 Advances in Linear Algebra & Matrix Theory, 2012, 2, 1-11 http://dx.doi.org/10.4236/alamt.2012.21001 Published Online March 2012 (http://www.SciRP.org/journal/alamt) Schur Complement of con-s-k-EP Matrices Bagyalakshmi Karuna Nithi Muthugobal Ramanujan Research Centre, Department of Mathematics, Government Arts College (Autonomous), Kumbakonam, India Email: bkn.math@gmail.com Received February 8, 2012; revised March 8, 2012; accepted March 15, 2012 ABSTRACT Necessary and sufficient conditions for a schur complement of a con-s-k-EP matrix to be con-s-k-EP are determined. Further it is shown that in a con-s-k-EPr matrix, every secondary sub matrix of rank “r” is con-s-k-EPr. We have also discussed the way of expr essing a matrix of r ank r as a produ ct of con- s-k-EP r matrices. Necessary and sufficient condi-tions for produ c t s of con -s- k-EPr partitioned matrices to be con-s-k-EPr are given. Keywords: con-s-k-EP Matrices; Partitioned Matrices; Schur Complements 1. Introduction Let be the space of n × n complex matrices of order n. Let be the space of all complex n-tuples. For nnnnCnCAC, let A, AT, A*, AS, SA, A†, R(A), N(A) and ρ(A) denote the conjugate, transpose, conjugate transpose, secondary transpose, conjugate secondary transpose, Moo- re-Penrose inverse, range space, null space and rank of A, respectively . A solution X of the equation AXA = A is called generalized inverses of A and is denoted by A. If nnAC, then the unique solution of the equations AXA = A, XAX = X, *[] ,AXAX*[]XAXA is called the moore penrose inverse of A and is denoted by †A. A matrix A is called con-s-k-EPr if Ar and N(A) = N (ATVK) or R(A) = R (KVAT). Throughout this paper let “k” be the fixed product of disjoint transposi-tion in Sn = {1, 2, ···, n} and K be the associated per-mute- tion matrix . Let us define the function . A matrix A = (aij) ,,,k1 k2knkxx xxCnxn is s-k symmetric if  ij nkj 1,nki 1 for i, j = 1, 2, · · · , n. A matrix ACnxn is said to be con-s-k-EP if it satisfies the condition or equivalently N(A) = N (ATVK). In addition to that A is con-s-k-EPaa0sAxA k ()x0KVA is con-EP or AVK i s c o n - E P a n d A i s con-s-k-EP  AT is con-s-k-EP. Moreover A is said to be con-s-k-EPr if A is con-s-k-EP and of rank r. For further properties of con-s-k-EP matrices one may ref er . In this paper we derive the necessary and sufficient conditions for a schur complement of a con-s-k-EP ma-trix to be con-s-k-EP. Further it is shown that in a con- s-k-EPr matrix, every secondary submatrix of rank r is con-s-k-EPr. We have also discussed the way of express-ing a matrix of rank r as a product of con-s-k-EPr matri-ces. Necessary and sufficient conditions for products of con-s-k-EPr partitioned matrices to be con-s-k-EPr are given. In this sequel, we need the following theorems. Theorem 1.1  Let nxnA,B C, then 1) for all {1}TTNANBRBRAB BAAAA   2) for all {1}TTNANBRBRAB AABAA Theorem 1.2  Let, ABMCD, then   †††††††††AABMA CAABMAMMA CAMA ,,and . TTTTNANC NANBNMANCNMA NB Also,   †††††††MD ABMAMDC M DMA    ,,and ,,,.TTT TTTT TNA NCNA NBNMA