Document downloaded from: http://hdl.handle.net/10251/103993 This paper must be cited as: Bru, R.; Corral Ortega, C.; Gimenez Manglano, MI.; Mas Marí, J. (2008). Classes of general H-matrices. Linear Algebra and its Applications. 429(10):2358-2366. doi:10.1016/j.laa.2007.10.030 The final publication is available at https://doi.org/10.1016/j.laa.2007.10.030 Copyright Elsevier Additional Information Classes of general H-matrices ∗ R. Bru, C. Corral, I. Giménez and J. Mas Institut de Matemàtica Multidisciplinar Universitat Politècnica de València, Spain {rbru, ccorral, igimenez, jmasm}@imm.upv.es Abstract Let M(A) denote the comparison matrix of a square H-matrix A, that is, M(A) is an M -matrix. H-matrices such that their comparison matrices are non-singular are well studied in the literature. In this paper, we study characterizations of H-matrices with singular or nonsingular comparison matrix. In particular, we analyze the case when A is irreducible and then give insights into the reducible case. The spectral radius of the Jacobi matrix of M(A) and the generalized diagonal dominance property are used in the characterizations. Finally, from these characterizations, a partition of the general H-matrix set in three classes is obtained. 1 Introduction In the literature on iterative methods of linear systems, H-matrices are widely used because they appear in many applications when discretizing certain nonlinear parabolic equations and when solving the linear complementarity problem. Furthermore, H-matrices are closely related to M -matrices [2, 19]. Matrices of this kind are currently the subject of much interest as noted in [6] and [5] and the references therein. For instance, to study the convergence of block iterative methods, the concepts of Z-matrix, M -matrix and H-matrix have been generalized to block matrices in [7, 17] while generalized H-matrices are defined in [14]. In [12] additional properties of generalized H-matrices are described. Another subject of recent attention is the determination of H-matrices. Most of the equivalent conditions given in [2] ∗ Supported by Spanish DGI grants MTM2004-02998 and MTM2007-64477. 1 are not of practical use to know if a given matrix is an H-matrix. To test this condition, several iterative algorithms based on the generalized diagonal dominance of the matrix, have been proposed (see [1, 4, 11, 13]), for a good discussion of this kind of algorithms we refer to the recent paper [1]. Further, direct criteria for H-matrices can be found in [9, 10]. Given the H-matrix A, if the comparison matrix M(A) is non-singular, then A is non-singular, and this fact has led many authors who consider only non-singular M -matrices to conclude that H-matrices are always nonsingular. In fact, it is known that if M(A) is a non-singular M -matrix then all equimodular matrices are non-singular (see [16]). We show here that Hmatrices can be singular. Furthermore, the converse of the above statement is not true, i.e., an H-matrix A can be invertible, while M(A) may be singular, as we show in Example 1 below. Moreover, if the M -matrix M(A) is singular, the invertible H-matrix may not satisfy all properties corresponding to the case of non-singular M(A). Different characterizations of singular and non-singular M -matrices are given in [2]. In the case of H-matrices we shall see that the non-singularity of the matrix A and of its comparison matrix M(A) may yield different types of H-matrices. Other classifications of Z-matrices, including M -matrices and inverses of Z-matrices, were given in [8] and [15]. The case of non-singular H-matrices with non-singular comparison matrices has been widely studied and characterized (see for instance [18, 2]). However, it seems that the remaining cases have not been studied. In addition, some conclusions may be uncertain as explained in the next section. We define three classes of H-matrices one of them with non-singular comparison matrix and the other two with singular comparison matrix. In one class