Generalized Irreducible α-Matrices and Its Applications

The class of generalized α-matrices is presented by Cvetković, L. (2006), and proved to be a subclass of H-matrices. In this paper, we present a new class of matrices-generalized irreducible α-matrices, and prove that a generalized irreducible α-matrix is an H-matrix. Furthermore, using the generalized arithmetic-geometric mean inequality, we obtain two new classes of H-matrices. As applications of the obtained results, three regions including all the eigenvalues of a matrix are given.


Introduction
H-matrices play a very important role in Numerical Analysis, in Optimization theory and in other Applied Sciences [1]- [7].Here we call a matrix , , is an M-matrix, i.e., ( ) ( ) One interesting problem involving on H-matrices is to identify whether or not a matrix is an H-matrix [2] [8].But it is not easy to do this by its definition.So researchers turned to study some subclasses of H-matrices, which are easy to identify [1] [2] [3] [4] [5] [8] [9] [10].One of the classical subclasses is strictly diagonally dominant matrices (see Definition 1) which was first presented by Lévy only for real matrices [11].And Minkowski [12] and Desplanques [13] ob-tained the general complex result.Definition 1.A matrix ∈ is called a strictly diagonally dominant matrix if for any i N ∈ , ( ) As is well known, a strictly diagonally dominant matrix is nonsingular.
This can lead to the following famous Geršgorin's Theorem.
Theorem 1. [12] Let and ( ) A σ be the spectrum of A. Then ( ) ( ) ( ) By considering the irreducibility of a matrix, Taussky [14] [15] extended the notion of a strictly diagonally dominant matrix, and given the following subclass of H-matrices (see Definition 2).A matrix A is irreducible if and only if its directed graph G (A) is strongly connected (for details, see [16] [17]).
Definition 2. A matrix and if strict inequality holds in (1) for at least one i.

and any
[ ] where ( ) ( ) Then A, which is called α 2 -matrices, is nonsingular and is an H-matrix.
By the nonsingularity of α 2 -matrices, one can easily obtain the corresponding eigenvalue localization theorem as below.
Theorem 4. [17] For any For irreducible matrices, Hadjidimos in [19] gave extensions of Theorem 4 by the nonsingularity of the so-called irreducible α 2 -matrices (see Theorems 5 and 6). :  and 1 N is the set of indices for which strict inequality holds in (3).
We remark here that although Hadjidimos in [19] pointed out that irreducible α 2 -matrices is nonsingular, he didn't give the relationship between α 2 -matrices and H-matrices.In fact, the class of α 2 -matrices is a subclass of H-matrices, which is showed by the following theorem.
Theorem 7.For an irreducible α 2 -matrix A, then A is an H-matrix.Proof.We let , , , nn D diag a a a =  , and prove that the spectral radius ( ) , is an irreducible α 2 -matrix, and hence it is nonsingular.
But this contradicts the fact that λ is an eigenvalue of the matrix Recently, Cvetković in [4] presented a new subclass of H-matrices, which is called generalized α-matrices defined as below, and given a new eigenvalue localization set by using the nonsingularity of generalized α-matrices (see Theorem 9).
holds, where ( ) is called a generalizaed α-matrices, is nonsingular, moreover it is an H-matrix.
We now present a new class of matrices-generalized irreducible α-matrix, which is different from the class of generalized α-matrices and will be proved to be an H-matrix in Section 2. Advances in Linear Algebra & Matrix Theory Definition 4. A matrix holds, with at least one inequality in (5) being strict.
The outline of this paper is given as follows.In Section 2, we prove that a generalized irreducible α-matrix is nonsingular, and is an H-matrix.

