H-matrix theory vs. eigenvalue localization

Transcription

1 Numer Algor 2006) 42: DOI /s H-matrix theory vs. eigenvalue localization Ljiljana Cvetković Received: 6 February 2006 / Accepted: 24 April 2006 / Published online: 31 August 2006 Springer Science+Business Media B.V Abstract The eigenvalue localization problem is very closely related to the H-matrix theory. The most elegant example of this relation is the equivalence between the Geršgorin theorem and the theorem about nonsingularity of SDD strictly diagonally dominant) matrices, which is a starting point for further beautiful results in the book of Varga [19]. Furthermore, the corresponding Geršgorin-type theorem is equivalent to the statement that each matrix from a particular subclass of H-matrices is nonsingular. Finally, the statement that all eigenvalues of a given matrix belong to minimal Geršgorin set defined in [19]) is equivalent to the statement that every H-matrix is nonsingular. Since minimal Geršgorin set remained unattainable, a lot of different Geršgorin-type areas for eigenvalues has been developed recently. Along with them, a lot of new subclasses of H-matrices were obtained. A survey of recent results in both areas, as well as their relationships, will be presented in this paper. Keywords H-matrices Eigenvalue localization AMS Subject Classification 2000) 15A18 65F15 1. Introduction In the first section a survey of pairs of equivalent statements about H-matrix characterizations and eigenvalue localization will be given. In the second section we are dealing with a possibility for finding a nonsingular diagonal matrix which scales a given H-matrix to strictly diagonally dominant SDD) form. Consequently, we get Communicated by Michael Neumann. Dedicated to Richard S.Varga. L. Cvetković B) Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad, Novi Sad Serbia and Montenegro lila@im.ns.ac.yu

2 230 Numer Algor 2006) 42: corresponding minimal Geršgorin sets with respect to some special classes of diagonal matrices. The third section reveals the relationships between obtained subclasses of H-matrices, as well as eigenvalue localization areas. Some new possibilities for getting H-matrix characterizations, which cannot be used for eigenvalue localization, will be presented in the Section 4. The Section 5 is related to irreducibility, which has its equivalent in obtaining some information about eigenvalues on the boundary. Finally, the Section 6 presents some necessary criteria for a matrix to be an H-matrix. Throughout the paper we will use the following notations: N := {1, 2,..., n} for the set of indices S for any nonempty subset of N S := N \ S for the complement of S r S i A) := k S,k =i r i A) := k N,k =i a ik for ith row sum and a ik for part of ith row sum, which corresponds to the subset S. Obviously, for arbitrary subset S and each index i N, r i A) = r S i A) + rs i A). It is important to emphasize that all the time we are dealing with nonsingular H-matrices, calling them shortly H-matrices. To be precise, we recall the definition of H-matrices. A matrix A = [a ij ] C n,n is called an H-matrix if its comparison matrix A = [m ij ] defined by is a M-matrix, i.e., A 1 0. m ii = a ii, m ij = a ij, i, j = 1, 2,..., n, i = j 2. Pairs of equivalent statements 2.1. SDD matrices As the class of strictly diagonally dominant SDD) matrices plays the central role in these considerations, we shall start with its definition. Definition 1. A matrix A = [a ij ] C n,n is called an SDD matrix if, for each i N, it holds that a ii > r i A). It is well known that Theorem 1. If a matrix A C n,n is an SDD matrix, then it is nonsingular, moreover it is an H-matrix.