NCNMA NBNDNBND NCNMD NBNMDNC When MA, then and ABMCCABTTT TTTT TAPAA PCMBPA BPC, Copyright © 2012 SciRes. ALAMT B. K. N. MUTHUGOBAL 2 where, TT TTPAABBAA ACC . Theorem 1.3  Let ,nnABC and nnUC be any nonsingular ma-trix, then, 1) () ()TTRA RBRUAURUBU  TT 2) () ()NA NBNUAUNUBU2. Schur Complements of con-s-k-EP Matrices In this section we consider a 2r × 2r matrix M Partitioned in the form, ABMCD (2.1) where A, B, C and D are all square matrices. If a parti-tioned matrix M of the form 2.1 is con-s-k-EP, then in general, the schur complement of C in M, that is (M/C) is not con-s-k-EP. Here, necessary and sufficient conditions for (M/C) to be con-s-k-EP are obtained for the class  MC and  MC, analogous to that of results in  . Now we consider the matrix MAMBSMCMB (2.2) the matrix formed by the Schur complements of M over A, B, C and D respectively. This is also a partitioned ma-trix. If a partitioned matrix S of the form 2.2 is con-s- k-EP, then in general, Schur complement of (M/C) in S, that is [S/(M/C)] is not con-s-k-EP. Here, the necessary and sufficient conditions for [S/(M/C)] to be con-s-k-EP are obtained for the class  SMC and  SMC, analogous to that of results in  As an application, a decomposition of a partitioned matrix into a sum of con-s-k-EPr matrices is obtained. Further it is shown that in a con-s-k-EPr matrix, every secondary sub matrix of rank r, is con-s-k-EPr. Through-out this section let k = k1k2 with. 1200KKK (2.3) where K1 and K2 are the permutation matrices relative to k1 and k2 and let “V” be the permutation matrix with units in its secondary diagonal of order 2r × 2r parti-tioned in such a way that 00vvV (2.4) Theorem 2.5 Let S be a matrix of the form 2.2 with NNMCMA and NS/N( )MCMD, then the following are equivalent: 1) S is a con-s-k-EPr matrix with k = k1k2 and V= 00νν. 2) (M/C) is a con-s-k-EP, S/M C is con-s-k2-EP.  TTMCMNND and  TTSMC MANN. 3) Both the matrices  0MCMAS MC and 0MC MDSMC are con-s-k-EPr. Proof: Since S is con-s-k-EPr with k=k1k2, KVS is Con-EP and where K1 and K2 are permutation matrices associated with k1 and k2 and . 12KKooKoνVνoConsider IMAMCPOI, SMCIOQMDI and OSMCLMC O. Clearly P and Q are non singular. Now,  121211 IOOSMCKOOνIMAMCKVPQL OK νOMDS MCIMCOOIIMAMCMDSMC MAMCOSMCOKνKνOMC OMDS MCIKνMC KνMDS MC           22SMCKνMA MCMCKνSMCMAMC MDSMCSMC      Copyright © 2012 SciRes. ALAMT B. K. N. MUTHUGOBAL 3 Since, NMC NMAhave , by Theorem 1.1 we -MA= MAM CM C, that is, 22-KνMA=KνMAMC MC. Since, NS MCN MD , we have by Theorem 1.1  MDMDSMC SMC. That is,   11KνMD KνMDSMCSMC. Also,  22.KνSMCMAMC MDSMCSMCKνMB Since,SMCMBMAMC MD , therefore,  11221212KνMC KνMDKVPQL KνMA KνMBOKνMA MB KνOMCMDKOMAMOν OKMCMDνO KVS B)).)L  as OMCLSMC O Thus KVS is factorized as KVS = KVPQL. Hence and ()(ρKVS ρL()(NKVS NLBut S is con-s-k-EP. Therefore, KVS is con-EP (By Theorem 2.11 ). ()() ()(TTN KVSN KVSN LNS VK Therefore, by using Theorem 1.1 again we get, TTSVKSVKLL holds for every . LWe choose   12211212TTTTTTTTTTTMAMBKOOνSVKMCMD OKνOMA MCOνK νKOMB MDMC νKMAνK MD νKMBνK As the equation (at the