Jacobi矩阵特征值反问题的一个数值解法

本文利用Hessenberg矩阵特征值配置的一个结果以及三对角矩阵的有关性质,提出了一个求解Jacobi矩阵特征值反问题的数值方法。     

关于矩阵An的特征值的研究

从一道线性代数习题出发,举例说明常见教材中关于由矩阵A的特征值确定ψ(A)的特征值的结论不够完备,进而分析问题关键,运用求解特征多项式的方法推导出矩阵多项式的特征值.     

特征多项式的系数

用一种新方法证明了方阵的特征多项式的一般项的系数与该方阵的主子式密切相关.利用该结论和盖尔圆盘定理,证明了0是一类特殊Laplace矩阵的单特征值.     

二次自伴矩阵多项式阵的特征值

系统地论证了二次自伴矩阵多项式特征值,特征向量的性质.给出了二次自伴矩阵多项式特征值与任一非零向量所对应的二次多项式根之间的大小关系;精确地给出了二次自伴矩阵多项式是负定时参数的界;简化了二次自伴矩阵多项式的符号特征是正(负)的特征值对应特征向量间可以是线性无关等定理的证明.     

关于正矩阵的最大特征值的包含定理及其应用

1　引　　言由于矩阵特征值问题在弹性动力学和自动控制等领域均已获得广泛的应用,所以关于矩阵特征值的计算方法及其上、下界的估计均为人们所关注.随着计算机的发展,有关矩阵特征值的各种有效算法应运而生[1].至于特征值的上、下界的估计问题,虽然也有很多成果[2-4],且它们在数学上都有一定的理论意义和应用价值,但常因其界限太宽而缺少工程价值.鉴于此,笔者利用文[3]引入的同步向量这一概念,讨论了正矩阵的最大特征值的上、下界的确定问题,获得了这类矩阵最大特征值的较为精确的包含定理,又与幂法[1]相结合,给出了非亏损正矩阵的最大特征…     

整系数多项式的"分解矩阵"及其应用

在数学上 ,求微分方程的特征根、矩阵的特征值时 ,都会遇到多项式的因式分解问题 ;在工程上 ,研究动态系统的稳定性等问题时 ,也会遇到多项式的因式分解问题。传统的因式分解法有一定的局限性 ,它只适合于一些低次多项式或较规则的高次多项式的分解 ,而对一般高次多项式的因式分解 ,传统的方法常显出它的缺陷。本文就整系数多项式的因式分解问题 ,给出了一个比较好用的方法——矩阵法。该方法的核心就是根据多项式构造一个“分解矩阵”,再用此“分解矩阵”对多项式进行因式分解。该方法具有简便、实用的特点 ,特别适用于高次多项式的因式分…     

矩阵与其友矩阵相似的一个充要条件及其证明

得到一个矩阵A与其特征多项式的友矩阵C相似的充要条件是对应于A的每个不同的特征值λi,Jordan标准形中只含有一个Jordan子矩阵,并给出证明.     

矩阵方幂的特征多项式

给出四种方法,分别根据特征多项式的性质,多项式根与系数之间的关系以及对称多项式的知识,k次本原单位根,特征多项式的伴侣阵,可在矩阵的特征多项式已知的情况下确定其矩阵方幂的特征多项式.     

利用矩阵的初等变换求方阵的特征值

高阶方阵的特征多项式以及特征值的求得,在计算上往往有一定的难度.本文首先从理论上分析了存在一个上三角矩阵或者下三角矩阵与一个方阵相似;接着,提出了相似变换的概念,分析了相似变换中初等矩阵的选择方法;然后指出了利用相似变换在求方阵的特征多项式以及特征值时的方法,并列举若干实例给予了说明.     

结构矩阵的三角表示及其应用

利用位移铁和交换Hessenberg矩阵代数给出结构矩阵的三角表示,并讨论在Toeplitz矩阵和Toeplitz Hankel矩阵方面的应用。     

Functions of a matrix and Krylov matrices

For a given nonderogatory matrix A, formulas are given for functions of A in terms of Krylov matrices of A. Relations between the coefficients of a polynomial of A and the generating vector of a Krylov matrix of A are provided. With the formulas, linear transformations between Krylov matrices and functions of A are introduced, and associated algebraic properties are derived. Hessenberg reduction forms are revisited equipped with appropriate inner products and related properties and matrix factorizations are given.     

