We denote by H0 the subclass of H-matrices consisting of all the matrices that lay simultaneously on the classes of doubly diagonally dominant (DDD) matrices (A = [aij ] ∈ Cn×n : |aii||ajj | ≥  k =i |aik| k =j |ajk|, i = j) and S-strictly diagonally dominant (S-SDD) matrices. Notice that strictly doubly diagonally dominant matrices (also called Ostrowsky matrices) are a subclass of H0. Strictly diagonally dominant matrices (SDD) are also a subclass of H0. In this paper we analyze some properties of the class H0 = DDD ∩ S-SDD

Bru García, Rafael

Cvetkovic, Ljiljana

Kostic, Vladimir

Pedroche Sánchez, Francisco

idUS. Depósito de Investigación Universidad de Sevilla

XX Congreso de Ecuaciones Diferenciales y AplicacionesX Congreso de Matema´tica AplicadaSevilla, 24-28 septiembre 2007(pp. 1–7)On the intersection of the classes of doubly diagonallydominant matrices and S-strictly diagonally dominantmatricesF. Pedroche1, R. Bru1, L. Cvetkovic´2, V. Kostic´21 Institut de Matema`tica Multidisciplinar, Universitat Polite`cnica de Vale`ncia. Camı´ de Vera s/n.46022 Vale`ncia. Spain. E-mails: pedroche@imm.upv.es, rbru@imm.upv.es.2 Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad. Serbia, 21000Novi Sad. E-mails: lilac@sbb.co.yu, vkostic@im.ns.ac.yu.Keywords: H-matrices, Doubly diagonally matrices, S-strictly diagonally dominant matricesAbstractWe denote by H0 the subclass of H-matrices consisting of all the matrices that laysimultaneously on the classes of doubly diagonally dominant (DDD) matrices (A =[aij ] ∈ Cn×n : |aii||ajj | ≥∑k =i |aik|∑k =j |ajk|, i = j) and S-strictly diagonallydominant (S-SDD) matrices. Notice that strictly doubly diagonally dominant matrices(also called Ostrowsky matrices) are a subclass of H0. Strictly diagonally dominantmatrices (SDD) are also a subclass of H0. In this paper we analyze some propertiesof the class H0 = DDD ∩ S-SDD.1 IntroductionIn this paper we analyze some properties of the matrices that lay simultaneously on theclasses of doubly diagonally dominant (DDD) matrices, see [11], and S-strictly diagonallydominant (S-SDD) matrices; see [4], [15]. This class, that we denote here by H0 =DDD ∩ S-SDD is a subclass of H-matrices. In several practical applications H-matricesplay a key role; e.g., in the numerical solution of Euler equations in ﬂuid dynamics [7],in nonlinear boundary problems and in the Lyapounov stability analysis for large scaleevolution systems (see [14] and the references therein, for more details). H-matriceswere deﬁned by Ostrowsky in [13] as a generalization of M -Matrices. H-matrices andM -matrices are called this way in homage to Hadamard and Minkowsky, respectively [15].We recall that a nonsingular matrix A having all non-positive oﬀ-diagonal entries iscalled an M -matrix if the inverse is (entry-wise) nonnegative, i.e., A−1 ≥ O; see, e.g.,1F. Pedroche, R. Bru, L. Cvetkovic´, V.Kostic´[1] for more characterizations. For any matrix A = (aij) ∈ Rn×n, its comparison matrix〈A〉 = (αij) can be deﬁned byαii = |aii|, αij = −|aij |, i = j.A matrix A is said to be an H-matrix if 〈A〉 is a nonsingular M -matrix. In particular,A is a nonsingular H-matrix if and only if it is (strictly) generalized (row) diagonallydominant, i.e.,|aii|wi >∑i=j|aij |wj, i = 1, . . . , n, (1)for some positive vector w = (w1, . . . , wn)T . This is equivalent to say that A is an H-matrixif and only if there exists a positive diagonal matrix W = diag(w1, w2, . . . , wn) such thatAW is an strictly (row) diagonally dominant (SDD) matrix. Some useful characterizationsof H-matrices (see, for example, [10], [8], [14], [9], [5]) are based on devising adequatescaling matrices