3 Numer Algor 2006) 42: It is also well known that this theorem is equivalent with the famous Geršgorin theorem, [11]: Theorem 2. All eigenvalues of the matrix A = [a ij ] C n,n belong to the Geršgorin set ƔA), given by ƔA) := i N Ɣ i A), where Ɣ i A) := {z C : z a ii r i A)}, i N Ostrowski matrices There is the following nonsingularity result of Ostrowski [16]: Theorem 3. For any A = [a ij ] C n,n, n 2, with a ii a jj > r i A) r j A), for all i = j, i, j N, 1) it follows that A is nonsingular, moreover it is an H-matrix. Its equivalent eigenvalue inclusion set is the following result of Brauer [1]: Theorem 4. For any A = [a ij ] C n,n, n 2, and for any eigenvalue λ of A, there is a pair of distinct integers i and j in N such that λ K i, j A) := {z C : z a ii z a jj r i A) r j A)}. Consequently, all the eigenvalues belong to the set KA) := K i, j A). i, j N i = j The class of matrices described by 1) is, in fact, a subclass of H-matrices, and in recent literature, see, for example, [13], they are called strictly doubly diagonally dominant matrices Dashnic Zusmanovich matrices Dashnic and Zusmanovich, [7], obtained both of the following equivalent results: Theorem 5. For any A = [a ij ] C n,n, n 2, for which there exists an index i N such that a ii a jj r j A) + a ji ) > r i A) a ji, for all j = i, j N, it follows that A is nonsingular, moreover it is an H-matrix.

4 232 Numer Algor 2006) 42: Theorem 6. All eigenvalues of a matrix A = [a ij ] C n,n, n 2, belong to the set where DA) := i N D i, j A), 2) j N i = j D i, j A) = { z C : z a ii z a jj r j A) + a ji ) r i A) a ji } S-SDD matrices The class of S-SDD matrices was defined in the present form in [4]. It can be easily shown that this class which is also a subclass of H-matrices) is the same one defined in [10] in a bit different way. But, the form presented here is more suitable for the formulation of the corresponding eigenvalue inclusion area. Definition 2. Given any matrix A=[a ij ] C n,n, n 2, and given any nonempty proper subset S of N, then A is an S-strictly diagonally dominant S-SDD) matrix if a ii > ri S A) for all i S and aii ri S A)) a jj r S j A)) > ri S A)rS j A) for all i S, j S. We note that if S = N, so that S =, then the above conditions reduce to a ii > r i A) for all i N, and this is just the familiar statement that A is strictly diagonally dominant. The above definition is, in fact, a slight modification of the corresponding definition in [4], which excludes this particular case. It is not a restriction, since if a matrix is an SDD matrix, then it is an S-SDD matrix for all nonempty proper subsets S of N. Further, it can be easily seen that the above definition can be equivalently reformulated as: Definition 3. Given any matrix A=[a ij ] C n,n, n 2, and given any nonempty proper subset S of N, then A is an S-strictly diagonally dominant S-SDD) matrix if a ii > ri S A) for at least one i S and aii ri S A)) a jj r S j A)) > ri S A)rS j A) for all i S, j S. More comments and another equivalent definition can be found in [15]. If there exists a nonempty proper subset S of N, such that A = [a ij ] C n,n, n 2 is an S-SDD matrix, then we will say that A belongs to the class of S-SDD matrices. The corresponding nonsingularity result is the following: Theorem 7. If a given matrix A = [a ij ] C n,n, n 2 is an S-SDD matrix, then A is nonsingular, moreover it is an H-matrix.

5 Numer Algor 2006) 42: and the equivalent eigenvalue inclusion result is: Theorem 8. Let S be any nonempty proper subset of N, and n 2. Then, for any A = [a ij ] C n,n, define the Geršgorin-type disks and the sets Ɣi S A) := {z C : z a ii ri S A)}, i S, Vi, S j A):= {z C : z a ii ri S A)) z a jj r S j A)) rs i A) rs j A), i S, j S. Then all the eigenvalues of A belong to set ) C S A) =: Ɣi S A) i S i S j S Vi, S j A) Obviously, an even better eigenvalue inclusion area can be represented by the set 2.5. Brualdi matrices CA) := S N S =,S =N. C S A). 3) In 1982 Brualdi, [2], discovered a new type of eigenvalue localization theorem, by using graph theory. In [19] Varga presented a modest extension of Brualdi s work. Here we will derive a weakened but simplified result, in order to avoid dealing with block triangular normal reduced form of A. Having the same notations as in [19], given any A = [a ij ] C n,n, if γ = i 1 i 2... i p ), with distinct elements and with p 2, is a strong cycle in directed graph GA), its associated Brualdi lemniscate B γ A) of order p is defined by B γ A) := z C : z a ii r i A). i γ i γ If γ = i) is a weak cycle in GA), its associated Brualdi lemniscate B γ A) is defined by B γ A) := {z C : z a ii = 0} = {a ii }. With BA) := B γ A), γ CA) where CA) is the set of all strong and weak cycles in GA), the following eigenvalue inclusion theorem is valid it is, in fact, a weakened version of Theorem 2.5 from [19]): Theorem 9. For any A = [a ij ] C n,n and any eigenvalue λ of A, there is a strong or weak) cycle γ in CA) such that λ B γ A).

