Research Article The Diagonally Dominant Degree and Disc Separation for the Schur Complement of Ostrowski Matrix

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1 Hindawi Pubishing Corporation Journa of Appied Mathematics Voume 2013, Artice ID , 10 pages Research Artice The Diagonay Dominant Degree and Disc Separation for the Schur Compement of Ostrowsi Matrix Jianxing Zhao, 1 Feng Wang, 1,2 and Yaotang Li 1 1 Schoo of Mathematics and Statistics, Yunnan University, Kunming, Yunnan , China 2 Department of Mathematics, Heze University, Heze, Shandong , China Correspondence shoud be addressed to Yaotang Li; iyaotang@ynueducn Received 25 Apri 2013; Accepted 31 August 2013 Academic Editor: Kazutae Komori Copyright 2013 Jianxing Zhao et a This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina wor is propery cited By appying the properties of Schur compement and some inequaity techniques, some new estimates of diagonay and douby diagonay dominant degree of the Schur compement of Ostrowsi matrix are obtained, which improve the main resuts of Liu and Zhang (2005 and Liu et a (2012 As an appication, we present new incusion regions for eigenvaues of the Schur compement of Ostrowsi matrix In addition, a new upper bound for the infinity norm on the inverse of the Schur compement of Ostrowsi matrix is given Finay, we give numerica exampes to iustrate the theory resuts 1 Introduction Let C n n denote the set of a n ncompex matrices, N= 1,2,,n},andA=(a ij C n n (n 2Denote R i = a ij (1 j =i We now that A is caed a stricty diagonay dominant matrix if a ii >R i, i N (2 A is caed a generaized Ostrowsi matrix if a ii a jj R i R j, i,j N, i=j (3 A is caed Ostrowsi matrix if a strict inequaities in (3 hod (see 1] SD n and OS n (GOS n wi be used to denote the sets of a n nstricty diagonay dominant matrices and the sets of a n n(generaized Ostrowsi matrices, respectivey Asshownin2], for a i N,weca a ii R i and a ii a jj R i R j the ith diagonay and douby diagonay dominant degree of A,respectivey The infinity norm of A is defined as A = max R i + 1 i n a ii } (4 For β N, denote by β the cardinaity of β and β= N/βIfβ, γ N,thenA(β, γ is the submatrix of A with row indices in β andcoumnindicesinγinparticuar,a(β, β is abbreviated to A(β Assuming that β=i 1,i 2,,i } N, β=n/β=j 1,j 2,,j } and the eements of β and β are both conventionay arranged in an increasing order For 1 t, we denote If A(β is nonsinguar, A t =A(β j t } (5 A β = A A(β =A(β A (β, β A (β] 1 A(β,β (6 is caed the Schur compement of A with respect to A(β The comparison matrix of A, μ = (α ij, is defined by α ij = a ij, if i=j, a ij, if i =j (7 AmatrixA=(a ij C n n is caed an M-matrix if there exist a nonnegative matrix B and a number s > ρ(b such that A=sI B,whereρ(B is the spectra radius of B Wenow that A is an H-matrix if and ony if μ is an M-matrix, and if