W . A diﬀerent strategy to the problem of ﬁnding classes of H-matricesresides in describing subclasses of H-matrices which are easily characterizable. Followingthis approach some new subclasses of H-matrices were introduced in [4]. In this paper wefocus on the subclass of H0-matrices. It is also interesting to note that SDD matrices arethe simplest case for this class; these ideas are depicted in Figure 1 below.S-SDDSDDOstrDDDDDDH0HFigure 1: DDD matrices and some subclasses of H-matrices2 S-SDD matricesWe begin with some deﬁnitions which can be found, e.g., in [2], [4], [6], [15].Definition 1 Given a matrix A = (aij) ∈ Cn×n, let us define the ith deleted absolute rowsum asri(A) =n∑j =i, j=1|aij |, ∀i = 1, 2, . . . , n,2On the intersection of the classes of DDD and S-SDDand the ith deleted absolute row-sum with columns in the set of indicesS = {i1, i2, . . .} ⊆ N := {1, 2, . . . n} asrSi (A) =∑j =i, j∈S|aij|, ∀i = 1, 2, . . . , n.Given any nonempty set of indices S ⊆ N we denote its complement in N by S¯ := N\S.Note that for any A = (aij) ∈ Cn×n we have that ri(A) = rSi (A) + rS¯i (A).Definition 2 Given a matrix A = (aij) ∈ Cn×n, n ≥ 2 and given a nonempty subset Sof {1, 2, . . . , n}, then A is an S-strictly diagonally dominant matrix if the following twoconditions holdi) |aii| > rSi (A) ∀i ∈ S,ii) (|aii| − rSi (A)) (|ajj | − rS¯j (A)) > rS¯i (A) rSj (A) ∀i ∈ S,∀j ∈ S¯.}(2)It was shown in [6] that an S-strictly diagonally dominant matrix (S-SDD) is a nonsin-gular H-matrix. In particular, when S = {1, 2, . . . n}, then A = (aij) ∈ Cn×n is a strictlydiagonally dominant matrix (SDD). It is easy to show that an SDD matrix is an S-SDDmatrix for any proper subset S, but the converse is not always true [3].Notice that condition 1) of deﬁnition 2 implies that the diagonal of any S-SDD matrixis nonzero. We also note that condition 1) can be substituted for |aii| > rSi (A), for somei ∈ S, since the condition 2) ensures that 1) will be satisﬁed for all i ∈ S; see [4].The class of S-SDD can be expressed equivalently in the following way. For arbitrarynonempty proper set of indices S let us deﬁne the interval JA(S) asJA(S) := (µS1 (A), µS2 (A)), (3)whereµS1 (A) := maxi∈SrSi (A)|aii| − rSi (A)and µS2 (A) := minj∈S,rSj (A)=0|ajj| − rSj (A)rSj (A). (4)By convention, when S = ∅ or S = N we deﬁne JA(S) = (0,+∞). Furthermore, whenrSj (A) = 0, ∀ j ∈ S then we take µS2 (A) = +∞.The next lemma, which is proved in [2], shows another characterization of S-SDDmatrices. Here we denote by A[S] the principal submatrix of A with indices from the setS.Lemma 1 Given S ∈ N , let A[S] and A[S] be strictly diagonally dominant matrices.Then A ∈ Cn×n is an S-SDD matrix if and only if the interval JA(S) given by (3) isnonempty.3 Doubly diagonally dominant matricesThe class of DDD matrices, see [11], is deﬁned as follows.{A = [aij] ∈ Cn×n : |aii||ajj | ≥ ri(A) rj(A), i = j} (5)3F. Pedroche, R. Bru, L. Cvetkovic´, V.Kostic´Example 1 The matrices[0 00 0]and[ −1 00 0]are DDD matrices.Example 2 The matrix 1 1 012 1120 12 1is DDD but it is not into the class H0, i.e., is not an S-SDD matrix for any S. But it isa nonsingular H matrix.We remark that we are interested in DDD matrices with at least one equality in (5).Otherwise, we would have SDDD (Ostrowsky) matrices or simply SDD matrices, whichare known classes.4 H0-matricesIn order to study the class H0 = DDD ∩ S-SDD we can adopt three points of view: a) wecan stay in the DDD class and look for conditions to be in the class S-SDD, b) we canstay in the class S-SDD and look for conditions to be in the DDD class and c) we canimpose all the conditions to be in the class DDD∩S-SDD and