6 234 Numer Algor 2006) 42: Consequently, all the eigenvalues of A belong to the set BA). The corresponding nonsingularity result readily stems from Theorem 2.6 in [19], and can be formulated as: Theorem 10. For any A = [a ij ] C n,n, with a ii > r i A) for all strong cycles in GA), i γ i γ and a ii > 0 for each i for which i) is a weak cycle in GA), it follows that A is nonsingular, moreover it is an H-matrix. Matrices with the above property we will call Brualdi matrices Generalized Brualdi matrices A new nonsingularity result, as well as a new eigenvalue inclusion set will be presented in this section. Let π be a partition of the set of indices N = {1, 2,..., n} into disjoint subsets S 1, S 2,..., S m. With G /π A) we will denote the factor graph associated with the partition π, and with C π A) we will denote the set of strong cycles γ π = S i1, S i2,..., S ip ), in G /π A). Theorem 11. If a given A = [a ij ] C n,n, n 2, satisfies the following two conditions: and a kk > r Si k A) for all k S i, and all i = 1, 2,..., m, ) ) ) a j1 j 1 r Si 1 j 1 a j2 j 2 r Si 2 j 2 a jp j p r Sip S i1 j p > r j 1 r Si 2 j 2 r Sip j p for all j k S ik, k = 1, 2,..., p and for all strong cycles γ π = S i1, S i2,..., S ip ) C π A), then it is nonsingular, moreover it is an H-matrix. Proof. Without loss of generality we can suppose that where S 1 = {1, 2,..., σ 1 }, S 2 = {σ 1 + 1,..., σ 2 },..., S m = {σ m 1 + 1,..., σ m }, σ i = i cards k ), i = 1, 2,..., m. k=1 Obviously, σ m = n. Then a matrix A can be considered in its block form A=[A ij ] m m, where diagonal blocks are A ii = A [S i ], i = 1, 2,..., m. Now, the comparison matrix A of the matrix A we split into A = D B,

7 Numer Algor 2006) 42: where D = Diag A 11, A 22,..., A mm ), and denote f = De, e = [1, 1,..., 1] T. Then, by the first condition of our Theorem, f i > 0, i = 1, 2,..., n, while the second condition can be rewritten as f j1 f j2 f jp > r j1 B)r j2 B) r jp B), for all j k S ik, k = 1, 2,..., p and for all strong cycles γ π = S i1, S i2,..., S ip ) C π A). Finally, we construct the matrix C = F 1 A D 1 F, where F = diag f 1, f 2,..., f n ). Obviously, C = E F 1 BD 1 F, and because of F 1 BD 1 Fe = F 1 Be, we have r i C) = r ib) f i, i = 1, 2,..., n. Now, it remains to observe that the matrix C preserve the block sparsity pattern of A, so the second condition of Theorem 11 implies that the first condition of Theorem 10 holds for the matrix C. The second condition of Theorem 10 also holds for the matrix C, because of f i > 0, i = 1, 2,..., n. Thus, by Theorem 10, C is a nonsingular M-matrix, and A 1 = D 1 FC 1 F 1 0, which means that A is an H-matrix. The matrix is called generalized Brualdi if there exists a partition π for which the conditions in the above theorem are satisfied. The corresponding eigenvalue inclusion area is given by: Theorem 12. For a matrix A = [a ij ] C n,n, n 2, all its eigenvalues belong to the set B π A) := Bγ π A) ) m Ɣ π Si A), where γ π C π A) Ɣ Si A) := Ɣ Si k A) with k S i i=1 and Ɣ Si k A) := {z C : z a kk r Si k A)} any k S i), Bγ π π A) :=:= 1 k p j k S ik B π γ π j 1,... j p )A), with γ π = S i1, S i2,..., S ip ), and B π γ π j 1,... j p )A) := {z C : z a j1 j 1 r Si 1 j 1 ) z a j2 j 2 r Si 2 j 2 ) ) z a jp j p r Sip S i1 j p r j 1 r Si 2 j 2 r Sip j p. Obviously, the better eigenvalue inclusion area let us denote it B A)) can be obtained if we take the intersection over all possible partitions π of the set of indices.

