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．     

Generic existence and uniqueness of positive eigenvalues and eigenvectors

We consider a closed cone of positive operators on an ordered Banach space and prove that a generic element of this cone has a unique positive eigenvalue and a unique (up to a positive multiple) positive eigenvector. Moreover, the normalized iterations of such a generic element converge to its unique eigenvector.     

Localization of the eigenvalues of a pencil of positive definite matrices

Let A and B be real square positive definite matrices close to each other. A domain S on the complex plane that contains all the eigenvalues λ of the problem Az = λBz is constructed analytically. The boundary ?S of S is a curve known as the limacon of Pascal. Using the standard conformal mapping of the exterior of this curve (or of the exterior of an enveloping circular lune) onto the exterior of the unit disc, new analytical bounds are obtained for the convergence rate of the minimal residual method (GMRES) as applied to solving the linear system Ax = b with the preconditioner B.     

Sufficient conditions for positive definiteness of tridiagonal matrices revisited

We review several sufficient conditions for the positive definiteness of a tridiagonal matrix and propose a different approach to the problem, recalling and comprising little-known results on chain sequences.     

On the eigenvalues of matrices

Summary Let A be an n×n matric with arbitrary complex elements and with eigen-values λ₁, λ₂, ..., λ_n. A method is described for the approximàte determination of max | λ_j | ; characteristical is that prescribing a percentual error the number of elementary operations of the process, necessary to reach such precision, depends only on n and not on the elements of A More general characteristical equations are also considered. To Enrico Bompiani on his scientific Jubilee     

Sufficient conditions for the convergent splittings of non-Hermitian positive definite matrices

We present sufficient conditions for the convergent splitting of a non-Hermitian positive definite matrix. These results are applicable to identify the convergence of iterative methods for solving large sparse system of linear equations.     

Estimates for eigenvalues of stochastic matrices

It is well-known that the eigenvalues of stochastic matrices lie in the unit circle and at least one of them has the value one. Let {1, r 2 , ··· , r N } be the eigenvalues of stochastic matrix X of size N × N . We will present in this paper a simple necessary and sufficient condition for X such that |r j | < 1, j = 2, ··· , N . Moreover, such condition can be very quickly examined by using some search algorithms from graph theory.     

Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices

The sum of the largest eigenvalues of a symmetric matrix is a nonsmooth convex function of the matrix elements. Max characterizations for this sum are established, giving a concise characterization of the subdifferential in terms of a dual matrix. This leads to a very useful characterization of the generalized gradient of the following convex composite function: the sum of the largest eigenvalues of a smooth symmetric matrix-valued function of a set of real parameters. The dual matrix provides the information required to either verify first-order optimality conditions at a point or to generate a descent direction for the eigenvalue sum from that point, splitting a multiple eigenvalue if necessary. Connections with the classical literature on sums of eigenvalues and eigenvalue perturbation theory are discussed. Sums of the largest eigenvalues in the absolute value sense are also addressed.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.The work of this author was supported by the National Science Foundation under grants CCR-8802408 and CCR-9101640.The work of this author was supported in part during a visit to Argonne National Laboratory by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under contract W-31-109-Eng-38, and in part during a visit to the Courant Institute by the U.S. Department of Energy under Contract DEFG0288ER25053.     

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.     

Inclusion domains for the eigenvalues of stochastic matrices

Summary Inclusion domains for the nontrivial eigenvalues of stochastic matrices are given which are closely related to a bound for the nontrivial eigenvalues given by Bauer, Deutsch and Stoer. The inclusion domains are constructed by adapting Bauer's concept of a field of values subordinate to norms to the more general case of seminorms.Parts of this paper were presented at the Gatlinburg Symposium on Numerical Algebra in 1969.     

Inequalities for generalized eigenvalues of quaternion matrices

Periodica Mathematica Hungarica - Studying eigenvalues of square matrices is a traditional and fundamental direction in linear algebra. Quaternion matrices constitute an important and extensively...     

Bounds for eigenvalues of certain stochastic matrices

We consider the class of stochastic matrices M generated in the following way from graphs: if G is an undirected connected graph on n vertices with adjacency matrix A, we form M from A by dividing the entries in each row of A by their row sum. Being stochastic, M has the eigenvalue λ=1 and possibly also an eigenvalue λ=-1. We prove that the remaining eigenvalues of M lie in the disk ¦λ¦?1–n^-3, and show by examples that the order of magnitude of this estimate is best possible. In these examples, G has a bar-bell structure, in which n/3 of the vertices are arranged along a line, with n/3 vertices fully interconnected at each end. We also obtain better bounds when either the diameter of G or the maximal degree of a vertex is restricted.     

Orlicz norm estimates for eigenvalues of matrices

Let φ be a supermultiplicative Orlicz function such that the function $t \mapsto \varphi \left( {\sqrt t } \right)$ is equivalent to a convex function. Then each complexn×n matrixT=(τ _ij ) _{i, j} satisfies the following eigenvalue estimate: $\left\| {\left( {\lambda _i \left( T \right)} \right)_{i = 1}^n } \right\|_{\ell _\varphi } \leqslant C\left\| ( \right\|\left( {\tau _{ij} } \right)_{i = 1}^n \left\| {_{_{\ell _{\varphi *} } } )_{j = 1}^n } \right\|\ell _{\bar \varphi } $ . Here, ?_* stands for Young’s conjugate function of φ, ?, $\bar \varphi $ is the minimal submultiplicative function dominating φ andC>0 a constant depending only on φ. For the power function φ(t)=t ^p ,p≥2 this is a celebrated result of Johnson, König, Maurey and Retherford from 1979. In this paper we prove the above result within a more general theory of related estimates.     

Geometric properties of positive definite matrices cone with respect to the Thompson metric

We shall discuss geometric properties of a quadrangle with parallelogramic properties in a convex cone of positive definite matrices with respect to Thompson metric.     

任意阶中立型方程正解存在的充要条件

给出了任意阶中立型微分方程（x(t)-p(t)g(x(τ(t)))^(n) ∫α^βt(t,ξ,x(g1(t,ξ））,x(g2(t,ξ））,…,x(gm(t,ξ）））dη（ξ）=0存在正解x(t)满足x(t)-p(t)g(x(τ(t)))/t^k→正常数（→∞)物条件,作为本文结果的特例,部分地解决了文[5]提出的公开问题2。     

Homotopy method for the eigenvalues of symmetric tridiagonal matrices

We will present the homotopy method for finding eigenvalues of symmetric, tridiagonal matrices. This method finds eigenvalues separately, which can be a large advantage on systems with parallel processors. We will introduce the method and establish some bounds that justify the use of Newton’s method in constructing the homotopy curves.     

On eigenvalues of rectangular matrices

Given a (k+1)-tuple A,B ₁, ..., B _k of m×n matrices with m ≤ n, we call the set of all k-tuples of complex numbers {λ ₁, ..., λ k} such that the linear combination A+λ ₁ B ₁+λ ₂ B ₂+ ... +λ _k B _k has rank smaller than m the eigenvalue locus of the latter pencil. Motivated primarily by applications to multiparameter generalizations of the Heine-Stieltjes spectral problem, we study a number of properties of the eigenvalue locus in the most important case k = n−m+1.     

Smallest eigenvalues of Hankel matrices for exponential weights

We obtain the rate of decay of the smallest eigenvalue of the Hankel matrices