2 2 Journa of Appied Mathematics A is an M-matrix, then the Schur compement of A is aso an M-matrix and det A>0(see 3] H n and M n wi denote the set of a n n H-matrices and the set of a n n M-matrices, respectivey The Schur compement has been proved to be a usefu too in many fieds such as contro theory, statistics, and computationa mathematics A ot of wor has been done on it (see 2, 4 15] It is we nown that the Schur compements of SD n and OS n are SD n and OS n, respectivey These properties have been used for the derivation of matrix inequaities in matrix anaysis and for the convergence of iterations in numerica anaysis (see 16 19] Meanwhie, estimating the upperboundfortheinfinitynormoftheinverseoftheschur compement is of great significance We now that the upper bound of A 1 pays an important roe in some iterations forargescaenonhomogeneoussystemofinearequation Ax=b(see 20] The paper is organized as foows In Section 2, wegive severa new estimates of diagonay and douby diagonay dominant degree on the Schur compement of matrices In Section3, new incusion regions for eigenvaues of the Schur compement are obtained A new upper bound of (A/β 1 is given in Section 4 InSection 5, we present numerica exampes to iustrate the theory resuts 2 The Diagonay Dominant Degree for the Schur Compement In this section, we give severa new estimates of diagonay anddoubydiagonaydominantdegreeontheschurcompement of OS n Lemma 1 (see 3] If A H n,thenμ] 1 A 1 Lemma 2 (see 3] If A SD n or A OS n,thena H n ;that is, μ M n Lemma 3 (see 6] If A SD n or A OS n and β N,then the Schur compement of A is in SD β or OS β,whereβ=n β is the compement of β in N and β is the cardinaity of β Lemma 4 (see 12] Let A SD n, β=i 1,i 2,,i } N, β=j 1,j 2,,j },and+=nforanyj t β,denote B jt x a i 1 j V ( μ(a (β ( a i j V a j t i 1 Then B jt GOS +1 if and ony if x max 1 w R iw a i w i w a j t i, x>0 (8 a j t i V (9 When the strict inequaity in (9 hods, B jt M +1,andthus det B jt >0Iftheequaityin(9 occurs, then det B jt 0 Lemma 5 Let A=(a ij OS n and β=i 1,i 2,,i } with an index i d (1 d satisfying a id i d R id, a id i d > iu β/i d } a i d i u, β=j 1,j 2,,j }, 1 <n,anda/β = (a ts Then,fora1 t, where a tt R t ( A β a j t j t R j t h=max max i N/i d } + a i d i d P i d a i d i d a j t j t P i d a R jt i d i d >0, a ii d a ii j N/i,i d } a ij, a j t i V a i d i d R id }, P id =hr id (10 (11 Proof From Lemmas 2 and 3, wenowthata(β H and μ(a(β M Further,byLemma 1,wehave Thus, for any 1 t, a tt R t ( A β = a tt μ(a (β] 1 A (β] 1 (12 a ts s=1, =t a i1 j t = a jt j t,,a jt i A(β] 1 ( a i j t a jt j s,,a jt i A(β] 1 ( s =t a j t j t R j t + a i d i d P i d a i d i d a i1 j s a i j s a j t i V + P i d a a j t i V i d i d a i 1 j s ( a j t i 1,, a j t i μ(a (β] 1 ( s=1 a i j s (13

3 Journa of Appied Mathematics 3 Further, a tt R t ( A β a j t j t R j t + a i d i d P i d a i d i d + 1 det μ(a (β] a j t i V P id a a j t i V a j t i 1 a j t i i d i d det a i 1 j s s=1 ( μ(a (β ( a i j s s=1 def = a j t j t R j t + a i d i d P i d a i d i d a j t i V 1 + det μ(a (β] det B (14 where a tt a ss +R t ( A β R s ( A β a j t j t + max P u ] P i =h a j s j s + max j N/i,i d } ], a ij + a ii d (i =i d, and P id and h are such as in Lemma 5 (17 (18 (b If a id i d >R id for any i d β (1 d,then,for a 1 s, t, t =s, a tt a ss R t ( A β R s ( A β a j t j t max R u ] a j s j s max ], (19 By Lemma 4, we can prove that det B 0Thus,inequaity (10hods Remar 6 Note that P id a R i d i d i d a (15 i d i d a tt a ss +R t ( A β R s ( A β a j t j t + max R u ] a j s j s + max ], (20 This shows that Lemma 5 improves Theorem 2 of 12] Theorem 7 Let A=(a ij OS n, β=i 1,i 2,,i } N, β=n/β=j 1,j 2,,j }, 1 <n,anda/β = (a ts (a If there exists an i d β(1 d such that a id i d R id,then,fora1 s, t, t =s, a tt a ss R t ( A β R s ( A β a j t j t max P u ] a j s j s max ], (16 where η=max max a i ω j V 1 ω a i ω i ω t =ω a, i ω i t max 1 ω 1 V a i ω j V } a }, i ω j V} Q iω =η a i ω i t + a i ω j V, t =ω 1 ω, (21 and if there exists some 1 ω such that a i ω j V =0, one denotes η=1