bottom of this page). and since   1111TTTρKνMCρKνMCρMC νKρMCNMCN MC νK Hence, (M/C) is con-s-k-EP. From  11 ,TTMD νKMDνKMC MC is follows that  111() TTTNMCN MD νKNMCνKNMDνK  (using (M/C) is con-s-k-EPr). Therefore  TTNMC NMD. After substituting  MBMCMAMCMS D and using   22 TTMA νKMAνKMC MCSS in   22 TTMB νKMBνKMC MCSS         1212121212121TTTTTTTTTTTTTTTMC νKMAνKSVK SVKLLMD νKMBνKOMCMC νKMAνKOSMCMC OSMC OMD νKMBνKMC νKMC MC AνKSMCSMCMD νKMCMC BνKSMCSMCMC νKMC   11111TTTTTνKMC MCKνMC KνMCMCMCNMC NKνMCNMC νK  Copyright © 2012 SciRes. ALAMT B. K. N. MUTHUGOBAL 4 We get,    22 TTMB νKMBνKMC MCSS  22TTMCMAMCMBνKMCMA MCMBνKMCMCSSSS    2222TTTTMC νKMAMCMBνKMC νKMC MCMA MCMBνKMC MCSSSS      222TTTSMC νKSMC νKSMC SMCNS MCNSMCνK    By Theorem 1.1 and since  2TTρKνSMC ρSMCρSMC we get, 2TNKνSMC NSMC 2NSMCνKNS MC SMC is con-s-k2-EPr. Further     2222222TTTTTTTTTMA νKMAνKSMC SMCNS MCNMAνKNKνSMC NMAνKNSMCνKNMAνKNS MCNMA Thus 2) holds 2) 1). Since NMC NMA,  TTNMC NMD, NS MCNMD and  TTNS MCNMA holds, according to the assumption by applying Theorem 1.2, †KVS is given by the formula  111 †11 2††221†††221 2† †††† †112† ††1 2 KνMC KνMCKνMDKνMC KνMD KνMCKνMCKνMA KνMCKVSKνMC KνMA KνMC KνMCKνMCMCMDMCMA KνMCMCMDMCνK MCM AKνMCMCνKSSSSSSSS       (2.6)†11 ††1†1††2†††11 12††1†2††21 2†2††12†21KνMC KνMCKνMC MCKνMCMC MDMCMDMC νK MA KνMCKνMD MCKνMDMC νKMA KνMCKVS KVSKνMA MCKνMAKνMC KνMA MCMDMC νKMDMCMDKνMCKνMB MCKνMBMCMA KνMCSSSSSSSS  2νK                    Copyright © 2012 SciRes. ALAMT B. K. N. MUTHUGOBAL 5 According to Theorem 1.1 the assumptions N(M/C)  N(M/A) and   TTMCMD MNN SC is invariant for every choice of MC Hence   22†211 KνMB KνMCKνMC KνMC KνMDS  Therefore  2†2211KνMCKνMB KνMA KνMC KνMDS †21 122 KνMAKνMC KνMDKνMB KνMCS †22 KνMB MCMDKνMB MCS  †MAMC MDMBMCS Further using 2†22 2KνMAKνMCKνMCKvMASS   and  †1111KνMD KνMC KνMCKvMD. That is  2†22†2KνMAKνSMCSMC νKKν2MAKK νMCMCMASS    †MAMCMCMSSA and   †11 11†1KνMD KνMC MC νKKνMDKνMC MCMD †MDMCMCMD, KVS†KVS reduces to the form, As the Equation (a) below. Again using †1†12 2KνMDKνMD KνMCKνMCSS and †2211KνMA KνMAKνMC KνMCthat is,  †MDMDMCMCSS   and ††,MAMAMCMCKVSKVS reduces to the form As the Equation (b) below. Since, MC is con-s-k1-EP 1KνMC is con-EP. Therefore we have   †11†11KνMC KvMCKvMC KvMC     Similarly, since MCS is con-s-k2-EPr. We have, †22†21KνMC KνMCKνMC KνMCSSS   Thus ††KVS KVSKVSKVS ††††††KVSS VKS VKKVSKVSSVKSSKVSSS SKV S is con-s-k-EP (by Theorem 2.11 ). Thus 1) holds 2)  3)  2220KνMCKνMA KνMCS is con-EP if and only if 1KνMC and 2KνMCS are con-EP. Therefore,  1200000MCKMA MCKSνν   †11††2200KνMCKνMCKVS KVSKνMC KνMCSS (a) †11††2200KνMCKνMCKVS KVSKνMC KνMCSS (b)Copyright © 2012 SciRes. ALAMT B. K. N. MUTHUGOBAL 6 is con-EP if and only if 1KνMC and 2KνMCare con-EP.  0MCMA MCS is con-s-k-EP if and only if MC is con-s-k1-EP and MCS is con-s-k2-EP . Further NMC NMA and  TTNMCNMDS Also 1120KνMC KνMDKνSMC is con-EP if and only if and 2KνSMC and con-EP. Therefore, 0MC MDSMCis con-s-k-EP if and only if MCis con-s-k1-EP and SMCis con-s-k2-EP further  TTNMC NMD and  TNS MCNMD . This proves the equivalence of 2) and 3). The proof is complete. Theorem 2.7 Let S be a matrix of the form (2.2) with  TTNMC NMDand  TTNMCNMAS , then the following are equivalent. 