8 236 Numer Algor 2006) 42: α matrices The following well known classes of nonsingular matrices, introduced by Ostrowski, too, here we will call α matrices: Definition 4. A matrix A = [a ij ] C n,n is called an α1 matrix if there exists α [0, 1], such that for each i N, it holds that a ii > αr i A) + 1 α)r i A T ). Definition 5. A matrix A = [a ij ] C n,n is called an α2 matrix if there exists α [0, 1], such that for each i N, it holds that a ii > r i A) α r i A T ). Due to Ostrowski, [16], the following nonsingularity result is valid: Theorem 13. If a matrix A C n,n is an α1 or α2 matrix, then it is nonsingular, moreover it is an H-matrix. These two classes are both generalizations of SDD property, and are both subclasses of H-matrices. As all the time before, they have their equivalent theorems in the field of eigenvalue localization: Theorem 14. All eigenvalues of the matrix A = [a ij ] C n,n belong to the set A 1 A) := Ɣi α1 A), where 0 α 1 i N Ɣ α1 i A) := {z C : z a ii αr i A) + 1 α)r i A T )}, i N. Theorem 15. All eigenvalues of the matrix A = [a ij ] C n,n belong to the set A 2 A) := Ɣi α2 A), where Ɣ α2 i 0 α 1 i N Ɣ α2 i A) := {z C : z a ii r i A) α r i A T ) }, i N. It is easy to see that the class of α1 matrices is contained in that of α2 matrices, and Ɣi α1 for all i N. Hence, Theorem 14 is a trivial consequence of Theorem Generalized α matrices First, we will show the following theorem: Theorem 16. If for a matrix A = [a ij ] C n,n there exists α [0, 1] and k N such that for each subset S N of cardinality k ) α a ii > ri S A) ri )) S AT + r S i A) for all i S 4)

9 Numer Algor 2006) 42: holds, then the matrix A we will call it generalized α matrix) is nonsingular, moreover it is an H-matrix. Proof. First of all, let us remark that the case k = 1 represents SDD matrices, while k = n represents α2 matrices, so in both cases the nonsingularity has already been shown. So, from now on, we suppose that 1 < k < n. We will also suppose that α [0, 1), since the case α = 1 represents the class of SDD matrices, again. If we suppose that A is a singular matrix, then there exists a nonzero vector x C n such that A x = 0, or equivalently, a ii x i = a ij x j, i N. j N\{i} Obviously, there exists a subset S N of cardinality k such that x i x j for each i S and each j S. Now we have and for all i S i.e. a ii x i a ij x j + a ij x j, i S, j S a ii x i a ij x j + ri S A) x i, aii ri S A)) x i a ij x j. Using Hölder s inequality we get a ii r S i A) ) x i a ij x j 1 ) a ij α a ij x j 1 = ri S A)) α a ij x j 1 a ij If we suppose that there exists an index i S such that ri S A) = 0, then the above inequality reduces to aii ri S A)) x i 0, α which contradicts the condition 4) for that particular i S. Hence, for all i S, ri S A) = 0, and ) a ii ri SA) r S i A) ) α x i a ij x j 1,