try to simplify the derivedrelations.In this communication we explore the options a) and c).Before giving suﬃcient conditions for a DDD matrix to be an S-SDD matrix we es-tablish the following result.Lemma 2 Let A ∈ Cn×n and S ⊆ N := {1, 2, . . . , n}. If1) A[S] and A[S] are SDD matrices2) rSi (A) rj(A) > rSj (A) |aii|, ∀i ∈ S, ∀j ∈ S3) ri(A) rSj (A) > rSi (A) |ajj |, ∀i ∈ S, ∀j ∈ Sthen A is an S-SDD matrix.Proof We ﬁrst note that 1) implies: |aii| > rSi (A), ∀i ∈ S and |ajj| > rSj (A), ∀j ∈ S.According to Lemma 1, we only have to show that the interval JA(S) given by equation(3) is nonempty. Note that condition 2) can be written asrSi (A) rSj (A) > rSj (A)[|aii| − rSi (A)] , ∀i ∈ S, ∀j ∈ S (6)and since A[S] is SDD, equation (6) implies that rSi (A) rSj (A) > 0, ∀i ∈ S, ∀j ∈ S.Now, from (6) and the deﬁnition of µS1 (A), see equation (4), we conclude thatµS1 (A) >rSi (A) rSj (A)rSi (A) rSj (A), ∀i ∈ S, ∀j ∈ S4On the intersection of the classes of DDD and S-SDDIn a similar way, condition 3) yields torSi (A) rSj (A) > rSi (A)[|ajj | − rSj (A)], ∀i ∈ S, ∀j ∈ S (7)and this equation jointly with the deﬁnition of µS2 (A), equation (4), leads toµS2 (A) <rSi (A) rSj (A)rSi (A) rSj (A), ∀i ∈ S, ∀j ∈ Sand the proof follows.In the following result we show that when A is a DDD matrix then we can replace thecondition 1) of Lemma 2 by the simple condition |aii| > rSi (A) for some i ∈ S.Proposition 1 Let A ∈ Cn×n be a DDD matrix. Let S ⊆ N := {1, 2, . . . , n}. If1) |aii| > rSi (A) for some i ∈ S2) rSi (A) rj(A) > rSj (A) |aii|, ∀i ∈ S, ∀j ∈ S3) ri(A) rSj (A) > rSi (A) |ajj |, ∀i ∈ S, ∀j ∈ Sthen A is an S-SDD matrix.Proof Since A is a DDD matrix we have that|aii||ajj | ≥ ri(A) rj(A), i = jNote that|aii||ajj| ≥ ri(A) rj(A)= [rSi (A) + rS¯i (A)] [rSj (A) + rS¯j (A)]= rSi (A)rSj (A) + rSi (A)rS¯j (A) + rS¯i (A)rSj (A) + rS¯i (A)rS¯j (A)= rSi (A)rj(A) + rSi (A)rS¯j (A)− rSi (A)rS¯j (A) + rS¯i (A)rSj (A) + rS¯i (A)rS¯j (A)= rSi (A)rj(A) + ri(A)rS¯j (A)− rSi (A)rS¯j (A) + rS¯i (A)rSj (A)(8)and using conditions 2) and 3) we conclude|aii||ajj | > rS¯j (A)|aii|+ rSi (A)|ajj | − rSi (A)rS¯j (A) + rS¯i (A)rSj (A)from which we obtain(|aii| − rSi (A)) (|ajj | − rS¯j (A)) > rS¯i (A)rSj (A)and this holds ∀i ∈ S, ∀j ∈ S. In conclusion, we have that A is an S-SDD matrix.5F. Pedroche, R. Bru, L. Cvetkovic´, V.Kostic´In the next section we will show some properties of the matrices that lay on the classH0.4.1 Set of pairs of indicesIn this section we consider N := {1, 2, . . . , n}, such that n ≥ 2. Let us deﬁne the setN2 = {(i, j) : i, j ∈ N, i = j}. Obviously, card(N2) = n(n−1)2 .Definition 3 Let A ∈ Cn×n be a DDD matrix such that n ≥ 2. We define the set of pairsof indicesE(A) = { (i, j) ∈ N2 : |aii||ajj| = ri(A) rj(A) } .We denote its complement by E(A) = N2\E(A).Example 3 Given the following DDD matrixA = 1 0.5 0.50.5 1 0.50 1 2we have N2 = {(1, 2), (1, 3), (2, 3)} and E(A) = {(1, 2)}.Definition 4 We define the class of matrices H0(S) which is formed by square matricesA of order n such that they are simultaneously DDD matrices and S-SDD matrices forsome proper subset S ⊆ N .Example 4 The matrix given by example 3 is DDD and {1, 2}-SDD, therefore it belongsto the class H0({1, 2}).Lemma 3 Let A ∈ Cn×n such that A ∈ H0(S) for some proper subset S and such thatthere exists i ∈ N : ri(A) = 0. Then (i, j) ∈ E(A), ∀j ∈ N .Proof Let us suppose that there exists j ∈ N : (i, j) ∈ E(A). Therefore |aii||ajj| =ri(A) rj(A) = 0 which implies aii = 0 or ajj = 0. But this is a contradiction because A isa nonsingular H-matrix.Remark 1 Note that this lemma still holds when A is a DDD matrix whose diagonalentries are nonzero.Lemma 4 Let