1) S is con-s-k-EP with k = k1k2 where 1200KKKand 00Vνν2) MC is con-s-k1-EP. Further and MCS is con-s-k2-EP. Further NMC NMA and  NMCNMDS 3) Both the matrices  0MCMA MCS and 0MC MDSMC are con-s-k-EP. Proof This follows from Theorem 2.5 and from the fact that S is con-s-k-EP ST is con-s-k-EP. In particular, when TMDMA, we got the fol-lowing. Corollary 2.8 Let S = TMAMBMCM A with NMC NMA and  TA. uivaNMCNMSThen the following are eqlent. 1) S is a con-s-k-EP matrix. d 2) (M/C) is con-s-k1-EP anMCS is con-s- k2e matrix -EP. 3) Th 0MCMA MCS is con-s-k- EP. Rons taken on S in Theorem 2.6 and Theo-emark 2.9 The conditirem 2.7 are essential. This is illustrated in the following example. Let ABMCD 1011A,1101B,1101C,1011D 10 1111 0111100111M  11 12,21 11MA MB   ,  121 1,112 1MCMD , MAMBSMCMD 111221 11121 111 21S 100001 0000100001K 0001001001 001000V 0001001001 001000KV Now KVS is symmetric of rank 3 s-k-EP. 11 21121 121 111112KVS , KVS is con-EPS is con-1MCM MDMCB MAS 1121MA, 1211MB  Copyright © 2012 SciRes. ALAMT B. K. N. MUTHUGOBAL 71121MD , 1121113MC 3303SMC  Hence 20333KνSMC  is con-EP, that is SMC is con-s-k2-EP. , Also 11211 KνM111 2MCC     is con-EP. 1KνMC is con-EP MC is con-s-EP. ver k1- MoreoC NMANM and  TTNMC . But NMD  NS MDNMD and  TNS MCNMAFurther . T 1200011 00113321 03KV MAS MCMC is not con-EP. Therefore, 0MCMDSMC is not con-s-k-EP. Thus the Theorem 2.5 as well as the corollary 2.8 fail. Rrem 2.5 and Theorem 2.7 that P matrix of the form 2.2 and k = k1k2 ivalent. nd the Theorem 2.7 aemarks 2.10 We conclude from Theofor a con-s-k-Ewhere 12k00kK and 00ννν the following are equ ,NMC NMANS MCNMD 2.11   ,TTTNMC NMDNS MCNMA T 2.12 However this fails if we omit the condition tcon-s-k-EP. hat S is For example, Let ABDMC, where , , 11, 101A010B1001C1101D 11 0101 1010 1101 01M    01,,, 10ABCDM A, 0210MB, 1111MC , 1012MD  MAMBSMCMD 010 2111 011 1011 12S    10 0001 0000 1000 01K 00 0100 1001 0010 00V 11 1211 1010100102KVS is not con-EP. Therefore S is not con-s-k-EP. Here 11KνMC111 P. is con-EMC is con-s-k-EP.  11TKνMD KνMD, 11TTKνMDMDνK, 11TMD νKA νK, νMC νMA, and  TTνMC νMD. HeSMC is iependent of the choice of ndnce MC. Now Copyright © 2012 SciRes. ALAMT B. K. N. MUTHUGOBAL Copyright © 2012 SciRes. ALAMT 8 Let S be of the form 2.2 with  .ρSρMCr Then S is con-s-k-EPr and K and V are of the form 2.3 and 2.4 if and only if (M/C) is con-s-k1-EPr and †SMCMBMAMC MD  02 01,10 10MB MA   ,  ††12TMAMC νKMCMDνK.  1101 11,121 12MD MC   Proof Let S be of the form 2.2 and let k= k1k2 with and then 1200kKk00ννν0111SMC  1122KνMC KνMDKVS KνMAKνMB. 21101KνSMC is not con-EP. Since ,ρSρMCr SMC  is not con-s-Also, k2-EP. 1ρKVS ρKνMCr by [ 6]  TTNS MCNMD. But ,NMC NMA TTNMC NMD and  NCNMD. Thus, 2.12 ho lds while 2.1 fails. Remark 2.13 r a con-s-k-EP mar- 2.6 gives S M 1200KVSKνMCKνSMC SMC1.  