10 238 Numer Algor 2006) 42: or, equivalently, Summing over all i S, we get i S aii ri SA)) 1 ) α x i r 1 i SA) aii ri SA)) 1 ) α x i r 1 i SA) i S Hence, there exists at least one k S such that akk r S k A)) 1 r Sk A) ) α a ij x j 1 r S k AT ), a ij x j 1. = j S r S j AT ) x j 1. which is in contradiction with 4). Hence, each matrix satisfying 4) is nonsingular. To prove that A is an H-matrix, set A = D B, where D = diag a 11, a 22,..., a nn ), and show that ρd 1 B) < 1. Indeed, if there exists an eigenvalue λ of the matrix D 1 B, such that λ 1, then the matrix DλE D 1 B) = λd B will satisfy the condition 4), and consequently it will be nonsingular. But this contradicts the fact that λ is an eigenvalue of the matrix D 1 B. Finally, since ρd 1 B) < 1, we get A 1 = k 0D 1 B) k D 1 0, which completes the proof. Now, it is easy to formulate the corresponding Geršgorin-type theorem: Theorem 17. All eigenvalues of the matrix A = [a ij ] C n,n belong to the set AA) := 0 α 1 k N S, S =k i S Ɣ α,k,s i, where { Ɣ α,k,s i := z C : z a ii ri S A)) α r S i A T ) ) } + r S i A). 3. Diagonal scaling According to the Geršgorin theorem, set ƔA) contains all eigenvalues of the matrix A = [a ij ] C n,n. Using a nonsingular matrix X one can obtain another set Ɣ X A) := ƔX 1 A X ) which also contains all the eigenvalues of the matrix A. Furthermore,

11 Numer Algor 2006) 42: denoting by B an arbitrary set of nonsingular matrices, we get the new eigenvalue inclusion area for the matrix A: Ɣ B A) = X B Ɣ X A). We will call this set the minimal Geršgorin set with respect to B. The case of B = R, where R is the set of all diagonal nonsingular matrices was studied in details by Varga, [19], where a lot of theoretical properties of this set were proved. Generally, how to find this set is a very hard problem, since there is no explicit calculable formula of it. This is the reason for considering the minimal Geršgorin sets with respect to some special subsets of R, for which a formula depending on the matrix entries only could be found. The first step in this direction was done by Dashnic and Zusmanovich, [8], where B = F and F is the union of all diagonal matrices, whose diagonal entries are equal to 1, all except one, which is an arbitrary positive number. The corresponding minimal Geršgorin set with respect to F) is described by Ɣ F A) = DA), where D is defined by 2). One step more has been done in the paper [6], where B = W has been chosen as the set of all diagonal matrices whose diagonal entries are either 1 or x, where x is an arbitrary positive number. Namely, W = S N W S, W S = {X = diagx 1, x 2,..., x n ) : x i = 1 for i S and x i = x > 0 otherwise}. With the same notation for CA) as in 3), the following theorem has been proved: Theorem 18. Ɣ W A) = CA). There is a property of H-matrices which is hiding behind these two results. It is a very well-known fact that a matrix A C n,n is an H-matrix if and only if there exists a nonsingular diagonal matrix X such that A X is an SDD matrix. The problem arises from the fact that such a matrix X could be found in a very few special cases. Up to now, we are aware of two such cases: Dashnic Zusmanovich matrices and S-SDD matrices which correspond precisely to the above two minimal Geršgorin sets!). 4. Relationships Let us stay, for a moment, in the field of matrix nonsingularity or, equivalently, in the field of H-matrices). The central position here belongs to the SDD class, and all

12 240 Numer Algor 2006) 42: other mentioned classes are generalizations of SDD property. They can be grouped into three directions : Subset direction Graph direction Alpha direction The subset direction form Ostrowski matrices, Dashnic Zusmanovich matrices and S-SDD matrices. As it has been shown in [4], SDD matrices are subset of Ostrowski matrices, which are subset of Dashnic Zusmanovich matrices, which are subset of S-SDD matrices. The graph direction form Ostrowski matrices, Brualdi matrices and generalized Brualdi matrices. SDD matrices are subset of Ostrowski matrices, which are subset of Brualdi matrices, which are subset of generalized Brualdi matrices. It is interesting to mention that subset direction and graph direction become the one at the level of generalized Brualdi matrices. Finally, the Alpha direction consists of α1 matrices, α2 matrices and generalized α matrices. It is obvious that SDD matrices are subset of α1 matrices, which are subset of α2 matrices, which are subset of generalized α matrices. All above relations are illustrated in Figure 1. If we switch to eigenvalue localization field, all above inclusions will become the opposite. More precisely, if a class A is a subclass of class B, then the corresponding Geršgorin-type area A will contain the corresponding Geršgorin-type area B. The Figure 1 Relations between subclasses of H-matrices.

