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1
Content available remote

New bounds for the minimum eigenvalue ofM-matrices

100%
Wang F., Sun D.
Open Mathematics
|
2016
|
tom 14
|
nr 1
1007-1013
EN
Some new bounds for the minimum eigenvalue of M-matrices are obtained. These inequalities improve existing results, and the estimating formulas are easier to calculate since they only depend on the entries of matrices. Finally, some examples are also given to show that the bounds are better than some previous results.
2
Content available remote

On the Spectral Characterizations of Graphs

100%
Huang J., Li S.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 3
729-744
EN
Several matrices can be associated to a graph, such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question “Which graphs are determined by their spectrum?” is still a difficult problem in spectral graph theory. Let [...] 𝒰p2q ${\cal U}_p^{2q}$ be the set of graphs obtained from Cp by attaching two pendant edges to each of q (q ⩽ p) vertices on Cp, whereas [...] 𝒱p2q ${\cal V}_p^{2q}$ the subset of [...] 𝒰p2q ${\cal U}_p^{2q}$ with odd p and its q vertices of degree 4 being nonadjacent to each other. In this paper, we show that each graph in [...] 𝒰p2q ${\cal U}_p^{2q}$ , p even and its q vertices of degree 4 being consecutive, is determined by its Laplacian spectrum. As well we show that if G is a graph without isolated vertices and adjacency cospectral with the graph in [...] 𝒱pp−1={H} ${\cal V}_p^{p - 1} = \{ H\}$ , then G ≅ H.
3
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An envelope for the spectrum of a matrix

100%
Psarrakos P., Tsatsomeros M.
Open Mathematics
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2012
|
tom 10
|
nr 1
292-302
EN
We introduce and study an envelope-type region ɛ(A) in the complex plane that contains the eigenvalues of a given n×n complex matrix A. ɛ(A) is the intersection of an infinite number of regions defined by cubic curves. The notion and method of construction of ɛ(A) extend the notion of the numerical range of A, F(A), which is known to be an intersection of an infinite number of half-planes; as a consequence, ɛ(A) is contained in F(A) and represents an improvement in localizing the spectrum of A.
4
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A Brauer’s theorem and related results

100%
Bru R., Cantó R., Soto R., Urbano A.
Open Mathematics
|
2012
|
tom 10
|
nr 1
312-321
EN
Given a square matrix A, a Brauer’s theorem [Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75–91] shows how to modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues. Older and newer results can be considered in the framework of the above theorem. In this paper, we present its application to stabilization of control systems, including the case when the system is noncontrollable. Other applications presented are related to the Jordan form of A and Wielandt’s and Hotelling’s deflations. An extension of the aforementioned Brauer’s result, Rado’s theorem, shows how to modify r eigenvalues of A at the same time via a rank-r perturbation without changing any of the remaining eigenvalues. The same results considered by blocks can be put into the block version framework of the above theorem.
5
Content available remote

A formula for all minors of the adjacency matrix and an application

76%
Bapat R. B., Lal A. K., Pati S.
Special Matrices
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2014
|
tom 2
|
nr 1
EN
We supply a combinatorial description of any minor of the adjacency matrix of a graph. This description is then used to give a formula for the determinant and inverse of the adjacency matrix, A(G), of a graph G, whenever A(G) is invertible, where G is formed by replacing the edges of a tree by path bundles.
6
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A convergence analysis of SOR iterative methods for linear systems with weakH-matrices

76%
Zhang C.-y., Xue Z., Luo S.
Open Mathematics
|
2016
|
tom 14
|
nr 1
747-760
EN
It is well known that SOR iterative methods are convergent for linear systems, whose coefficient matrices are strictly or irreducibly diagonally dominant matrices and strong H-matrices (whose comparison matrices are nonsingular M-matrices). However, the same can not be true in case of those iterative methods for linear systems with weak H-matrices (whose comparison matrices are singular M-matrices). This paper proposes some necessary and sufficient conditions such that SOR iterative methods are convergent for linear systems with weak H-matrices. Furthermore, some numerical examples are given to demonstrate the convergence results obtained in this paper.
7
Content available remote

New iterative codes for𝓗-tensors and an application

76%
Wang F., Sun D.
Open Mathematics
|
2016
|
tom 14
|
nr 1
212-220
EN
New iterative codes for identifying 𝓗 -tensor are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor, i.e., an even-degree homogeneous polynomial form are given. Advantages of results obtained are illustrated by numerical examples.
8
Content available remote

Some new bounds of the minimum eigenvalue for the Hadamard product of anM-matrix and an inverseM-matrix

76%
Zhao J., Sang C.
Open Mathematics
|
2016
|
tom 14
|
nr 1
81-88
EN
Some convergent sequences of the lower bounds of the minimum eigenvalue for the Hadamard product of a nonsingular M-matrix B and the inverse of a nonsingular M-matrix A are given by using Brauer’s theorem. It is proved that these sequences are monotone increasing, and numerical examples are given to show that these sequences could reach the true value of the minimum eigenvalue in some cases. These results in this paper improve some known results.
9
Content available remote

AnS-type upper bound for the largest singular value of nonnegative rectangular tensors

76%
Zhao J., Sang C.
Open Mathematics
|
2016
|
tom 14
|
nr 1
925-933
EN
An S-type upper bound for the largest singular value of a nonnegative rectangular tensor is given by breaking N = {1, 2, … n} into disjoint subsets S and its complement. It is shown that the new upper bound is smaller than that provided by Yang and Yang (2011). Numerical examples are given to verify the theoretical results.
10
Content available remote

