Partly zero eigenvectors and matrices that satisfy Au = b

For given vectors u,b ∈ F ⁿ (where F is a field with at least 3 elements), we establish criteria for deciding whether a digraph allows in its pattern class a matrix A which satisfies Au = b. As corollaries to this we give necessary and sufficient conditions for a pattern class to allow a matrix which has an eigenvector with a particular zero/nonzero pattern. Moreover we establish whether or not that eigenvector can correspond to a zero or nonzero eigenvalue. We use these results to establish the analogous results for loopfree digraphs, and thus we obtain results additional to work already done by Maybee, Olesky, and Van Den Driessche in this area. We then use bipartite graphs to generalize our criteria for solutions to Au = b to the rectangular case.     

分配格上矩阵的特征向量

设(L,≤,∨,∧)为一分配格。满足Ax=x的向量x称为方阵A的特征向量。本文的主工目的是通过矩阵的伴随有向图来刻画矩阵的特征向量并给出矩阵特征向量的界。同时我们将定义矩阵A的上基本特征向量并讨论它的性质。     

On least eigenvalues and least eigenvectors of real symmetric matrices and graphs

We give an upper bound for the least eigenvalue of a principal submatrix of a real symmetric matrix with zero diagonal, from which we establish an upper bound for the least eigenvalue of a graph when some vertices are removed using the components of the least eigenvector(s). We give lower and upper bounds for the least eigenvalue of a graph when some edges are removed. We also establish bounds for the components of the least eigenvector(s) of a real symmetric matrix and a graph.     

A-posteriori residual bounds for Arnoldi’s methods for nonsymmetric eigenvalue problems

Convergence of the implicitly restarted Arnoldi (IRA) method for nonsymmetric eigenvalue problems has often been studied by deriving bounds for the angle between a desired eigenvector and the Krylov projection subspace. Bounds for residual norms of approximate eigenvectors have been less studied and this paper derives a new a-posteriori residual bound for nonsymmetric matrices with simple eigenvalues. The residual vector is shown to be a linear combination of exact eigenvectors and a residual bound is obtained as the sum of the magnitudes of the coefficients of the eigenvectors. We numerically illustrate that the convergence of the residual norm to zero is governed by a scalar term, namely the last element of the wanted eigenvector of the projected matrix. Both cases of convergence and non-convergence are illustrated and this validates our theoretical results. We derive an analogous result for implicitly restarted refined Arnoldi (IRRA) and for this algorithm, we numerically illustrate that convergence is governed by two scalar terms appearing in the linear combination which drives the residual norm to zero. We provide a set of numerical results that validate the residual bounds for both variants of Arnoldi methods.     

First eigenvalue and first eigenvectors of a nonsingular unicyclic mixed graph

Let G be a mixed graph and let L(G) be the Laplacian matrix of the graph G. The first eigenvalue and the first eigenvectors of G are respectively referred to the least nonzero eigenvalue and the corresponding eigenvectors of L(G). In this paper we focus on the properties of the first eigenvalue and the first eigenvectors of a nonsingular unicyclic mixed graph (abbreviated to a NUM graph). We introduce the notion of characteristic set associated with the first eigenvectors, and then obtain some results on the sign structure of the first eigenvectors. By these results we determine the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order and fixed girth, and the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order.     

Backward errors for eigenvalues and eigenvectors of Hermitian,skew-Hermitian,H-even and H-odd matrix polynomials

We discuss the perturbation analysis for eigenvalues and eigenvectors of structured homogeneous matrix polynomials with Hermitian, skew-Hermitian, H-even and H-odd structure. We construct minimal structured perturbations (structured backward errors) such that an approximate eigenvalue and eigenvector pair (finite or infinite eigenvalues) is an exact eigenvalue eigenvector pair of an appropriately perturbed structured matrix polynomial. We present various comparisons with unstructured backward errors and previous backward errors constructed for the non-homogeneous case and show that our results generalize previous results.     