A ∈ Cn×n such that A ∈ H0(S) for some proper subset S and such thatthere exists i ∈ S : |aii| = ri(A). Then (i, j) ∈ E(A), ∀j ∈ S.Proof Let us suppose that there exists j ∈ S : (i, j) ∈ E(A). Therefore |aii||ajj | =ri(A) rj(A) and using the hypothesis |aii| = ri(A) we conclude that |ajj| = rj(A). There-fore we have(|aii| − rSi (A)) (|ajj | − rS¯j (A)) = rS¯i (A) rSj (A), with i ∈ S, j ∈ Sand the condition ii) of the deﬁnition of S-SDD matrices is not satisﬁed. Therefore Adoes not belong to H0(S), which is a contradiction.The counterpart of the previous lemma is the following.6On the intersection of the classes of DDD and S-SDDLemma 5 Let A ∈ Cn×n such that A ∈ H0(S) for some proper subset S and such thatthere exists i ∈ S : |aii| = ri(A). Then (i, j) ∈ E(A), ∀j ∈ S.As a consequence of the two previous results we have the following.Proposition 2 Let A ∈ Cn×n such that A ∈ H0(S) and let T be the set of indicesT = {i ∈ N : |aii| = ri(A)}.Then T ⊆ S or T ⊆ S.AcknowledgementF. Pedroche and R. Bru are supported by the Spanish DGI grant MTM2004-02998. L.Cvetkovic´ and V. Kostic´ are supported by the Provincial Secretariat of Science and Tech-nological Development of Vojvodina, Serbia, grant 01123.Bibliography[1] A. Berman, and R. J. Plemmons, Nonnegative matrices in the mathematical sciences, AcademicPress, New York. Reprinted and updated, SIAM, Philadelphia, 1994.[2] R. Bru, L. Cvetkovic., V. Kostic and F. Pedroche. Sums of S-strictly diagonally dominant matricesElectron. Trans. Numer. Anal., 2007 (submitted).[3] R. Bru, F. Pedroche, and D. B. Szyld. Subdirect sums of S-Strictly Diagonally Dominant matrices,Electron. J. Linear Algebra, 15:201–209, 2006.[4] L. Cvetkovic. H-matrix theory vs. eigenvalue localization Numerical Algorithms, 42: 229-245, 2006.[5] L. Cvetkovic and V. Kostic. New criteria for identifying H-matrices, J. Comput. Appl. Math.,180:265–278, 2005.[6] L. Cvetkovic, V. Kostic and R. S. Varga. A new Gersˇgoring-type eigenvalue inclusion set, Electron.Trans. Numer. Anal., 18:73–80, 2004.[7] L. Elsner and V. Mehrmann. Convergence of Block-Iterative Methods for Linear Systems Arising inthe Numerical Solution of Euler Equations. Numerische Mathematik, Vol. 59, pp. 541-560, 1991.[8] T. B. Gan, and T. Z. Huang. Simple criteria for nonsingular H-matrices, Linear Algebra Appl.,374:317–326, 2003.[9] T-Z Huang, J-S Leng, E.L. Wachspress and Y. Y. Tang. Characterization of H-matrices. Computers& Mathematics with applications, 48 (10-11): 1587-1601, 2004.[10] B. Li, L. Li, M. Harada, H. Niki, M. J. Tsatsomeros, An iterative criterion for H-matrices, LinearAlgebra Appl., 271:179–190, 1998.[11] B. Li and M. J. Tsatsomeros. Doubly diagonally dominant matrices. Linear Algebra and its Appli-cations, 261:221–235, 1997.[12] J. Liu and Y. Huang. Some properties on Schur complements of H-matrices and diagonally dominantmatrices. Linear Algebra and its Applications, 389:365–380, 2004.[13] A. M. Ostrowski, (1937), U¨ber die Determinanten mit u¨berwiegender Hauptdiagonale, ComentariiMathematici Helvetici 10 pp. 69–96.[14] P. Spiteri. A new characterization of M-matrices and H-matrices. BIT Numerical Mathematics,43:1019-1032, 2003.[15] R. S. Varga. Gersˇgorin and his circles. Springer Series in Computational Mathematics, vol. 36.Springer, Berlin, Heidelberg, 2004.7

On the intersection of the classes of doubly diagonally dominant matrices and S-strictly diagonally dominant matrices

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On the intersection of the classes of doubly diagonally dominant matrices and S-strictly diagonally dominant matrices

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