实对称带状矩阵逆特征值问题

研究了一类实对称带状矩阵逆特征值问题：给定三个互异实数λ,μ和v及三个非零实向量x,y和z,分别构造实对称五对角矩阵T和实对称九对角矩阵A,使其都具有特征对(λ,x),(μ,y)和(v,z)．给出了此类问题的两种提法,研究了问题的可解性以及存在惟一解的充分必要条件,最后给出了数值算法和数值例子．     

The property of maximal eigenvectors of trees

Let T be a tree on n vertices and L(T) be its Laplacian matrix. The eigenvalues and eigenvectors of T are respectively referred to those of L(T). With respect to a given eigenvector Y of T, a vertex u of T is called a characteristic vertex if Y [u] = 0 and there is a vertex w adjacent to u with Y [w] ≠ 0; an edge e = (u, w) of G is called a characteristic edge if Y [u]Y [w] < 0. By 𝒞(T, Y) we denote the characteristic set of T with respect to the vector Y, which is defined as the collection of all characteristic vertices and characteristic edges of T corresponding to Y. Merris shows that 𝒞(T, Y) is fixed for all Fiedler vectors of the tree T. An eigenvector of T is called a k-vector (k ≥ 2) of T if this eigenvector corresponds to an eigenvalue λ _k with λ _k > λ _k?1, where λ₁, λ₂, …, λ _n are the eigenvalues of T arranged in non-decreasing order. A k-vector Y of T is called k-maximal if 𝒞(T, Y) has maximum cardinality among all k-vectors of T. We prove that (1) the characteristic set of T with respect to an arbitrary k-vector is contained in that with respect to any k-maximal vector; and consequently (2) the characteristic sets of T with respect to any two k-maximal vectors are same. Our result may be considered as a generalization of Merris' result as Fiedler vectors are 2-maximal.     

对称矩阵的两特征值问题

介绍了对称矩阵的两特征值问题,并给出了计算公式.     

On the jackknife statistics for eigenvalues and eigenvectors of a correlation matrix

This paper deals with some problems of eigenvalues and eigenvectors of a sample correlation matrix and derives the limiting distributions of their jackknife statistics with some numerical examples.     

对称双边对角矩阵特征值问题的计算

1 引 言 大型稀疏矩阵在工程上有广泛的应用.例如,结构工程的有限元分析、电力系统的分析、流体力学及图像数据压缩等应用中常遇到求大型稀疏矩阵的特征值问题.因而矩阵特征值计算问题成为数值代数领域长期关注的问题,如[6][7].最近M.Gu与S.C.Eisenstat     

Computation of multiple eigenvalues and generalized eigenvectors for matrices dependent on parameters

The paper develops Newton's method of finding multiple eigenvalues with one Jordan block and corresponding generalized eigenvectors for matrices dependent on parameters. It computes the nearest value of a parameter vector with a matrix having a multiple eigenvalue of given multiplicity. The method also works in the whole matrix space (in the absence of parameters). The approach is based on the versal deformation theory for matrices. Numerical examples are given. Copyright © 2005 John Wiley & Sons, Ltd.     

On the characteristic polynomial,eigenvectors and determinant of heptadiagonal matrices

In this paper, the characteristic polynomial of general heptadiagonal matrices is derived as well as eigenvectors associated to a prescribed eigenvalue. A symbolic algorithm to compute the determinant of heptadiagonal matrices is also presented allowing a suite implementation through computational software programs.     

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 .     

Iterative computation of derivatives of repeated eigenvalues and the corresponding eigenvectors

Recently the authors proposed a simultaneous iteration algorithm for the computation of the partial derivatives of repeated eigenvalues and the corresponding eigenvectors of matrices depending on several real variables. This paper analyses the properties of that algorithm and extends it in several ways. The previous requirement that the repeated eigenvalue be dominant is relaxed, and the new generalized algorithm given here allows the simultaneous treatment of simple and repeated eigenvalues. Methods for accelerating convergence are examined. Numerical results support our theoretical analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

IRAM‐based method for eigenpairs and their derivatives of large matrix‐valued functions

Based on the implicitly restarted Arnoldi method for eigenpairs of large matrix, a new method is presented for the computation of a few eigenpairs and their derivatives of large matrix‐valued functions. Eigenpairs and their derivatives are calculated simultaneously. Equation systems that are solved for eigenvector derivatives are greatly reduced from the original matrix size. The left eigenvectors are not required. Hence, the computational cost is saved. The convergence theory of the proposed method is established. Finally, numerical experiments are given to illustrate the efficiency of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree

We study the maximum possible multiplicity of an eigenvalue of a matrix whose graph is a tree, expressing that maximum multiplicity in terms of certain parameters associated with the tree.     

The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree

We study the maximum possible multiplicity of an eigenvalue of a matrix whose graph is a tree, expressing that maximum multiplicity in terms of certain parameters associated with the tree.     

一类混合图的结构及其特征空间

主要讨论具有如下性质的一类连通混合图G:其所有非奇异圈恰有一条公共边,且除了该公共边的端点外,任意两个非奇异圈没有其它交点.本文给出了图G的结构性质,建立了其最小特征值λ1(G)(以及相对应的特征向量)与某个简单图的代数连通度(以及Fiedler向量)之间联系,并应用上述联系证明了λ1(■)≤α(G),其中G是由G通过对其所有无向边定向而获得,α(■)为■的代数连通度.     

特征值反问题的唯一性

证明了由特征值及特征向量反求矩阵时,特征值在对角矩阵中的排序可以是任意的,只须将对应特征向量作相应排序,所得矩阵唯一。对于重特征值的线性无关的特征向量可任意选取,所得矩阵唯一。     

Sparse random graphs: Eigenvalues and eigenvectors

In this paper, we prove the semi‐circular law for the eigenvalues of regular random graph G_n,d in the case d →∞, complementing a previous result of McKay for fixed d. We also obtain a upper bound on the infinity norm of eigenvectors of Erd?s–Rényi random graph G(n,p), answering a question raised by Dekel–Lee–Linial. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

一类块三对角阵特征值的求解及应用

提出了求一类块三对角矩阵A的特征值和特征向量的方法,求得了该类矩阵的特征值和特征向量的表达式,并写出了用迭代法解该类方程组Au=f时迭代矩阵的特征值.