4 4 Journa of Appied Mathematics Proof (a If there exists an i d βsuch that a id i d R id, then, for a j t β, P max u u N/j t} a By Lemma 5,fora1 t, = max a = P i d a (22 i d i d a tt R t ( A β a j t j t P i d a R jt >0 (23 i d i d Thus, for a 1 t, s, t =s, a tt R t ( A β ] a ss R s ( A ]>0 (24 β a js j V (a js i 1,,a js i V =s a i1 j V A(β] 1 ( ] a i j V ] a jt j u,,a jt i u =t a i1 j u A(β] 1 ( ] a i j u ] From Lemma 3, A/β is in OS β ;thatis,fora1 t, s, t =s, a tt a ss R t ( A β R s ( A >0 (25 β Further, for a 1 t, s, t =s, a tt a ss R t ( A β R s ( A β a tt R t ( A β ] a ss R s ( A β ] a j t j t max P u ] a j s j s max ] (26 Therefore, inequaity (16 hods Simiary, we can prove inequaity (17 (b If a id i d >R id for any i d β (1 d, then, from Lemmas 1 and 2,fora1 t, s, t =s, a tt a ss R t ( A β R s ( A β a i1 j s = a js j s (a js i 1,,a js i A(β] 1 ( a i j s a i1 j t a jt j t,,a jt i A(β] 1 ( a i j t a j s j s ( a j s i 1,, a j s i a i 1 j s μ(a (β] 1 ( ] ] a i j s ] a j t j t ( a j t i 1,, a j t i a i 1 j t μ(a (β] 1 ( ] ] a i j t ] def a j s j V +( a j s i 1,, a j s i V =s a i 1 j V μ(a (β] 1 ( ] } ] a } i j V ]} a j t j u +( a j t i 1,, a j t i u =t a i 1 j u μ(a (β] 1 ( ] } ] a } i j u ]} =ξ= a j s j s ( a j s i 1,, a j s i

5 Journa of Appied Mathematics 5 Therefore, a i 1 j s μ(a (β] 1 ( ] ] a i j s ] a j t j t max R u R + max u a i 1 j t ( a j t i 1,, a j t i μ(a (β] 1 ( ] ] a i j t ] a j s j V ( a j s i 1,, a j s i V =s a i 1 j V μ(a (β] 1 ( ] } ] a } i j V ]} a j t j u ( a j t i 1,, a j t i u =t ξ= a j s j s ( a j s i 1,, a j s i a i 1 j u μ(a (β] 1 ( ] } ] a } i j u ]} a i 1 j s μ(a (β] 1 ( ] ] a i j s ] a j t j t max R u ] + a j s j s ( a j s i 1,, a j s i a i 1 j s μ(a (β] 1 ( ] ] a i j s ] (27 Further, max u N/j t} R u a R jt a i 1 j t ( a j t i 1,, a j t i μ(a (β] 1 ( ] ] a i j t ] a j s j V ( a j s i 1,, a j s i V =s a i 1 j V μ(a (β] 1 ( ] } ] a } i j V ]} a j t j u ( a j t i 1,, a j t i u =t a i 1 j u μ(a (β] 1 ( ] } ] a } i j u ]} a i 1 j s a j s j s ( a j s i 1,, a j t i μ(a (β] 1 ( = a j s j s max a a j s i V + max a i j s a a j s i V a i 1 j s (η a j s i 1,,η a j s i ημ(a (β] 1 ( = a j s j s max det ( def ( max = a j s j s max a a a j s i V η a j s i 1 i V i V a i j s a j s i V + 1 det ημ(a (β] a i 1 j s ημ (A (β a i j s a η a j s i (28 a j s i V + 1 det ημ(a (β] det B 1 (29