Characterization of α1 and α2-matrices

76%
Bru R., Cvetković L., Kostić V., Pedroche F.
Open Mathematics
|
2010
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tom 8
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nr 1
32-40
EN
This paper deals with some properties of α1-matrices and α2-matrices which are subclasses of nonsingular H-matrices. In particular, new characterizations of these two subclasses are given, and then used for proving algebraic properties related to subdirect sums and Hadamard products.
11
Content available remote

New bounds for the minimum eigenvalue of 𝓜-tensors

76%
Zhao J., Sang C.
Open Mathematics
|
2017
|
tom 15
|
nr 1
296-303
EN
A new lower bound and a new upper bound for the minimum eigenvalue of an 𝓜-tensor are obtained. It is proved that the new lower and upper bounds improve the corresponding bounds provided by He and Huang (J. Inequal. Appl., 2014, 2014, 114) and Zhao and Sang (J. Inequal. Appl., 2016, 2016, 268). Finally, two numerical examples are given to verify the theoretical results.
12
Content available remote

On the Yang-Baxter-like matrix equation for rank-two matrices

76%
Zhou D., Chen G., Ding J.
Open Mathematics
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2017
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tom 15
|
nr 1
340-353
EN
Let A = PQT, where P and Q are two n × 2 complex matrices of full column rank such that QTP is singular. We solve the quadratic matrix equation AXA = XAX. Together with a previous paper devoted to the case that QTP is nonsingular, we have completely solved the matrix equation with any given matrix A of rank-two.
13
Content available remote

The maximum multiplicity and the two largest multiplicities of eigenvalues in a Hermitian matrix whose graph is a tree

76%
Fernandes R.
Special Matrices
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2015
|
tom 3
|
nr 1
EN
The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, M1, was understood fully (froma combinatorial perspective) by C.R. Johnson, A. Leal-Duarte (Linear Algebra and Multilinear Algebra 46 (1999) 139-144). Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree, we focus upon M2, the maximum value of the sum of the two largest multiplicities when the largest multiplicity is M1. Upper and lower bounds are given for M2. Using a combinatorial algorithm, cases of equality are computed for M2.
14
Content available remote

A note on certain ergodicity coeflcients

76%
Tudisco F.
Special Matrices
|
2015
|
tom 3
|
nr 1
EN
We investigate two ergodicity coefficients ɸ ∥∥ and τn−1, originally introduced to bound the subdominant eigenvalues of nonnegative matrices. The former has been generalized to complex matrices in recent years and several properties for such generalized version have been shown so far.We provide a further result concerning the limit of its powers. Then we propose a generalization of the second coefficient τ n−1 and we show that, under mild conditions, it can be used to recast the eigenvector problem Ax = x as a particular M-matrix linear system, whose coefficient matrix can be defined in terms of the entries of A. Such property turns out to generalize the two known equivalent formulations of the Pagerank centrality of a graph.
15
Content available remote

Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l 2 by the use of finite submatrices

76%
Malejki M.
Open Mathematics
|
2010
|
tom 8
|
nr 1
114-128
EN
We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l 2(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].
16
Content available remote

Two new eigenvalue localization sets for tensors and theirs applications

76%
Zhao J., Sang C.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1267-1276
EN
A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Qi (J. Symbolic Comput., 2005, 40, 1302-1324) and Li et al. (Numer. Linear Algebra Appl., 2014, 21, 39-50). As an application, a weaker checkable sufficient condition for the positive (semi-)definiteness of an even-order real symmetric tensor is obtained. Meanwhile, an S-type E-eigenvalue localization set for tensors is given and proved to be tighter than that presented by Wang et al. (Discrete Cont. Dyn.-B, 2017, 22(1), 187-198). As an application, an S-type upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.
17
Content available remote

Patterns with several multiple eigenvalues

65%
Dorsey J., Johnson C. R., Wei Z.
Special Matrices
|
2014
|
tom 2
|
nr 1
EN
Identified are certain special periodic diagonal matrices that have a predictable number of paired eigenvalues. Since certain symmetric Toeplitz matrices are special cases, those that have several multiple 5 eigenvalues are also investigated further. This work generalizes earlier work on response matrices from circularly symmetric models.
18
Content available remote

Bound for the largest singular value of nonnegative rectangular tensors

65%
He J., Liu Y.-M., Ke H., Tian J.-K., Li X.
Open Mathematics
|
2016
|
tom 14
|
nr 1
761-766
EN
In this paper, we give a new bound for the largest singular value of nonnegative rectangular tensors when m = n, which is tighter than the bound provided by Yang and Yang in “Singular values of nonnegative rectangular tensors”, Front. Math. China, 2011, 6, 363-378.
19
Content available remote

Equality in Wielandt’s eigenvalue inequality

65%
Friedland S.
Special Matrices
|
2015
|
tom 3
|
nr 1
EN
In this paper we give necessary and sufficient conditions for the equality case in Wielandt’s eigenvalue inequality.
20
Content available remote

Perron-Frobenius and Krein-Rutman theorems for tangentially positive operators

65%
Kanigowski A., Kryszewski W.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2240-2263
EN
We study several aspects of a generalized Perron-Frobenius and Krein-Rutman theorems concerning spectral properties of a (possibly unbounded) linear operator on a cone in a Banach space. The operator is subject to the so-called tangency or weak range assumptions implying the resolvent invariance of the cone. The further assumptions rely on relations between the spectral and essential spectral bounds of the operator. In general we do not assume that the cone induces the Banach lattice structure into the underlying space.
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