Maximal and minimal entry in the principal eigenvector for the distance matrix of a graph

Let G=(V,E) be a simple, connected and undirected graph with vertex set V(G) and edge set E(G). Also let D(G) be the distance matrix of a graph G (Jane?i? et al., 2007) [13]. Here we obtain Nordhaus–Gaddum-type result for the spectral radius of distance matrix of a graph.A sharp upper bound on the maximal entry in the principal eigenvector of an adjacency matrix and signless Laplacian matrix of a simple, connected and undirected graph are investigated in Das (2009) [4] and Papendieck and Recht (2000) [15]. Generally, an upper bound on the maximal entry in the principal eigenvector of a symmetric nonnegative matrix with zero diagonal entries and without zero diagonal entries are investigated in Zhao and Hong (2002) [21] and Das (2009) [4], respectively. In this paper, we obtain an upper bound on minimal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs. Moreover, we present the lower and upper bounds on maximal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs.     

Newton's method for the common eigenvector problem

In El Ghazi et al. [Backward error for the common eigenvector problem, CERFACS Report TR/PA/06/16, Toulouse, France, 2006], we have proved the sensitivity of computing the common eigenvector of two matrices A and B, and we have designed a new approach to solve this problem based on the notion of the backward error.If one of the two matrices (saying A) has n eigenvectors then to find the common eigenvector we have just to write the matrix B in the basis formed by the eigenvectors of A. But if there is eigenvectors with multiplicity >1, the common vector belong to vector space of dimension >1 and such strategy would not help compute it.In this paper we use Newton's method to compute a common eigenvector for two matrices, taking the backward error as a stopping criteria.We mention that no assumptions are made on the matrices A and B.     

Functions of small growth with no unbounded Fatou components

We prove a form of the cos πρ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition for a transcendental entire function to have no unbounded Fatou components. These two results enable us to show that there is a large class of entire functions of order zero which have no unbounded Fatou components. On the other hand, we give examples which show that there are in fact functions of order zero which not only fail to satisfy Hinkkanen’s condition but also fail to satisfy our more general condition. We also give a new regularity condition that is sufficient to ensure that a transcendental entire function of order less than 1/2 has no unbounded Fatou components. Finally, we observe that all the conditions given here which guarantee that a transcendental entire function has no unbounded Fatou components also guarantee that the escaping set is connected, thus answering a question of Eremenko for such functions.     

A fixed point technique to refine a simple approximate eigenvalue and a corresponding eigenvector

Let T be a bounded operator on a Banach space X. Let λ₀ be a nonzero simple eigenvalue of a ‘nearby’ operator T₀ and let ?₀ be a corresponding eigenvector. Several modified versions of a fixed point scheme are given for iteratively refining the initial approximations λ₀ and ?₀ of an eigenvalue λ of T and a corresponding eigenvector ? Convergence of these schemes is proved by considering error bounds for the iterates. These bounds hold if a compact operator T is approximated in the norm or in a Collectively compact manner by a sequence (T₀) of bounded operators, and λ₀ and ?₀ are eigenelements of T_n0 for a fixed _n0 of ‘moderate’ size. Numerical examples are no included to illustrate the performation of various iteration schemes.     

On Boundary Conditions for Wedge Operators on Radial Sets

We present a theorem about calculation of fixed point index for k-ψ-contractive operators with 0 ≤ k < 1 defined on a radial set of a wedge of an infinite-dimensional Banach space. Then, results on the existence of eigenvectors and nonzero fixed points are obtained.     

用简单反迭代法计算三对角矩阵的特征向量

与特征值计算的算法丰富多彩相比,在已知比较精确的特征值的情况下,求其相应的特征向量的算法却不多见,已有的算法有基本反迭代法[1][2][4][5]、交替法[3]等.到目前为止,计算特征向量的算法都是基于反迭代法的,衡量算法是否收敛都是以残量的大小为标准,本文的算法也不例外.本文的目的就是计算不可约实对称三对角矩阵T=[bj-1,aj,bj]的相应于某个特征值λi（已得到其近似λ）的特征向量.首先我们来看下面的例子:例1 我们取T为201阶的Wilkinson负矩阵,λ取计算的最大特征值,分别令迭代的初始向量是e1,e100,e201,e=（1,1,…,1）T.图1反映了反迭代的收敛速度.     