It is clear by Remark 2.10 that fo†By Theorem 1.1 these relation equivalent to 22 ,KνMA KνMAMC trix S, formulaKVS if and only if either 2. of the form 2.2 with K and V are of d 2.4 respectively, for wh ich 11 or 2.12 holds. Corollary 2.14 Let S be a matrix †11KνMD KνMCMCMD and  †22KνMBKνMAMC MD †KVSthe forms 2.3 an is given by the formula then S is con-s-k-EP if and only if both (M/C) and SMC and con-s-k-EPProof This follows em 2.5 and using . Let us consider the matrices 0IMAMCPI from TheorRemark ow we proceed to prove the most important Th2.13. N†0IMC MDQI and 000LMC eorem. Theorem 2.15     ††12††12†††1†2†11†220000000000000000KνIMA MCIMCMDKVPLQ MCKνIIKνMA MCMCIMCMDKνMC IMAMCMCMAMC MCMCMCKνKνMCMC MCMDKνMC KνMC MCMDKνMA MCMCK     †1122120000νMA MCMDKνMC KνMDKνMA KνMBKMAMBνKMCMDνKVS B. K. N. MUTHUGOBAL 9 Thus KVS can be factorized as KVS = KVPLQ. Since KVP = (KVQ)T. We have KVPTVK=Q. Therefore, TTTKVSKVPLKVP VKKVP LKVKVPKVP KVL KVP [since LVK = KVL]. Since (M/C) is con-s-k1-EPr. We have k1v(M/C) is con-EPr. Therefore (Theorem 2.11 of ) By Theorem 1.3 assume that S is con-s-k-EPr. Since S is con-s-k-EPr, KVS is con-EPr. Since KVS = KVPLQ, one choic e o f () TNLN LVK TNKVL NKVL  TTNKVP KVLKVPNKVP KVLKVP T()TNKVSN KVS TNS NSVK S is con-s-k-EP (Theorem 2.11 of ). Since ρSr, S is con-s-k-EPr. Conversely, let us  11†00 0KVSQPVK KVSMC is con-EP ()TNKVSN KVS By Theorem 1.1 T.TKVS KVSKVSKVS That is,     1122νMAKν1122TTMC KνMDKMBKνMC KνMDKνMAKνMB Kν1†000QMC   1211KνMC Kν2MDPνMAVK KKνMB As the equation (at the bottom of this page). or conversely, †11TTKνMC KνMCMCMC †21TTKνMC KνMCMCMD and †11TTKνMC KνMCMCMC From it follows that  1TNMCN KνMC  1TNMCNMC νKMC is con-s-k-EP. Since .ρMCrMC is con-s-k-EPr. From †21TTKνMA KνMCMCMD it follows that. Now, †2††1††1††1†1†1TTTTTTTTTTKνMA MCMD MCKνMC MCMCMC MCKνMDMCMC MCνKMD MCνKKνMCMD (By theorem 2.11 [) TMD1]††21TKνMA MCMCMDνK  ††12TMA MCνKKνMCMD  †12TνKMCMDνK†MA MC Mark 2.16 hen (M/A) is non singlular, KV(M/A) is automti-cally con-EPr and (M/A) is con-s-k-EPr and Theorem 2.15reo the following. Let S be of the form 2.2 with C non singular and Wa duces tCorollary 2.17 []ρSρMC. Then S is con-s-k-EP with K = k1k2 and†1†200νTνMAMCνKνMCMDνK .  †11 1†22 11TT TTTT TTKνMC KνMC Kν†MC MC1KνMAM†CMCMDKνMDKνMB KνMDMC MCKνMCMCMD     Copyright © 2012 SciRes. ALAMT B. K. N. MUTHUGOBAL 10 Remark 2.18 When k(i) = i, we have K1 = K2 = I, then the Theorem 2.15 reduces to the result for con-s-EP matrices. When KV = I then Theorem 2.15 reduces to Theorem 3 of . Remark 2.19 Theorem 2.15 fails if we relax the condition on the rank of S. For example, let us consider the matrix S and K given in Remark 2.10, 2But [][]ρKVSρS. 11,ρKVM CρMC 1() ()ρKVS ρKνMA ρSρ.MA KVS is not con-EP Therefore S is not Con-s-k-EP. However, 110011101 1011011 111 10 1111KVM C is con-EP. Therefore (M/C) is con-s-k1-EP and 1111112MC, 11111112A MCνK, M110MC MDνK. 210 = K1K2, where Thus the theorem fails. Corollary 2.20 Les S be a 2r x 2r matrix of rank r. Thus S is con-s-k-EPr with K1200KKand V = 00νν every secondary sub matrix of S of rank r is con-s-k-EPr. trix then KVS is an co ix by Theorem 2.11 . Let Proof Suppose S is con-s-k-EPr man-EPr matr1KνMC such thatbe any Principal submatrix of KVS 1[]ρKVS ρKν,MCrtation matrix P such that, then there exists a permu-   1122TTKνMC KνMDKVSPKVSPMA νKνKMB with 1.