A method for separating nearly multiple eigenvalues for Hermitian matrix

In this paper, we propose a numerical method to verify for nearly multiple eigenvalues of a Hermitian matrix not being strictly multiple eigenvalues. From approximate eigenvalues computed, it seems to be difficult to distinguish whether they are strictly multiple eigenvalues or simple ones, and if they are very close each other, the verification method for simple eigenvalues may fail to enclose them separately, because of singularity of the system in the verification. There are several methods for enclosing multiple and nearly multiple eigenvalues (e.g., [Rump, Computational error bounds for multiple or nearly multiple eigenvalues, Linear Algebra Appl. 324 (2001) 209–226]), For such cases, there is no result to decide the enclosed eigenvalues are nearly multiple or strictly multiple, up to now. So, for enclosed eigenvalues, we propose a numerical method to separate nearly multiple eigenvalues.     

基于矩阵幂运算的重特征值存在性定理

对于判断矩阵重特征值的存在性问题,运用“若λ是矩阵A的特征值,则入“是Ak的特征值”这一性质,通过矩阵的迹与特征值的关系,得到了实数域上矩阵重特征值的存在性定理并给出了证明．定理实现了“由矩阵幂运算来判断矩阵重特征值的存在性”这样一个计算过程,对讨论矩阵特征值问题具有一定的启示意义．     

二部图的单特征值

一个图的特征值通常指的是它的邻接矩阵的特征值,在图的所有特征值中,重数为1的特征值即所谓的单特征值具有特殊的重要性.确定一个图的单特征值是一个比较困难的问题,主要是没有一个通用的方法.1969年,Petersdorf和Sachs给出了点传递图单特征值的取值范围,但是对于具体的点传递图还需要根据图本身的特性来确定它的单特征值.给出一类正则二部图,它们是二面体群的凯莱图,这类图的单特征值中除了它的正、负度数之外还有0或者±1,而它们恰好是Petersdorf和Sachs所给出的单特征值范围内的中间取值.     

Singular left-definite Sturm-Liouville problems

We study singular left-definite Sturm-Liouville problems with an indefinite weight function. The existence of eigenvalues is established based on the existence of eigenvalues of corresponding right-definite problems. Furthermore, for each singular left-definite problem with limit-circle non-oscillatory endpoints we construct a regular left-definite problem with the same eigenvalues and use it to obtain properties of eigenvalues and eigenfunctions. Inequalities among eigenvalues recently established for regular left-definite problems are extended to the singular case.     

Calculation of eigenvalues of a discrete self-adjoint operator perturbed by a bounded operator

We generalize the method of regularized traces which calculates eigenvalues of a perturbed discrete operator for the case of an arbitrary multiplicity of eigenvalues of the unperturbed operator. We obtain a system of equations, enabling one to calculate eigenvalues of the perturbed operator with large ordinal numbers. As an example, we calculate eigenvalues of a perturbed Laplace operator in a rectangle.     

The Eigenvalues of Mathieu's Equation and their Branch Points

A comprehensive account is given of the behavior of the eigenvalues of Mathieu's equation as functions of the complex variable q. The convergence of their small-q expansions is limited by an infinite sequence of rings of branch points of square-root type at which adjacent eigenvalues of the same type become equal. New asymptotic formulae are derived that account for how and where the eigenvalues become equal. Known asymptotic series for the eigenvalues apply beyond the rings of branch points; we show how they can now be identified with specific eigenvalues.     

Inequalities of Eigenvalues for the Dirac Operator on Compact Complex Spin Submanifolds in Complex Projective Spaces

For a compact complex spin manifold M with a holomorphic isometric embed- ding into the complex projective space,the authors obtain the extrinsic estimates from above and below for eigenvalues of the Dirac operator,which depend on the data of an isometric embedding of M.Further,from the inequalities of eigenvalues,the gaps of the eigenvalues and the ratio of the eigenvalues are obtained.     

矩阵特征值的一类新的包含域

用盖尔圆盘定理来估计矩阵的特征值是一个经典的方法,这种方法仅利用矩阵的元素来确定特征值的分布区域.本文利用相似矩阵有相同的特征值这一理论,得到了矩阵特征值的一类新的包含域,它们与盖尔圆盘等方法结合起来能提高估计的精确度.     

Spectral Analysis of an Operator Arising in Fluid Dynamics

Eigenmode solutions are very important in stability analysis of dynamical systems. The set of eigenvalues of a non-self-adjoint differential operator originated from the linearization of some Cauchy problem is investigated. It is shown that the eigenvalues are purely imaginary, and that they are related to the eigenvalues of Heun's differential equation. These two results are used to derive the asymptotic behavior of the eigenvalues and to compute them numerically.     

Isoperimetric control of the Steklov spectrum

We prove that the normalized Steklov eigenvalues of a bounded domain in a complete Riemannian manifold are bounded above in terms of the inverse of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above. On a compact surface with boundary, we obtain uniform bounds for the normalized Steklov eigenvalues in terms of the genus. We also establish a relationship between the Steklov eigenvalues of a domain and the eigenvalues of the Laplace-Beltrami operator on its boundary hypersurface.     