13 Numer Algor 2006) 42: Figure 2 Subclasses of H-matrices vs. Geršgorin-type sets. situation in the subset direction, including the scaling approach Section 3), is given in Figure 2. How this works on a simple 4 4 example can be seen in Figure More criteria for identifying H-matrices Several characterizations of H-matrices has been obtained in [5]. Let us show here two of them. From now on, for the matrix A = [a ij ] C n,n, satisfying a ii = 0, for all i N, we will use the notation R S i A) = k S\{i} r S k A) a kk a ik, i N, for arbitrary nonempty subset S N. We also set R i A) = 0. Figure 3 ƔA) - black; DA) - gray; CA) - light gray; MGS - white.

14 242 Numer Algor 2006) 42: Theorem 19. Let A = [a ij ] C n,n, n 2, be a matrix with nonzero diagonal entries. If there exists a nonempty subset S N such that ri S A) > RS i A) for each i S, and r S i A) R S i A)) r S j A) RS j A)) > R S i A)RS j A) for each i S and each j S, then A is an H-matrix. Theorem 20. Let A=[a ij ] C n,n, n 2, be a matrix with nonzero diagonal entries. If then A is an H-matrix. r i A)r j A) > R N i A)R N j A) for all i, j N, i = j, Let us observe that Theorem 19 is a trivial implication of Theorem 7, while Theorem 20 readily stems from Theorem 3. Both Theorems 19 and 20 are based on proper choices of a scaling matrix for more such choices we refer to [3, 9, 12]). We have chosen to present these two particular characterizations of H-matrices in order to illustrate the following situation: In the H-matrix theory we have obtained a qualitatively new result the relationships between these new classes and their origins S-SDD matrices and Ostrowski matrices are shown in Figure 4, see [5]); Although both Theorems 19 and 20 can be used to describe eigenvalue inclusion sets, such sets are not nice. Figure 4 Relations between two new classes and their origins.

15 Numer Algor 2006) 42: Irreducibility direction There is a fourth direction for generalization of SDD property. This is the concept of irreducibility, see [17]. To be precise, we have to emphasize that in such a way we will not obtain supersets of SDD matrices. Nevertheless, we can consider it as a kind of generalization. In fact, almost all previous nonsingularity results can be extended by letting all, but at least one, considered inequalities not to be strict. Here we will present two of them, given in [5]. Definition 6. A matrix A = [a ij ] C n,n is an irreducibly diagonally dominant IDD) matrix if A is irreducible, if a ii r i A) for all i N, and if strict inequality holds for at least one index i N. Theorem 21. Let A = [a ij ] C n,n, n 2 be an irreducible matrix. If there is a nonempty proper subset S N such that and a ii ri S A) for each i S, aii ri S A)) a jj r S j A)) ri S A)rS j A) for each i S and j S, where the last inequality becomes strict one for at least one pair of indices i S, and j S, then A is an H-matrix. Theorem 22. Let A = [a ij ] C n,n, n 2, be an irreducible matrix with nonzero diagonal entries. If there exists a nonempty proper subset S N such that ri S A) RS i A) for each i S, and r S i A) R S i A)) r S j A) RS j A)) R S i A)RS j A) for each i S and j S, where the last inequality becomes strict one for at least one pair of indices i S, and j S, then A is an H-matrix. The knowledge of such a characterization of H-matrices could be useful for getting more information about eigenvalues on the boundary, like it has primarily been done by Taussky, [18]. 7. Necessary conditions for H-matrices Along with the efforts to find out whether a given matrix is an H-matrix, it is interesting to find various ways to conclude that a given matrix cannot be an H-matrix.