with singular comparison matrix all equimodular matrices are singular while in the other class there are both singular and non-singular matrices. to define the classes we determine the properties that identify these three types of H-matrices. Facts related with the nullity of diagonal elements, irreducibility or generalized diagonal dominance are used to obtain necessary or sufficient conditions to conclude that a given matrix is an H-matrix, and if so, to which of three types it belongs. The structure of the paper is as follows. In Section 2, we recall concepts, results, notations we may need in the sequele. In particular, two examples will illustrate the singularity of some H-matrices. In Section 3, we discuss the results characterizing H-matrices and study irreducible H-matrices in more detail. Finally, in section 4 we obtain three types or classes of H-matrices providing the last characterization of the singular type of H-matrices; in addition, we summarize all properties of any type of H-matrices that we have obtained. 2 2 Preliminaries and motivation We recall that a square real matriz A is said to be Z-matrix if aij ≤ 0 for all i 6= j, i, j = 1, 2, . . . , n. The comparison matrix of the (complex) matrix A ∈ Cn×n is defined as the Z-matrix ( − |aij | , if i 6= j M(A) = 2 |DA | − |A| = , i, j = 1, 2, . . . , n, (1) |aii | , if i = j where DA denotes the diagonal matrix DA = diag(aii ). The set of equimodular matrices associated with A, denoted by Ω(A), is Ω(A) ≡ {B ∈ Cn×n : M(B) = M(A)}. (2) Note that both A and M(A) are in Ω(A). Let us recall that a splitting A = M − N , where M is invertible, is called regular if M −1 ≥ O and N ≥ 0. Associated with the splitting A = DA − (−E − F ), we consider the Jacobi iteration matrix −1 JA = −DA (E + F ), (3) where E and F are the strictly lower and upper triangular parts of A, respectively. With these notations, if τ = maxi {aii }, a Z-matrix A can be written as A = τ I − C where the matrix C is nonnegative. In particular, the matrix A is an M -matrix if A = sI − B with B ≥ 0 and s ≥ ρ(B), (4) where ρ(B) denotes the spectral radius of matrix B. Recall that an M matrix A has aii ≥ 0, i = 1, 2, . . . , n, s ≥ τ and A is invertible if and only if s > ρ(B); in this case, aii > 0, i = 1, 2, . . . , n. Finally, A is said to be an H-matrix if its comparison matrix M(A) is an M -matrix. Properties and characterizations of H-matrices such that their comparison matrices are non-singular M -matrices are obtained by characterizations of non-singular M -matrices. It is well-known that M(A) is an invertible M matrix if and only if A is generalized strictly diagonally dominant (GSDD), that is, if there exists a positive diagonal matrix D = diag(di ) such that AD is strictly diagonally dominant (SDD), i.e., X |aij | dj < |aii | di , i = 1, 2, . . . , n, j6=i 3 or, there exists a positive vector d = (di ) such that the above inequalities hold. Another characterization considered is that M(A) is an invertible M matrix if and only if ρ(JM(A) ) < 1. In this study we consider general H-matrices, i.e., when the comparison matrix may be singular. From points (X) and (XI) of Theorem 1 in Varga [18] (and from the original paper by Ostrowski [16]), one may deduce that if any B ∈ Ω(A) is singular then M(A) is singular, i.e., if M(A) is invertible then all matrices in Ω(A), including A, are invertible. However, when the matrix A is non-singular, the non-singularity of M(A) is not guaranteed as the following example shows. Example 1. The matrix   2 −2 A= −2 −2 is non-singular, while its comparison matrix M(A) is a singular M -matrix. Therefore, A is an H-matrix and it is non-singular. Thus, the singularity of M(A) does not imply the singularity of all matrices in Ω(A). Example 1 proves somewhat confusing when working with general Hmatrices without taking into account the invertibility of M(A). One example of this confusion appears in Theorem 7.5.14, page 185, of [2], where statement (1) assures that a non-singular H-matrix A is such that