Polynomial Fibonacci–Hessenberg matrices

A Fibonacci–Hessenberg matrix with Fibonacci polynomial determinant is referred to as a polynomial Fibonacci–Hessenberg matrix. Several classes of polynomial Fibonacci–Hessenberg matrices are introduced. The notion of two-dimensional Fibonacci polynomial array is introduced and three classes of polynomial Fibonacci–Hessenberg matrices satisfying this property are given.     

Existence of matrices with prescribed entries

It is proved that, apart from for some exceptional cases, there always exists an n×n nonderogatory matrix over an arbitrary field with n prescribed entries and prescribed characteristic polynomial.     

GMRES and the Arioli,Pták,and Strakoš parametrization

In this paper we study the parametrization proposed by Arioli, Pták and Strakoš (BIT Numerical Mathematics, v 38, 1998) for the class of matrices having the same GMRES residual norm convergence curve. We give expressions for the Hessenberg matrix and the orthonormal basis vectors constructed by GMRES as well as for iterates and error vectors. The iterates do not depend on the eigenvalues in the sense that changing the coefficients of the characteristic polynomial in the parametrization does not change the GMRES iterates as well as the residuals. However, the error vectors do depend on these coefficients.     

On sufficient conditions for the existence of a unitary congruence transformation of a given complex matrix into a real one

A complex n × n matrix A is said to be nonderogatory if the degree of its minimal polynomial is equal to the degree of the characteristic polynomial. The aim of the paper is to prove the following assertion: Let A[`(A)] A\bar{A} be a nonderogatory matrix with real positive spectrum. Then A can be made real by a unitary congruence transformation if and only if A and [`(A)] \bar{A} are unitarily congruent. Bibliography: 5 titles.     

Eigenvalue computation for unitary rank structured matrices

In this paper we describe how to compute the eigenvalues of a unitary rank structured matrix in two steps. First we perform a reduction of the given matrix into Hessenberg form, next we compute the eigenvalues of this resulting Hessenberg matrix via an implicit QR-algorithm. Along the way, we explain how the knowledge of a certain ‘shift’ correction term to the structure can be used to speed up the QR-algorithm for unitary Hessenberg matrices, and how this observation was implicitly used in a paper due to William B. Gragg. We also treat an analogue of this observation in the Hermitian tridiagonal case.     

A canonical form for nonderogatory matrices under unitary similarity

A square matrix is nonderogatory if its Jordan blocks have distinct eigenvalues. We give canonical forms for

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nonderogatory complex matrices up to unitary similarity, and

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pairs of complex matrices up to similarity, in which one matrix has distinct eigenvalues.

The types of these canonical forms are given by undirected and, respectively, directed graphs with no undirected cycles.     

An algorithm for computing the eigenvalues of block companion matrices

In this paper we propose a method for computing the roots of a monic matrix polynomial. To this end we compute the eigenvalues of the corresponding block companion matrix C. This is done by implementing the QR algorithm in such a way that it exploits the rank structure of the matrix. Because of this structure, we can represent the matrix in Givens-weight representation. A similar method as in Chandrasekaran et al. (Oper Theory Adv Appl 179:111–143, 2007), the bulge chasing, is used during the QR iteration. For practical usage, matrix C has to be brought in Hessenberg form before the QR iteration starts. During the QR iteration and the transformation to Hessenberg form, the property of the matrix being unitary plus low rank numerically deteriorates. A method to restore this property is used.     

Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems

Summary. The eigenproblem method calculates the solutions of systems of polynomial equations . It consists in fixing a suitable polynomial and in considering the matrix corresponding to the mapping where the equivalence classes are modulo the ideal generated by The eigenspaces contain vectors, from which all solutions of the system can be read off. This access was investigated in [1] and [16] mainly for the case that is nonderogatory. In the present paper, we study the case where have multiple zeros in common. We establish a kind of Jordan decomposition of reflecting the multiplicity structure, and describe the conditions under which is nonderogatory. The algorithmic analysis of the eigenproblem in the general case is indicated. Received May 20, 1994