proc Gambridge Philos Soc 51, (1955) MR 16: 18 [] Pearl M H, On generalized inverse for matrices, Proc Gambridge Phil Soc 6, Pp (1966) MR : 565 [] C R Rao S K Mitra, Generalized Inverse of Matrices Its Applications, Wiley, New York, 1971 MR 49: 78 [4] Burns F, Carlson C, Haynsworth E Markham T H, Generalized inverse formulas using the schur complement, SIAM J Appl Math 6 (1974), MR 48: 8519 [5] Carlson C, Haynsworth E Markham T H, A generalization of the schur complement by means of the Moore-Penrose inverse, SIAM J Appl Math 6 (1974), MR 5: 44 [6] Meenakshi A R, On sums of EP matrices, Houston JMath 9 (198), 6-69 MR 84k: 154 [7] Meyer C D, Generalized inverse of block triangular matrices, SIAM J Appl (1996), MR 16: 18 International Journal of Scientific Innovative Mathematical Research (IJSIMR) Page 89

12 Dr Bagyalakshmi Karuna Nithi Muthugobal [8] Robert P, On the group inverse of a linear transformation, J Math Anal Appl (1968), MR 7: 5 [9] Katz I J Pearl M H, On EPr normal EPr matrices, J Res Nat Bur Stds 7b (1966), [1] Drazin M P, Pseudo-inverses in associative rings semi groups, Amer Math Monthly, 65 (1958), 56-5 [] Baskett T S Katz I J, Theorems on products of EP matrices, Linear Algebra its Appl (1969), 87- MR 4:48 [1] H Schwerdtefger, Introduction to Liner Algebra the Theory of Matrices, Noord-Hoff, Groningen, 195 MR 1: 47 AUTHOR S BIOGRAPHY DrBKNMuthugobal is an Assistant professor in Mathematics, Bharathidasan University Constituent College, Nannilam He is a stalwart of Mathematics His papers more than 4 were published in various esteemed reputable International Journals He is a Member of Various Professional Bodies He had conducted many seminars He received so many prestigious rewards He is doing research in artificial genetic code with the application of mathematics It is a new work conducted by him which paves the way to introduce drugs in medicine field He is well versed in the fields such as graph theory, matrix theory, linear algebra International Journal of Scientific Innovative Mathematical Research (IJSIMR) Page 9