M(A) satisfies any one of the 50 equivalent conditions of a non-singular M -matrix given in Theorem 6.2.3 of [2]. However, M(A) in Example 1 does not satisfy conditions of said Theorem 6.2.3. In fact, in this statement the non-singularity should be imposed on matrix M(A) instead of on the H-matrix A. Moreover, if A is an H-matrix such that M(A) is non-singular, then it is clearly understood that all diagonal entries of A are non-zero. However, there are H-matrices with some zero diagonal element:       0 −1 0 0 0 0 0 a 0 Example 2. Matrices A = ,B= and C =  0 −a 0 −a b −1 −1 b with a, b > 0, are singular H-matrices and some diagonal entries are null. As the three matrices of Example 2 have singular comparison matrices, it is not possible to compute their Jacobi matrices and the GSDD property is not satisfied. Additionally, these matrices are reducible and only the matrix B satisfies the GDD property, i.e., B is generalized diagonally dominant 4 (but not strictly): there exists a positive diagonal matrix D such that BD is diagonally dominant, That is, X |aij | dj ≤ |aii | di , i = 1, 2, . . . , n. j6=i Specifically, D = diag(b, a) proves that matrix B of Example 2 is GDD. These examples illustrate the complexity of the set of general H-matrices, which we shall study in the following section. 3 Characterization and properties of general H-matrices Let us start by characterizing general H-matrices. Theorem 1. Let A ∈ Cn×n . The following statements are equivalent: 1. A is an H-matrix 2. for each B ∈ Cn×n , M(B) ≥ M(A) ⇒ B is an H-matrix. Proof. (2 ⇒ 1) It is clear taking B = A. (1 ⇒ 2) Since M(B) ≥ M(A), |bii | ≥ |aii | and |bij | ≤ |aij | for j 6= i and i, j = 1, 2, . . . , n. Let us write M(B) = mI − C with m = maxi |bii | and C ≥ 0. Then M(A) = mI − P where m ≥ maxi |aii | and then P ≥ 0. Note that, ρ(P ) ≤ m since M(A) is M -matrix. Since 0 ≤ C ≤ P we have ρ(C) ≤ ρ(P ) ≤ m and hence B is an H-matrix. The property of generalized diagonally dominance, without any strict inequality, does not characterize general H-matrices. Note that the H-matrix of Example 1 is GDD, but the H-matrices A and C of Example 2 are not. So, we first restrict our analysis to the case of non-zero diagonal elements. Now, we recall the following well-known result. Lemma 1. Let A be a Z-matrix. Then A is an M -matrix, if and only if DA is an M -matrix for each positive diagonal matrix D. Theorem 2. Let A ∈ Cn×n be such that aii 6= 0, for i = 1, 2, . . . , n. The following statements are equivalent: 1. A is an H-matrix
2. ρ JM(A) ≤ 1 5 3. for any B ∈ Ω(A), ρ(JB ) ≤ 1. Proof. Let DM(A) = diag(M(A)) and consider the regular splitting of M(A) −1 yielding the Jacobi matrix JM(A) = I−DM(A) M(A) ≥ 0. Then, by Lemma 1, A is an H-matrix, or equivalently M(A) is an M -matrix, if and only if −1 the matrix DM(A) M(A) = I − JM(A) is an M -matrix. Note that this is
equivalent to ρ JM(A) ≤ 1 by (4). This proves the equivalence of the first two statements. Statement 2 implies 3 since for any B ∈ Ω(A), we have ρ(JB ) ≤ ρ(|JB |) = ρ(JM(A) ). The converse (3 implies 2) follows taking B = M(A) ∈ Ω(A). Now, we shall characterize the irreducible H-matrices using the GDD property. First, we prove the following result. Theorem 3. Let A ∈ Cn×n be an irreducible H-matrix. Then aii 6= 0, for i = 1, 2, . . . , n. Proof. Consider the splitting of the comparison matrix M(A) = mI − C, C ≥ 0, ρ(C) = ρ ≤ m. Since C is an irreducible non-negative matrix, there exists a positive vector u such that Cu = ρu. Then, M(A)u = mu − Cu = (m − ρ)u. Thus, for each row we have X |aij | uj = (m − ρ)ui ≥ 0 (5) |aii | ui − j6=i and, if aii = 0, the corresponding row will be zero, and so, A becomes reducible. The proof follows. With this result, we obtain the following characterization. Theorem 4. Let A ∈ Cn×n be an irreducible matrix. Then, A is an Hmatrix if and only if A is GDD, that is, there exists a positive vector d such that X |aii | di ≥ |aij | dj , i = 1, 2, . . . , n. (6) j6=i Proof. Let us suppose that A is an H-matrix. Since