16 244 Numer Algor 2006) 42: All subclasses mentioned in Section 2 could be a source for obtaining corresponding necessary conditions for a matrix to be an H-matrix. The main idea is based on the well known fact: Every H-matrix has at least one SDD row; see, for example, [14]. For example, due to S-SDD class, we can get the following necessary condition: Theorem 23. Let A = [a ij ] C n,n, n 2 be an H-matrix. Then for every nonempty proper subset S N there exist i S, j S such that aii r S i A)) a jj r S j A)) > r S i A)rS j A). Since for every H-matrix A, and every nonempty proper subset S N, the matrices A [S] and A [S] are H-matrices, too, we have another necessary condition: Theorem 24. Let A = [a ij ] C n,n, n 2 be an H-matrix. Then for every nonempty proper subset S N there exist i S, j S such that a ii > r S i A), a jj > r S j A). At the very end, I would like to conclude this survey with an observation that until the minimal Geršgorin set in the sense of Varga s definition) is to be found, which, by nature, has to be in iterative way, it would be worth to find more and more Geršgorintype theorems, and consequently, more and more easily checkable subclasses of H-matrices. Acknowledgments The author is grateful to the referees for their useful and constructive suggestions. This work is partly supported by Republic of Serbia, Ministry of Science and Environmental Protection, grant References [1] Brauer, A.: Limits for the characteristic roots of a matrix II, Duke Math. J. 14, ) [2] Brualdi, R.: Matrices, eigenvalues and directed graphs. Linear Multilinear Algebra 11, ) [3] Cvetković, Lj.: Convergence theory for relaxation methods to solve systems of equations. MB-5 PAMM, Technical University of Budapest 1998) [4] Cvetković, Lj., Kostić, V., Varga, R.: A new Geršgorin-type eigenvalue inclusion area. ETNA 18, ) [5] Cvetković, Lj., Kostić, V.: New criteria for identifying H-matrices. J. Comput. Appl. Math. 180, ) [6] Cvetković, Lj., Kostić, V.: Between Geršgorin and minimal Geršgorin sets. J. Comput. Appl. Math., in press [7] Dashnic, L.S., Zusmanovich, M.S.: O nekotoryh kriteriyah regulyarnosti matric i lokalizacii ih spectra. Zh. vychisl. matem. i matem. fiz. 5, ) [8] Dashnic, L.S., Zusmanovich, M.S.: K voprosu o lokalizacii harakteristicheskih chisel matricy. Zh. vychisl. matem. i matem. fiz. 106), ) [9] Gan, T.B., Huang, T.Z.: Simple criteria for nonsingular H-matrices. Linear Algebra Appl. 374, ) [10] Gao, Y.M., Xiao, H.W.: Criteria for generalized diagonally dominant matrices and M-matrices. Linear Algebra Appl. 169, )

17 Numer Algor 2006) 42: [11] Geršgorin, S.: Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk SSSR Ser. Mat. 1, ) [12] Huang, T.Z.: A note on generalized diagonally dominant matrices. Linear Algebra Appl ) [13] Li, B., Tsatsomeros, M.J.: Doubly diagonally dominant matrices. Linear Algebra Appl ) [14] Li, B., Li, L., Harada, M., Niki, H., Tsatsomeros, M.J.: An iterative criterion for H-matrices. Linear Algebra Appl. 271, ) [15] Morača, N., Cvetković, Lj., Algorithm for checking S-SDD matrices, ICNAAM 2005 Extended Abstracts. Simos, T.E., Psihoyios, G., Tsitouras, Ch. eds.), Wiley, ) [16] Ostrowski, A.M.: Über die Determinanten mit überwiegender Hauptdiagonale. Comment. Math. Helv. 10, ) [17] Plemmons, R.J., Berman, A.: Nonnegative matrices in mathematical sciences. SIAM, Philadelphia, 1994) [18] Taussky, O.: A recurring theorem on determinants. Amer. Math. Monthly 56, ) [19] Varga, R.S.: Geršgorin and His Circles. Springer Series in Computational Mathematics, Vol )