Classification Results for λ-Biminimal Surfaces in 2-Dimensional Complex Space Forms

We completely classify λ-biharmonic slant surfaces and λ-biminimal Lagrangian surfaces in 2-dimensional complex space forms, under the condition that the mean curvature is nonzero constant. In addition, we provide some examples of λ-biminimal slant surfaces with nonzero constant mean curvature, which are neither Lagrangian nor λ-biharmonic.     

Aggregating nonnegative eigenvectors of the adjacency matrix as a measure of centrality for a directed graph

Eigenvector centrality is a popular measure that uses the principal eigenvector of the adjacency matrix to distinguish importance of nodes in a graph. To find the principal eigenvector, the power method iterating from a random initial vector is often adopted. In this article, we consider the adjacency matrix of a directed graph and choose suitable initial vectors according to strongly connected components of the graph instead so that nonnegative eigenvectors, including the principal one, can be found. Consequently, for aggregating nonnegative eigenvectors, we propose a weighted measure of centrality, called the aggregated-eigenvector centrality. Weighting each nonnegative eigenvector by the reachability of the associated strongly connected component, we can obtain a measure that follows a status hierarchy in a directed graph.     

On generating eigenvectors of multiparameter polynomial matrices

The notion of a generating eigenvector of a multiparameter polynomial matrix, generalizing the notion of an eigenvector of a one-parameter polynomial matrix, is introduced. Some properties of generating eigenvectors are established, and two methods for constructing them are suggested. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 165–186. Translated by V. B. Khazanov.     

Zero-Term Ranks of Real Matrices and Their Preservers

Zero-term rank of a matrix is the minimum number of lines (rows or columns) needed to cover all the zero entries of the given matrix. We characterize the linear operators that preserve zero-term rank of the m× nreal matrices. We also obtain combinatorial equivalent condition for the zero-term rank of a real matrix.     

Nowhere-zero eigenvectors of graphs

A vector is called nowhere-zero if it has no zero entry. In this article we search for graphs with nowhere-zero eigenvectors. We prove that distance-regular graphs and vertex-transitive graphs have nowhere-zero eigenvectors for all of their eigenvalues and edge-transitive graphs have nowhere-zero eigenvectors for all non-zero eigenvalues. Among other results, it is shown that a graph with three distinct eigenvalues has a nowhere-zero eigenvector for its smallest eigenvalue.     

Almost sure,L_1-and L_2-growth behavior of supercritical multi-type continuous state and continuous time branching processes with immigration

Under a first order moment condition on the immigration mechanism, we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration(CBI process) converges almost surely. If an x log(x) moment condition on the branching mechanism does not hold, then the limit is zero. If this x log(x) moment condition holds, then we prove L_1 convergence as well. The projection of the limit on any left non-Perron eigenvector of the branching mean matrix is vanishing.If, in addition, a suitable extra power moment condition on the branching mechanism holds, then we provide the correct scaling for the projection of a CBI process on certain left non-Perron eigenvectors of the branching mean matrix in order to have almost sure and L_1 limit. Moreover, under a second order moment condition on the branching and immigration mechanisms, we prove L_2 convergence of an appropriately scaled process and the above-mentioned projections as well. A representation of the limits is also provided under the same moment conditions.     

Nonexistence of formal first integrals for general nonlinear systems under resonance

In this paper, we consider the nonintegrability for nonlinear systems under the simple resonant case, i.e., the Jacobian matrix of vector field at some fixed point has some single multiply zero eigenvalues, and some nonzero eigenvalues which are N-independent. By using the Poincaré-Dulac normal form theory, we give a necessary condition for the system under consideration to have formal first integral.     

On the Computation of Particular Eigenvectors of Hamiltonian Matrix Pencils

We discuss a structure-preserving algorithm for the accurate solution of generalized eigenvalue problems for skew-Hamiltonian/Hamiltonian matrix pencils λN − ℋ. By embedding the matrix pencil λ𝒩 − ℋ into a skew-Hamiltonian/Hamiltonian matrix pencil of double size it is possible to avoid the problem of non-existence of a structured Schur form. For these embedded matrix pencils we can compute a particular condensed form to accurately compute the simple, finite, purely imaginary eigenvalues of λ𝒩 − ℋ. In this paper we describe a new method to compute also the corresponding eigenvectors by using the information contained in the condensed form of the embedded matrix pencils and associated transformation matrices. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)