6 6 Journa of Appied Mathematics In B 1,forap=1,2,3,,, η a i p i p max a a j s i V η a Q ip i p i p a a j s i V i p i p =ηq ip a j s i V =η(η a i p i V + a i p j V a j s i V V =p (η a i p i V + a i p j s ηa j s i V V =p And for a p,q=1,2,3,,, p =q, η a i p i p η a i q i q >ηr i p ηr iq (30 =(η a i p i V +η ai j V (η p a i q i V +η a i q j V V =p V =q (η ai i V + p ai j s (η p a i q i V + a i q j s V =p V =q (31 Hence, by (30and(31, we have B 1 GOS +1 and so det B 1 0Further,by(29, we obtain a i 1 j s a j s j s ( a j s i 1,, a j t i μ(a (β] 1 ( a j s j s max a a j s i V a i j s (32 a j s j s max By (28 and a simiar method as the proof of Theorem 21 in 2], we can prove ξ>0 Therefore, by (29 and(32, we obtain inequaity (19 Simiary, we can prove inequaity (20 Remar 8 Note that 0 h, η 1 (33 This shows that Theorem 7 improves Theorem 21 of 2] 3 Eigenvaue Incusion Regions of the Schur Compement In this section, we present new incusion regions for eigenvauesoftheschurcompementofos n Lemma 9 (Brauer Ovas theorem Let A=(a ij C n n Then the eigenvaues of A areintheunionofthefoowingsets: U ij =z C z a ii z a jj R i R j }, i, j = N, i =j (34 Theorem 10 Let A=(a ij OS n, β=i 1,i 2,,i } N, β=n/β=j 1,j 2,,j }, 1 <n,anda/β = (a ts,andet λ be eigenvaue of A/β (a If there exists an i d β (1 d such that a id i d R id, then there exist 1 t, s, t =s,suchthat λ det (A t λ det (A s det A(β det A(β 2 a j s j s max u N/j t } + a j t j t max P u a R jt ], λ det (A t λ det (A s det A(β det A(β a j t j t + max P u ] a j s j s + max ], (35 (36 where P id is such as in Lemma 5 and (V =d is suchas intheorem 7 (b If a id i d >R id for any i d β (1 d, then there exist 1 t, s, t =s,suchthat λ det (A t λ det (A s det A(β det A(β 2 a j s j s max u N/j t} + a j t j t max R u a R jt ], λ det (A t λ det (A s det A(β det A(β a j t j t + max R u ] a j s j s + max ], where is such as in Theorem 7 (37 (38

7 Journa of Appied Mathematics 7 Proof By Lemma 9, we now that there exist 1 t, s, t =s,suchthat λ a tt λ a ss R t ( A β R s ( A β (39 (a If there exists i d βsatisfying a id i d R id,by(16, we have λ a tt On the other hand, for a 1 t, = λ a jt j t +(a jt i 1,,a jt i A(β] 1 ( a i j t = λ det ( A t A(β = λ det (A t det A(β a i1 j t (41 R t ( A β R s ( A β a tt a ss a j t j t max P u ] a j s j s max ] a i1 j s = a js j s (a js i 1,,a js i A(β] 1 ( a i j s a i1 j t a jt j t,,a jt i A(β] 1 ( a i j t a j t j t max P u ] a j s j s max ] a j t j t + max P u ] a j s j s + max ] a j t j t max P u ] a j s j s max =2 a j s j s max u N/j t} + a j t j t max ] P u a R jt ] (40 Therefore, by (39, (40, and (41, we obtain inequaity (35 With a Simiar method, we can prove inequaity (36 (b If a id i d >R id for any i d β (1 d, then by (19, (32, and a simiar method as the part (a, we obtain inequaity (37 Simiary, we can prove inequaity (38 4 Upper Bound for the Infinity Norm on the Inverse of the Schur Compement In this section, we present a new upper bound of (A/β 1 Lemma 11 (see 2] Let A=(a ij OS n and M=(m ij C n n Then, A 1 M max 1 i,j n i =j a jj n m iv +R i n m jv a ii a jj R i R j (42 Theorem 12 Let A=(a ij SD n, M=(m ij C, β= i 1,i 2,,i } N, β=n/β=j 1,j 2,,j }, 1 <n,and A/β = (a ts Then, ( A 1M β max (Δ jt j s 1 t,s t =s (( a j t j t max R u ( a j s j s max 1, (43

