特征值与特征向量

特征值与特征向量的两种不同定义是一致的：线性变换/A与其对应的n阶矩阵A有相同的特征值,而n阶矩阵A的特征向量x是/A的特征向量ξ在基ε1,…εn下的坐标。     

二阶矩阵的特征值和特征向量常考题型例析

矩阵的特征值和特征向量是矩阵与变换的一个非常重要的内容,利用矩阵的特征值和特征向量,可以方便地计算多次矩阵变换的结果,而且在实际工程计算和工程控制中也发挥着重要作用.二阶矩阵的特征值和特征向量有两个基本内容.一是二阶矩阵的特征值和特征向量的概念：设A是一个二阶矩阵,如果对于实数λ,存在一个非零向量α,使得Aα=λα,那么λ称为A的一个特征值,而α称为A的属于特征值λ的一个特征向量.     

实对称矩阵、特征值、特征向量之间的关系及其几何意义的一点注记

本文提出并证明命题:设n 阶实对称矩阵A 的特征值中有一个是单根,其余是n-1重根,且已知属于单根的特征向量,则所有与属于单根的特征向量正交的非零向量都是属于n-1重根的特征向量,进而确定A,且以三阶实对称矩阵为例说明特征值与特征向量的几何意义。     

广义特征向量几种算法的比较

关于求一个矩阵 Jordan标准型的变换矩阵问题须计算矩阵的广义特征向量 ,目前常见的矩阵理论教材中有下述两种方法 .下设 A是 n× n阶矩阵 ,λ是 A的某特征值 ,其代数重数为 r,几何重数为 s且r>s.方法 1　由方程 ( A- λI) x=0解出 A的特征向量 p,由 ( A- λI) x=p解出第一个广义特征向量 q,这里的 p不能是任一特征向量而应是 A的 λ特征子空间内使方程 ( A- λI) x=p有解的那一个 .设 A的 λ特征子空间的基是 p1,p2 ,… ,ps,当 s>1时正确的说法是 :通过选择系数 k1,k2 ,… ,ks由方程 ( A- λI) x=k1p1+ k2 p2 +… + ksps解出第一个广…     

人工智能中的数学基础之一(线性代数篇)

<正>人工智能技术是建立在数学模型之上,包含有许多数学基础知识,其中线性代数研究的是以向量和矩阵的形式来研究抽象化的万事万物的变化规律.来看两个问题:(1)特征值和特征向量的意义;(2)范数的意义.可以这样来描述向量和矩阵:向量x是n维线性空间中的静止点;线性变换描述了向量的变化,用矩阵     

乘幂法求矩阵特征向量与特征值的初始向量及循环控制

矩阵迭代法是求矩阵的第一阶特征值与特征向量的一种数值方法 .本文讨论了用矩阵迭代法求解矩阵的特征值与特征向量时的初始向量选取和循环控制条件     

几类非自治系统的指数渐近稳定性

§1.预备知识对向量及矩阵引进模的概念如下:向量x的模记为||x|| ||X|| sum from i=1 to n |x_i|矩阵A的模记为||A|| ||A||sum from i.j=1 to n |a_(ij)|引理1设A为n×n阶常数矩阵,且它的所有特征根λ_k(k=1,2,…,n)均具有负     

一类二次特征值反问题的中心对称解及其最佳逼近

1引言给定n阶实矩阵M,C和K,二次特征值问题:求数λ和非零向量x使得Q(λ)x=0, (1．1)其中Q(λ)=λ2M λC K称为二次束．数λ和相应的非零向量x分别称为二次束Q(λ)的特征值和特征向量．Tisseur和Meerbergen概述了二次特征值问题的各种应用、数学理论和数值方法．在工程技术,特别是结构动力模型修正技术领域经常遇到与二次特征值问题相反的问题(称之为二次特征值反问题)．对阻尼结构进行动力分析时,应用有限元方法可得到系统的质量矩阵M,阻尼矩阵C和刚度矩阵K,从而可求得二次特征值问题的特征值(频率)和特征向量(振型)．但是有限元模型毕竟是实际结构系统的离散化,并且     

调和Arnoldi方法的一种变形

讨论求解大规模非对称矩阵内部特征问题的一种方法,与标准的调和A rnold i方法相比,该方法仍用调和R itz值作为特征值的近似,而在近似特征向量选取方面,我们充分利用A rnold i过程所提供的最末一个基向量的信息,在多1维K ry lov子空间中选取一个向量-称之为改进的调和R itz向量-作为所求的特征向量的近似.理论分析和数值试验均表明这种变形的调和A rnold i方法的可行性和有效性.     