A is irreducible, the inequality (5) holds for i = 1, 2, . . . , n, and thus A is generalized diagonally dominant, for d = u. 6 Conversely, since A is GDD and irreducible then aii 6= 0, for all i = 1, 2, . . . , n and we can construct the Jacobi matrix JM(A) . Then the inequalities (6) may be written as X |aij | dj j6=i |aii | di ≤ 1, i = 1, 2, . . . , n which means that the spectral radius of the nonnegative irreducible matrix D−1 JM(A) D is bounded by 1, where D = diag(di ). Therefore, ρ(JM(A) ) ≤ 1 and using Theorem 2 we deduce that A is an H-matrix. Note that in the proof of the converse, the irreducibility is needed only to assure the non-nullity of the diagonal elements of the matrix. Thus, we can conclude the following more general statement: if A is a GDD matrix with aii 6= 0 for i = 1, 2, . . . , n, then A is an H-matrix. The case when H-matrices are reducible is studied in the following result. First, we shall recall that the normal form of a reducible matrix A is given by a block triangular matrix P AP T = (Rij ), i, j = 1, . . . , p, in which each square diagonal block Rii is either irreducible or a 1 × 1 null matrix and P is a permutation matrix. Theorem 5. Let A ∈ Cn×n be a reducible matrix. Then, A is an H-matrix if and only if in the normal form of A, P AP T = (Rij ), each square diagonal block is an H-matrix. Proof. The only part follows from the fact that P AP T is an H-matrix and hence all its principal submatrices. To prove the if part, we construct the normal form of M(A), P M(A)P T = (Sij ) from the normal form of A, P AP T = (Rij ). Now consider as usual the splitting P M(A)P T = mI − C, where C ≥ 0. Then, we obtain the splittings of the M -matrices Skk = mI − Ckk , where the identity matrix I has an adequate order, satisfying ρ(Ckk ) ≤ m for k = 1, . . . , p. Since ρ(C) = maxk ρ(Ckk ), then, P AP T is an H-matrix and so is A. Note that by Theorem 4 the irreducible diagonal blocks of the normal form of an H-matrix are GDD. We obtain the following converse result. Theorem 6. Let A ∈ Cn×n . If A is GDD, then A is an H-matrix. Proof. The case that A is irreducible follows from Theorem 4. The reducible case is studied for each diagonal block of the normal form, which is either GDD and irreducible and then, by the same theorem, an H-matrix, or a 1×1 null matrix which is also an H-matrix. Finally, by Theorem 5, we conclude that A is an H-matrix. 7 Remark: Symmetric H-matrices are characterized as GDD matrices in [3] (Theorem 8). This characterization is also deduced from our results. In the irreducible case, H-matrices are characterized as GDD in Theorem 4. In the reducible case, a symmetric H-matrix is a block diagonal matrix; then by Theorem 5, the diagonal blocks are irreducible H-matrices and GDD matrices and thus, the whole matrix. In addition, our Theorem 6 leads to the converse in this case. 4 Classification of H-matrices With the results given in the previous section, we can observe an initial partition in the family of H-matrices. The first set contains all H-matrices such that its comparison matrix is non-singular. These H-matrices are invertible and are characterized as GSDD matrices or as those matrices such that the spectral radius of the corresponding Jacobi matrix is less than 1, in addition to all characterizations of non-singular M -matrices on the comparison matrix (see [2, 18]). This class will be called “invertible class”. The second set contains H-matrices with a singular comparison matrix. In this set we observe from the aforestated two examples that in Ω(A) there are singular, and maybe non-singular, matrices. Thus, we shall study this second set in order to differenciate it in subclasses. Theorem 7. Let A be an H-matrix. Then, A has some null diagonal element if and only if B is singular for all B ∈ Ω(A). Proof. Assume null the diagonal element aii of A and let B ∈ Ω(A). Then, B is an H-matrix and bii = 0, and, by Theorem 3, B is reducible. If Q = P BP T is the normal form of B, the irreducible diagonal blocks