8 8 Journa of Appied Mathematics Since A SD n,thena OS n Thus,byTheorem 7,wehave ( A β 1 where max (( a j t j t +R j s Δ jt j s 1 t,s t =s + max a (R jt +R js (( a j t j t max R u =( a j t j t + max +(R js + max and Q iv is such as in Theorem 7 Proof By Lemma 11,wehave ( A 1M β max 1 t,s t =s Simiar to (29, we obtain ( a j s j s max, (44 a R jt m sv m tv, a tt m sv +R s (A/β a tt a ss R t (A/β R s (A/β a i1 j t m tv a tt = a jt j t,,a jt i A(β] 1 ( a i j t 1 (45 (46 a i 1 j t a j t j t +( a j t i 1,, a j t i μ(a (β] 1 ( a j t j t + max a R jt Thus, by Theorem 1 of 12], we have R s ( A β a ss a j s j s +R j s max +R js a i j t (47 (48 a tt a ss R t ( A β R s ( A β a j t j t max R u ] a j s j s max ] (49 Further, by (46, (47, (48, and (49, we obtain inequaity (43 Let M=I=diag(1,1,,1;wecanproveinequaity (44 5 Numerica Exampes In this section, we present severa numerica exampes to iustrate the theory resuts Exampe 1 (see Exampe 2 in 2] Let A=( , β=1, 2} By Theorem 10,theeigenvauesofA/β are in the set Γ 1 = λ λ 187 λ } λ λ 187 λ } λ λ 278 λ } (50 (51 From Theorem 31 of 2], the eigenvaues of A/β are in the set Γ 1 = λ λ 187 λ } λ λ 187 λ } λ λ 278 λ } (52 Evidenty, Γ 1 Γ 1,andweuseFigure 1 to show this fact And the eigenvaues of A/β are denoted by + in Figure 1 Exampe 2 Let A=( , β=2, 4} (53

9 Journa of Appied Mathematics Figure 1: The red dotted ine and bac dashed ine denote the corresponding discs Γ 1 and Γ 1,respectivey Figure 3: The red dotted ine and bac dashed ine denote the corresponding discs Γ 3 and Γ 3,respectivey Figure 2: The red dotted ine and bac dashed ine denote the corresponding discs Γ 2 and Γ 2,respectivey By Theorem 10,theeigenvauesofA/β areintheset Γ 2 = λ λ 149 λ } λ λ 149 λ } λ λ 164 λ } (54 From Theorem 31 of 2], the eigenvaues of A/β are in the set Γ 2 = λ λ 149 λ } λ λ 149 λ } λ λ 164 λ } (55 Evidenty, Γ 2 Γ 2,andweuseFigure 2 to show this fact And the eigenvaues of A/β are denoted by + in Figure 2 Exampe 3 Let A=(, ( β=1, 3, 5} By Theorem 10,theeigenvauesofA/β are in the set Γ 3 = λ λ 116 λ } λ λ 116 λ } λ λ 168 λ } (56 (57 From Theorem 31 of 2], the eigenvaues of A/β are in the set Γ 3 = λ λ 116 λ } λ λ 116 λ } λ λ 168 λ } (58 Evidenty, Γ 3 Γ 3,andweuseFigure 3 to show this fact And the eigenvaues of A/β are denoted by + in Figure 3 Meanwhie, by Theorem 12, ( A β (59 From Theorem 42 of 2], ( A β (60 Remar 13 Numerica exampes show that the new eigenvaue incusion set is tighter than that in Theorem 31 of 2] and the new upper bound of (A/β 1 is sharper than that in Theorem 42 of 2]