Location for the right eigenvalues of quaternion matrices

This paper aims to discuss the location for right eigenvalues of quaternion matrices. We will present some different Gerschgorin type theorems for right eigenvalues of quaternion matrices, based on the Gerschgorin type theorem for right eigenvalues of quaternion matrices (Zhang in Linear Algebra Appl. 424:139?C153, 2007), which are used to locate the right eigenvalues of quaternion matrices. We shall conclude this paper with some easily computed regions which are guaranteed to include the right eigenvalues of quaternion matrices in 4D spaces.     

Computable eigenvalue bounds for rank-k perturbations

We investigate lower bounds for the eigenvalues of perturbations of matrices. In the footsteps of Weyl and Ipsen & Nadler, we develop approximating matrices whose eigenvalues are lower bounds for the eigenvalues of the perturbed matrix. The number of available eigenvalues and eigenvectors of the original matrix determines how close those approximations can be, and, if the perturbation is of low rank, such bounds are relatively inexpensive to obtain. Moreover, because the process need not be restricted to the eigenvalues of perturbed matrices, lower bounds for eigenvalues of bordered diagonal matrices as well as for singular values of rank-k perturbations and other updates of n×m matrices are given.     

Eigenvalues of large sample covariance matrices of spiked population models

We consider a spiked population model, proposed by Johnstone, in which all the population eigenvalues are one except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the non-unit population ones when both sample size and population size become large. This paper completely determines the almost sure limits of the sample eigenvalues in a spiked model for a general class of samples.     

Dependence of eigenvalues of Dirac system on the parameters

The dependence of eigenvalues of Dirac system with general boundary conditions is studied. It is shown that the eigenvalues of Dirac operators depend not only continuously but also smoothly on the coefficients, the boundary conditions, and the endpoints of the problem. Furthermore, the differential expressions of the eigenvalues as regards these parameters are given. The results obtained in this paper would provide theoretical support for the numerical calculations of eigenvalues of the corresponding problems.     

Numerical enclosure for multiple eigenvalues of an Hermitian matrix whose graph is a tree

In this paper we consider a numerical enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. If an Hermitian matrix A whose graph is a tree has multiple eigenvalues, it has the property that matrices which are associated with some branches in the undirected graph of A have the same eigenvalues. By using this property and interlacing inequalities for Hermitian matrices, we show an enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. Since we do not generally know whether a given matrix has exactly a multiple eigenvalue from approximate computations, we use the property of interlacing inequalities to enclose some eigenvalues including multiplicities.In this process, we only use the enclosure of simple eigenvalues to enclose a multiple eigenvalue by using a computer and interval arithmetic.     

矩阵特征值、特征向量的确定

首先对由 A的特征值、特征向量求 A- 1 ,AT,A* ( A的伴随矩阵 )、P- 1 AP以及 A的多项式φ( A)的特征值和特征向量的结论作了个归纳 ;对相反的情形 ,我们给出了部分已有的结果 ,并通过四道例题着重讨论了如何由 φ( A)的特征值来求 A的特征值 .     

具有周期系数的左定Sturm-Liouville问题的特征值不等式

该文利用了左定问题与右定问题的联系,得到了具有周期系数的左定Sturm-Liouville问题在区间[a,a+kh]上的周期和半周期特征值的描述,阐明了周期特征值之间的不等式关系,并明确给出了区间[a,a+kh]上的周期、半周期特征值和区间[a,a+h]上特征值的一一对应关系.     

A correction method for finding lower bounds of eigenvalues of the second‐order elliptic and Stokes operators

In this paper, for the second‐order elliptic and Stokes eigenvalue problems with variable coefficients, we propose a correction method to nonconforming eigenvalue approximations and prove that the corrected eigenvalues converge to the exact ones asymptotically from below. In particular, the asymptotic lower bound property of corrected eigenvalues is always valid whether the eigenfunctions are smooth or singular. Finally, we prove that the convergence order of corrected eigenvalues is still the same as that of uncorrected eigenvalues.     

多参数二次特征值问题重特征值的灵敏度分析

本文研究解析依赖于多参数的二次特征值问题重特征值的灵敏度分析,得到了重特征值的方向导数,证明了相应的特征向量矩阵和特征值平均值的解析性,给出了其一阶偏导数的表达式．然后以这些结论为基础,定义了二次特征值问题重特征值及其不变子空间的灵敏度,并给出了确定二次特征值问题所含矩阵中敏感元素的方法．     

Symmetric nonnegative realizability via partitioned majorization

A sufficient condition for symmetric nonnegative realizability of a spectrum is given in terms of (weak) majorization of a partition of the negative eigenvalues by a selection of the positive eigenvalues. If there are more than two positive eigenvalues, an additional condition, besides majorization, is needed on the partition. This generalizes observations of Suleǐmanova and Loewy about the cases of one and two positive eigenvalues, respectively. It may be used to provide insight into realizability of 5-element spectra and beyond.