ρKVSρKνMCr By  TKVS2.15 that is con-EP . Now we conclude from Theorem r 1KνMC is con-EPr. That is (M/C) isr Sis under which a partitioned matrix is dek-EP matrices ale-mentary summands of S if S = S1 + S2 and con-s-k1-EPnce [M/C] is arbitrary it follows that every secondary submatrix of rank r is con-s-k-EPr. The converse is ob-vious. The conditioncomposed into complementary sum and S of con-s- re given. S1 and S2 and called comp12.ρSρSρS Theorem 2.21 Let S be of the form 2.2 with ,ρSρMC ρSMC †(SMCMBMAMCMD where and K is of the form 2.3 and V is of the fo rm 2.4. If ( M/C) is con-s-k1-EP and SMCis con-s-k2-EP matrices such that  ††12TMA MCνKMCMDνKand  ††21TMDS MCνKSMCMCνK thProof Let us consider the matrices, en S can be decomposed into complementary sum-mands of con-s-k-EP matrices.  †††1MCMCMCMDSMAMC MCMAMC MDand  †0IMCMC2†)MDSMASMCIMCMC. toTaking in account that  †MCNMAMCMA CvKNMA MCMCvKand †TNM 11†11†††1††††SMCMAMCMD MAMMAMC MDMAMCMCMC MDMA MCMDMAMCMCMAMC MMAMCMD ††CMDMC MCMCMD†MC MD0MC MCDCopyright © 2012 SciRes. ALAMT B. K. N. MUTHUGOBAL 11 ain by  that We obt 1.ρSρMC Since (M/C) is con-s-k1-EP and ††1†1†1††TTC MCMCMC νKMA MCνKMC MDνK at is S is con-s-k1-EP. ††1M AMCMνKMA MC2MC MCMCMD νKWe have by Theorem 2.15, th1Since  ,ρSρMC ρSMC Theorem 1 of , gives  †,NS MCNIMCMCMD  †TTNS MCNMCIMCMC  and †††0IMCMC MDSMCIMC MC  Therefore, 2SSMC0. Thus by  we get  2.ρSρSMC Thus 12.ρSρSρS Further using  †11MC MCKνKνMCMC We obtain,      ††2††††11† †† †11††11(TTTTTTTTIMCMCMDSMCνKIMCMCSMC MAνKSMCvAνKIMC MCS MCvAIMCMCKνSMC vAKνMC MC KνSMCMAKνKνMC MC         1  AIMCC νK††1†1T TTTTSMCMAKνIMCMCM  REFERENCES  S. Krishnamoorthy, K. Gunasekaran and B. K. N. Mut-hugobal, “con-s-k-EP Matries,” Journal of Mathematical Sciences and Engineering Applications, Vol. 5, No. 1, 2011, pp. 353-364.  C. R. Rao and S. K. Mitra, “Generalized Inverse of Ma-trices and Its Applications,” Wiley and Sons, New York, 1971.  ar ix Equations,” Mathematical Proceedings of the , Vol. 521, 1959,  T. S. Baskett and I. J. Katz, “Theorems on Products of EPr Matrices,” Linear Algebra and Its Applicationsol. 2, No. 1, 1969, pp. 87-103. A. R. Meenakshi, “On Schur Complements in an EP Ma-trix, Periodica, Mathematica Hungarica,” Periodica Mathematica Hungarica, Vol. 16, No. 3,, pp. 193- 200.  D. H. Carlson, E. Haynesworth and T. H. Markham, “A ation of the Schur Complme nt by Means of t he nrose Inverse,” SIAM Journal on Applied Ma- , 1974, pp. 169-175.  Greviue, “Generalized InversTheory and Applications,” Wiley and Sons, New York,  y, K. Gunasekaran and B. K. N. Mut-hugobal, “On Sums of con-s-k-EP Matrix Thai Journal atics, in Press, 2012. †SMC M R. Penrose, “On Best Approximate Solutions of LineMatr , No. Cambridge Philosophical Societypp. 17-19. , V 1985Generaliz eMoore-Pethematics, Vol. 26, No. 1 A. B. Isral and T. N. E.es 1974. S. Krishnamoorth,”of MathemCopyright © 2012 SciRes. ALAMT