10 10 Journa of Appied Mathematics Acnowedgments The authors woud ie to than the anonymous referees for their vauabe suggestions and comments This research is supported by the Nationa Natura Science Foundation of China ( , , IRTSTYN, and Foundation of Yunnan University (2012CG017 References 1] L Cvetović, V Kostić, and S Rauši, A new subcass of Hmatrices, Appied Mathematics and Computation, vo208,no 1,pp ,2009 2] JLiu,JZhang,andYLiu, TheSchurcompementofstricty douby diagonay dominant matrices and its appication, Linear Agebra and its Appications, vo437,no1,pp , ]RAHornandCRJohnson,Topics in Matrix Anaysis, Cambridge University Press, New Yor, NY, USA, ] D Carson and T L Marham, Schur compements of diagonay dominant matrices, Czechosova Mathematica Journa, vo 29, no 2, pp , ] K D Iramov, Invariance of the Brauer diagona dominance in gaussian eimination, Moscow University Computationa Mathematics and Cybernetics,vo2,pp91 94,1989 6] B Li and M J Tsatsomeros, Douby diagonay dominant matrices, Linear Agebra and its Appications,vo261,pp , ] JLiu,JLi,ZHuang,andXKong, SomepropertiesofSchur compements and diagona-schur compements of diagonay dominant matrices, Linear Agebra and its Appications, vo 428, no 4, pp , ] J Liu, Y Huang, and F Zhang, The Schur compements of generaized douby diagonay dominant matrices, Linear Agebra and its Appications,vo378,pp ,2004 9] J Liu and Y Huang, Some properties on Schur compements of H-matrices and diagonay dominant matrices, Linear Agebra and its Appications,vo389,pp , ] J Liu and Z Huang, The Schur compements of γ-diagonay and product γ-diagonay dominant matrix and their disc separation, Linear Agebra and its Appications, vo432,no4, pp , ] J Liu, Z Huang, and J Zhang, The dominant degree and disc theorem for the Schur compement of matrix, Appied Mathematics and Computation,vo215,no12,pp , ] J Liu and F Zhang, Disc separation of the Schur compement of diagonay dominant matrices and determinanta bounds, SIAM Journa on Matrix Anaysis and Appications, vo27,no 3, pp , ] Y-T Li, S-P Ouyang, S-J Cao, and R-W Wang, On diagona- Schur compements of boc diagonay dominant matrices, Appied Mathematics and Computation,vo216,no5,pp , ] R L Smith, Some interacing properties of the Schur compement of a Hermitian matrix, Linear Agebra and its Appications, vo 177, pp , ] F Z Zhang, The Schur Compement and Its Appications, Springer,NewYor,NY,USA, ] J W Demme, Appied Numerica Linear Agebra, Society for Industria and Appied Mathematics (SIAM, Phiadephia, PA, ] G H Goub and C F van Loan, Matrix Computationss, vo 18, Johns Hopins University Press, Batimore, Md, USA, 3rd edition, ] R Kress, Numerica anaysis, vo 181ofGraduate Texts in Mathematics, Springer-Verag, New Yor, ] S Xiang and S Zhang, A convergence anaysis of boc acceerated over-reaxation iterative methods for wea boc Hmatrices to partition π, Linear Agebra and its Appications, vo 418, no 1, pp 20 32, ] N Morača, Upper bounds for the infinity norm of the inverse of SDD and S-SDD matrices, Journa of Computationa and Appied Mathematics,vo206,no2,pp ,2007

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