of Q are H-matrices, by Theorem 5, and so its diagonal elements are different from zero by Theorem 3. Then, the 1 × 1 submatrix (bii ) = (0) is a diagonal block of Q. Thus, Q is singular and so is B. Conversely, consider now that M(A) is singular and suppose that aii 6= 0 for all i. We shall construct a non-singular matrix B ∈ Ω(A), in fact we shall construct by induction a matrix B with all leading principal submatrices Bk non-singular. Obviously, the 1 × 1 leading principal submatrix is non-singular. Suppose now that a k ×k non-singular matrix Bk equimodular with the principal k ×k submatrix of A has been constructed. Then consider the (k + 1) × (k + 1) matrix,   Bk ak+1 B̃k+1 = ak+1 ak+1,k+1 8 where ak+1 (ak+1 ) is the column (row) formed by the first k components of the (k + 1)th column (row) of A. If B̃k+1 is non-singular then Bk+1 = B̃k+1 . Otherwise, construct the matrix   Bk ak+1 Bk+1 = ak+1 ak+1,k+1 − 2ak+1,k+1 whose determinant is  det Bk+1 = det B̃k+1 + 2 det Bk 0 ak+1 −ak+1,k+1  = −2ak+1,k+1 det Bk 6= 0, ˜ is singular and Bk is non-singular joint with ak+1,k+1 is not null. since Bk+1 This result leads us to consider two separate classes of the second set: the class of H-matrices in which all matrices B in Ω(A) are singular, which we will call the “singular class”, and the class of H-matrices such that Ω(A) contains singular and non-singular matrices, which we will call the “mixed class”. The singular class of H-matrices is characterized by the existence of null diagonal entries (Theorem 7). In addition, H-matrices of this class have the following properties. Corollary 1. Let A be an H-matrix of the singular class. Then (i) A is reducible. (ii) If A is GDD then the ith row is null whenever aii = 0. Proof. (i) The proof follows from Theorem 3. (ii) Obvious. Note that the H-matrices in the third class, the mixed class, all have diagonal elements different from zero, but their comparison matrices are singular. These matrices may be singular or not, as well as reducible or not. In the irreducible case, they are GDD. In thereducible case, all irreducible diagonal blocks of their normal form are GDD and the non-singular diagonal blocks are GSDD; thus, the values of the elements in the non-zero off-diagonal blocks do not play any role in the fact that the matrix is of this class, i.e., if P AP T = (Rij ) is the normal form of the reducible H-matrix A of the mixed class, the block diagonal matrix D = diag(Rii ) 9 is a GDD H-matrix, and all block triangular matrices with the same block diagonal D are H-matrices of the mixed class. As a result, we obtain a complete classification of the set of H-matrices, denoted by H, in the three following classes: • Invertible class: {A ∈ H : B ∈ Ω(A) ⇒ B is non-singular} = {A ∈ H : M(A) is non-singular}. • Singular class: {A ∈ H : B ∈ Ω(A) ⇒ B is singular}. • Mixed class: {A ∈ H : M(A) is singular and ∃B ∈ Ω(A) non-singular}. We note that the matrix of Example 1 belongs to the mixed class and matrices of Example 2 are in the singular class. Finally, the properties of these three classes are summarized in Table 1. M(A) = sI − C M(A) Diag. elements B ∈ Ω(A) Reducibility Diag. dominance J = JM(A) Invertible class ρ(C) < s Invertible Non-zero Invertible GSDD ρ(J ) < 1 Singular class ρ(C) = s Singular ∃aii = 0 Singular Yes J does not exist Mixed class ρ(C) = s Singular Nonzero Inv. and sing. Irred. ⇒ GDD ρ(J ) = 1 Table 1: Main properties of each class of H-matrices. The bold properties determine the classes. Acknowledgement: The authors thank Debra Westall for editing the manuscript and the referees for their helpful comments. References [1] M. Alanelli and A. Hadjidimos, A new iterative criterion for H-matrices, SIAM J. Matrix Anal. Appl., 29(1), 160–176, 2006. [2] A. Berman and R. J. Plemmons, Nonnegative matrices in the mathematical sciences. Academic Press, London, 1979 (Reprinted and updated, SIAM, Philadelphia, 1994). [3] E. G. Boman, D. Chen, O. Parekh and S. 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