On the distribution of real eigenvalues in linear viscoelastic oscillators

In this paper, a linear viscoelastic system is considered where the viscoelastic force depends on the past history of motion via a convolution integral over an exponentially decaying kernel function. The free‐motion equation of this nonviscous system yields a nonlinear eigenvalue problem that has a certain number of real eigenvalues corresponding to the nonoscillatory nature. The quality of the current numerical methods for deriving those eigenvalues is directly related to damping properties of the viscoelastic system. The main contribution of this paper is to explore the structure of the set of nonviscous eigenvalues of the system while the damping coefficient matrices are rank deficient and the damping level is changing. This problem will be investigated in the cases of low and high levels of damping, and a theorem that summarizes the possible distribution of real eigenvalues will be proved. Moreover, upper and lower bounds are provided for some of the eigenvalues regarding the damping properties of the system. Some physically realistic examples are provided, which give us insight into the behavior of the real eigenvalues while the damping level is changing.     

Error estimate of eigenvalues of discrete linear Hamiltonian systems with small perturbation

In this paper, eigenvalues of perturbed discrete linear Hamiltonian systems are considered. A new variational formula of eigenvalues is first established. Based on it, error estimates of eigenvalues of systems with small perturbation are given under certain non-singularity conditions. Small perturbations of the coefficient functions, the weight function and the coefficients of the boundary condition are all involved. As a direct consequence, continuous dependence of eigenvalues on boundary value problems is obtained under the non-singularity conditions. In addition, two examples are presented to illustrate the necessity of the non-singularity conditions and the complexity of the problem in the singularity case.     

Generalization of some results concerning eigenvalues of a certain class of matrices and some applications

In this note, we present a generalization of some results concerning the spectral properties of a certain class of block matrices. As applications, we study some of its implications on nonnegative matrices and doubly stochastic matrices as well as on graph spectra and graph energy.     

Main eigenvalues of a graph

An eigenvalue of a graph is said to be a main eigenvalue if it has an eigenvector not orthogonal to the main vector j=(1,1,…,1). In this paper we shall study some properties of main eigenvalues of a graph.     

Upper bounds on the maximum modulus of subdominant eigenvalues of nonnegative matrices

Upper bounds for the maximum modulus of the subdominant roots of square nonnegative matrices are obtained. We provide a unified approach that yields or improves upon most of the bounds that have been obtained so far.     

On the eigenvalues of quaternion matrices

This article is a continuation of the article [F. Zhang, Ger?gorin type theorems for quaternionic matrices, Linear Algebra Appl. 424 (2007), pp. 139–153] on the study of the eigenvalues of quaternion matrices. Profound differences in the eigenvalue problems for complex and quaternion matrices are discussed. We show that Brauer's theorem for the inclusion of the eigenvalues of complex matrices cannot be extended to the right eigenvalues of quaternion matrices. We also provide necessary and sufficient conditions for a complex square matrix to have infinitely many left eigenvalues, and analyse the roots of the characteristic polynomials for 2?×?2 matrices. We establish a characterisation for the set of left eigenvalues to intersect or be part of the boundary of the quaternion balls of Ger?gorin.     

Implicit construction of multiple eigenvalues for trees

We are generally concerned with the possible lists of multiplicities for the eigenvalues of a real symmetric matrix with a given graph. Many restrictions are known, but it is often problematic to construct a matrix with desired multiplicities, even if a matrix with such multiplicities exists. Here, we develop a technique for construction using the implicit function theorem in a certain way. We show that the technique works for a large variety of trees, give examples and determine all possible multiplicities for a large class of trees for which this was not previously known.     

Normal essential eigenvalues in the boundary of the numerical range

A purely geometric property of a point in the boundary of the numerical range of an operator on Hilbert space is examined which implies that such a point is the value at of a multiplicative linear functional of the -algebra, , generated by and the identity operator. Roughly speaking, such a property means that the boundary of the numerical range (of ) has infinite curvature at that point. Furthermore, it is shown that if such a point is not a sharp linear corner of the numerical range of , then the multiplicative linear functional vanishes on the compact operators in .