逻辑矩阵的特征向量及应用

特征值与特征向量描述了线性变换的基本性质.特征向量是线性变换的作用下保持方向不变的向量,特征值体现了特征向量在线性变换中的伸缩性.讨论了一类布尔矩阵在布尔空间中的特征值与特征向量问题,证明了逻辑矩阵只有1特征值,所有1特征值构成1特征子空间,并且1特征子空间由唯一的一组基本特征向量布尔生成.最后,将逻辑矩阵特征向量的相关结果用于研究布尔网络极限环个数等拓扑性质.     

Computing eigenvectors of block tridiagonal matrices based on twisted block factorizations

New methods for computing eigenvectors of symmetric block tridiagonal matrices based on twisted block factorizations are explored. The relation of the block where two twisted factorizations meet to an eigenvector of the block tridiagonal matrix is reviewed. Based on this, several new algorithmic strategies for computing the eigenvector efficiently are motivated and designed. The underlying idea is to determine a good starting vector for an inverse iteration process from the twisted block factorizations such that a good eigenvector approximation can be computed with a single step of inverse iteration.     

Partly zero eigenvectors

This paper is concerned with the forced presence or absence of zero components in an eigenvector. Relative to a fixed matrix Awith eigenvalue λ, we characterize the strictly nonzero part of a partly zero eigenvector associated with λ. We also give a sufficient condition for a fixed matrix to have a partly zero eigenvector, and discuss several examples in which a matrix has one or more partly zero eigenvectors. Our main results, however, are qualitative in nature. We associate a zero/nonzero pattern class of matrices with a digraph, and characterize the set of pattern classes which requires all eigenvectors to be strictly nonzero. A sufficient condition is also given that identifies the components of a partly zero eigenvector which may be nonzero.     

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.     

COMPUTING EIGENVECTORS OF NORMAL MATRICES WITH SIMPLE INVERSE ITERATION

It is well-known that if we have an approximate eigenvalue λ- of a normal matrix A of order n,a good approximation to the corresponding eigenvector u can be computed by one inverse iteration provided the position,say kmax,of the largest component of u is known.In this paper we give a detailed theoretical analysis to show relations between the eigenvecor u and vector xk,k=1,…,n,obtained by simple inverse iteration,i.e.,the solution to the system(A-λI)x=ek with ek the kth column of the identity matrix I.We prove that under some weak conditions,the index kmax is of some optimal properties related to the smallest residual and smallest approximation error to u in spectral norm and Frobenius norm.We also prove that the normalized absolute vector v=|u|/||u||∞ of u can be approximated by the normalized vector of (||x1||2,…||xn||2)^T,We also give some upper bounds of |u(k)| for those “optimal“ indexeds such as Fernando‘s heuristic for kmax without any assumptions,A stable double orthogonal factorization method and a simpler but may less stable approach are proposed for locating the largest component of u.     

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.     

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.     

A control theory for stochastic eigenvectors of stochastic matrices through variations of entries

This paper describes the set of stochastic eigenvectors which can be obtained from a stochastic matrix S by variations of specified entries of S. Further, given any such eigenvector, the variations of entries of S needed to obtain this eigenvector are specified.     

Eigenvectors of interval matrices over max–plus algebra

The behaviour of a discrete-event dynamic system is often conveniently described using a matrix algebra with operations max and plus. Such a system moves forward in regular steps of length equal to the eigenvalue of the system matrix, if it is set to operate at time instants corresponding to one of its eigenvectors. However, due to imprecise measurements, it is often unappropriate to use exact matrices. One possibility to model imprecision is to use interval matrices. We show that the problem to decide whether a given vector is an eigenvector of one of the matrices in the given matrix interval is polynomial, while the complexity of the existence problem of a universal eigenvector remains open. As an aside, we propose a combinatorial method for solving two-sided systems of linear equations over the max–plus algebra.     

一类4×4无界算子矩阵的本征向量组的块状基性质及其在弹性力学中的应用*

本文讨论了力学中出现的一类4×4无界Hamilton算子矩阵的本征向量组的块状Schauder基性质.在一定的条件下, 考虑了此类Hamilton算子矩阵的本征值问题, 进而给出了其本征向量组是某个Hilbert空间的一组块状Schauder基的一个充要条件,并通过矩形薄板的自由振动和弯曲问题验证了所得结果的有效性.     

Optimal expansion of subspaces for eigenvector approximations

We consider the problem of how to expand a given subspace for approximating an eigenvalue and eigenvector of a matrix A. Specifically, we consider which vector in the subspace, after multiplied by A, provides optimal expansion of the existing subspace for the eigenvalue problem. We determine the optimal vector, when the quality of subspace for approximation is measured by the angle between the subspace and the eigenvector. We have also derived some characterization of the angle that might lead to more practically useful choice of the expansion vector.