Nonsingularity of Generalized Irreducible α-Matrices
In this section, we prove that a generalized irreducible α-matrix is nonsingular, and obtain a new eigenvalue localization set by using its nonsingularity.
Theorem 10.If a matrix Proof.First, Apparent we remark that the case k = 1 represents the class of irreducibly diagonally dominant matrices, while k = n represents irreducible α 2 -matrices, so in both cases the nonsingularity has already been shown in Theorem 2 and Theorem 5, respectively.So, from now on, we suppose that 1 < k < n.
Suppose on the contrary that A is singular.Then there exists a nonzero vector ( ) , for each i N ∈ Taking absolute values in the above equation and using the triangle inequality gives , 1 , , for each i N ∈ Note that for the nonzero vector ( ) and each j S ∈ .Hence, for each i S ∈ .
( ) Furthermore, by (5) with at least one strict inequality holds above.Using Höder's inequality (see Lemma 2.1 in [19]) we get ( ) ( ) ( ) ( ) without loss of generality, suppose that for any i S ∈ , ( ) then from ( 7),we have ( ) ( ) Note that 0 i x ≠ for each i S ∈ .then ( ) ( ) Since A is a generalized irreducible α-matrix, we Furthermore, by ( 6) and ( 9), we get that ( ) For every i S ∈ , ( ) r A α on both sides of (8)and raising both sides of (8) to the power where strict inequality holds above for at least one i S ∈ .Summing on all i in S in the above inequalities gives ( ) ( ) . This is a contradiction.Therefore, A is nonsingular.Moreover, similar to the proof of Theorem 7, we can easily prove that A is an H-matrix.
From Theorem 10, we easily get the corresponding eigenvalue localization set as below.
Corollary 1.For any

and any
[ ] and 2

\ S S S
= with 1 S is the set of indices for which strict inequality holds in (5).

Applications
Combining the nonsingularity of generalized (irreducible) α-matrices with the generalized arithmetic-geometric mean inequality: ( ) holds, then A, which is called a generalized sum α-matrix, is nonsingular, moreover it is an H-matrix.Proof.By the generalized arithmetic-geometric mean inequality, we have ( This implies that A is generalized α-matrix.Hence A is nonsingular.Furthermore, similar to the proof of Theorem 7, we can obtain easily that A is an H-matrix.
From Theorem 11, we also get a corresponding eigenvalue localization set.Corollary 2. For any

and any
[ ] According to Theorem 10 and the generalized arithmetic-geometric mean inequality, we can obtain easily the following subclass of H-matrices and the corresponding eigenvalue localization set.
Theorem 12.If for an irreducible matrix and k N ∈ such that for each subset S N ⊂ of cardinality k.
( ) ( ) ( ) ( ) holds, with at least one inequality in (11) being strict, then A is nonsingular, moreover it is an H-matrix.
Corollary 3.For any

and any
[ ] = with 1 S is the set of indices for which strict inequality holds in (11).

Simplifications of Eigenvalue Localization Sets
The eigenvalue localization sets in Theorem 9 and Corollary 2 are not of much practical use because of the restriction of α.To solve this problem, we in this section use the method provided in [5] [6], and obtain more convenient forms of the two eigenvalue localization sets.First, the sufficient and necessary conditions of generalized α-matrices and generalized sum α-matrices are given.
For a matrix ∈ with 2 n ≥ , and for S N ⊆ of cardinality k N ∈ , we partition the set of indices S into three sets:

i S ∈
Hence by Theorem 5 in [5], A in this case is an α 2 -matrix.The case 1 < k < n: Similar to the proof of Theorem 5 in [5], the conclusion in this case follows easily.
Similar to the proof of Lemma 13, for generalized sum α-matrices we also obtain easily its sufficient and necessary condition by Theorem 4 in [5]

(
other subclasses of H-matrices, consequently, two new eigenvalue localization set.Theorem 11.If for a matrix each i R ∈ and each j C ∈ .We now establish two eigenvalue localization sets by Lemmas 13 and 14, which are the equivalent forms of the sets in Theorem 9 and Corollary 2 respec-et al. DOI: 10.4236/alamt.2018.83010 .