     

New regions including eigenvalues of Toeplitz matrices

Two new eigenvalue inclusion regions for matrices with a constant main diagonal are given. We then apply these results to Toeplitz matrices, and obtain two regions including all eigenvalues of Toeplitz matrices. Furthermore, it is proved that the new regions are tighter than those in [Melman A. Ovals of Cassini for Toeplitz matrices, Linear and Multilinear Algebra. 2012;60:189–199].     

二部竞赛矩阵的实特征值

令ζm,n表示所有的不可约m×n二部竞赛矩阵,M∈ζm,n和实数k≠0,本文主要获得了下述结论:首先研究k是M的特征值时k的几何重数,然后研究k是M的特征值的一些充要条件,最后讨论k是M的特征值时M的性质.     

Asymptotic behavior of eigenvalues and random updating schemes

For a stochastic matrix (Q _ij ^T ) _i,j=1 ^M withQ _ij ^T exp(–U(ij)/T) at the off-diagonal positions, we develop an algorithm to evaluate the asymptotic convergence rate of all eigenvalues ofQ _ij ^T asT 0 using Ventcel's optimal graphs. As an application we can compare the convergence rates of some random updating schemes used in image processing.This research was partially supported by the National Science Council, Taiwan and Air Force Office of Scientific Research Contract No. F49620 S5C 0144, and was completed while Tzuu-Shuh Chiang was visiting the Center for Stochastic Processes, Department of Statistics, University of North Carolina, Chapel Hill, NC 27599-3260, USA.     

Sturm-Liouville问题的特征值对势函数的依赖性

研究了定义在[0,1]上的Sturm-Liouville问题的特征值对势函数的连续依赖性.应用比较定理和定义区间单调性证明了:当部分区间[x0,1]上的势函数趋于无穷大时,[0,1]区间上的特征值渐进趋近于[0,x0]区间上的某个特征值.推广了一些作者对Sturm-Liouville问题研究的相应结果,并为其相应问题的研究提供了一个新的视角.     

Estimates for eigenvalues of Laplacian operator with any order

Let D be a bounded domain in an n-dimensional Euclidean space Rn. Assume that 0 ＜ λ1 ≤λ2 ≤ … ≤ λκ ≤ … are the eigenvalues of the Dirichlet Laplacian operator with any order l{(-△)lu=λu, in D u=(δ)u/(δ)(→n)=…(δ)l-1u/(δ)(→n)l-1=0,on (δ)D.Then we obtain an upper bound of the (k 1)-th eigenvalue λκ 1 in terms of the first k eigenvalues.k∑i=1(λκ 1-λi) ≤ 1/n[4l(n 2l-2)]1/2{k∑i=1(λκ 1-λi)1/2λil-1/l k∑i=1(λκ 1-λi)1/2λ1/li}1/2.This ineguality is independent of the domain D. Furthermore, for any l ≥ 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang.     

恰有两个Q-主特征值的三圈图

图G的无符号拉普拉斯矩阵定义为图G的邻接矩阵与度对角矩阵的和,其特征值称为图G的Q-特征值.图G的一个Q-特征值称为Q-主特征值,如果它有一个特征向量其分量的和不等于零.确定了所有恰有两个Q-主特征值的三圈图.     

A difference-analytical method for the computation of eigenvalues of the fourth-order equations with separated boundary conditions

We propose a method for the computation of eigenvalues with odd multiplicities for spectral boundary-value problems of the 4th order with separated boundary conditions. In this method approximate eigenvalues appear as zeros of a certain function f(λ) which admits an explicit representation.     

The largest two Laplacian eigenvalues of a graph

In this article, we present lower bounds for the largest eigenvalue, the second largest eigenvalue and the sum of the two largest eigenvalues of the Laplacian matrix of a graph.