实对称区间矩阵特征值确界的交错定理及其应用

本文将实对称矩阵特征值的交错定理推广到实对称区间矩阵,给出了实对称区间矩阵特征值确界的交错定理,并应用该定理构造了估计实对称三对角区间矩阵特征值界的算法.文中数值例子表明,本文所给算法与一些现有算法相比在使用范围、计算精度和计算量等方面都具有一定的优越性.     

由谱数据构造实双反对称矩阵

讨论实完全反对称矩阵的一个特秆值反问题．研究了实完全反对称矩阵的一些特征性质，构造一个实反对称矩阵使其各阶顺序主子矩阵具有指定的特征值．证明了：给定满足一定分隔条件的两组数，存在一个实完全反对称矩阵，使其各阶中心主子矩阵具有相应的特征值．     

由特征值唯一确定的实3×3Hankel矩阵(英文)

本文研究了由特征值唯一确定的3×3实Hankel矩阵.借助于M.Fielder[1]的结论并经过细致的讨论,得到3×3实Hankel矩阵由其特征值唯一确定的充分必要条件,刻画了3×3实Hankel矩阵的一种特征值性质.     

广义实对称矩阵及有关性质

给出了广义实对称矩阵的定义,得到的基本运算结果仍然是广义实对称矩阵,并讨论了它的特征值和特征向量.     

正则化HSS预处理鞍点矩阵的特征值估计

最近,Bai和Benzi针对鞍点问题提出了一类正则化HSS（Regularized Hermitian and skew-Hermitian splitting,RHSS）预处理子（BIT Numer.Math.,57（2017）287-311）.为了进一步分析RHSS预处理子的效果,本文重点研究了RHSS预处理鞍点矩阵特征值的估计,分析了复特征值实部和模的上下界、实特征值的上下界,还给出了特征值均为实数的充分条件.当正则化矩阵取为零矩阵时,RHSS预处理子退化为HSS预处理子,分析表明本文给出的复特征值实部的界比已有的结果更精确.数值算例验证了本文给出的理论结果.     

实对称矩阵基本定理的存在性证明

对于实对称矩阵A,通过考虑欧氏空间?~n中的连续函数f(X)=X~TAX在一些有界闭集上的最大值,构造相应子空间上的半正定矩阵,进而得到实对称矩阵A的实特征值和相应的特征向量.最终可得实对称矩阵A可以正交相似对角化.     

实规范矩阵的特征值问题

实对称矩阵的特征值问题,无论是低阶稠密矩阵的全部特征值问题,或高阶稀疏矩阵的部分特征值问题,都已有许多有效的计算方法,迄今最重要的一些成果已总结在[5]中。本文利用规范矩阵的一些重要性质将对于Hermite矩阵(特别是对弥矩阵)特征值问题的一些有效算法推广到规范矩阵的特征值问题,由于对复规范阵的推广是简单的,而且实际上常遇到的是实矩阵(这时常要求只用实运算),因此我们着重讨论实规范矩阵的特征值问题。     

矩阵具有正实部特征值的判定

利用根与系数的关系.证明了特征方程没有零实部根的充要条件,给出了矩阵特征值至少具有一个正实部的充分条件,最后通过示例计算验证了该方法的有效性.     

关于离散型随机变量次序统计量分布矩阵对称性猜想的探索

本文研究了离散型随机变量次序统计量的分布矩阵的对称性 ,获得了二个定理 .定理 1　服从等概率二点分布或等概率三点分布的离散型随机变量的次序统计量的分布矩阵是对称矩阵 .定理 2　取值有限且等概率的离散型随机变量的次序统计量的分布矩阵具有中心对称性 .     

时标上具有分布势函数的Sturm-Liouville问题的矩阵表示

本文讨论时标上具有分布势函数的二阶Sturm-Liouville问题的矩阵表示.通过分析得出所研究的具有分布势函数的Sturm-Liouville问题与一类矩阵特征值问题之间的等价关系.文章针对分离型和实耦合型自共轭边界条件分别进行了讨论.     

On inverse eigenvalue problems for block Toeplitz matrices with Toeplitz blocks

We propose an algorithm for solving the inverse eigenvalue problem for real symmetric block Toeplitz matrices with symmetric Toeplitz blocks. It is based upon an algorithm which has been used before by others to solve the inverse eigenvalue problem for general real symmetric matrices and also for Toeplitz matrices. First we expose the structure of the eigenvectors of the so-called generalized centrosymmetric matrices. Then we explore the properties of the eigenvectors to derive an efficient algorithm that is able to deliver a matrix with the required structure and spectrum. We have implemented our ideas in a Matlab code. Numerical results produced with this code are included.     

A global harmonic Arnoldi method for large non-Hermitian eigenproblems with an application to multiple eigenvalue problems

The global Arnoldi method can be used to compute exterior eigenpairs of a large non-Hermitian matrix A, but it does not work well for interior eigenvalue problems. Based on the global Arnoldi process that generates an F-orthonormal basis of a matrix Krylov subspace, we propose a global harmonic Arnoldi method for computing certain harmonic F-Ritz pairs that are used to approximate some interior eigenpairs. We propose computing the F-Rayleigh quotients of the large non-Hermitian matrix with respect to harmonic F-Ritz vectors and taking them as new approximate eigenvalues. They are better and more reliable than the harmonic F-Ritz values. The global harmonic Arnoldi method inherits convergence properties of the harmonic Arnoldi method applied to a larger matrix whose distinct eigenvalues are the same as those of the original given matrix. Some properties of the harmonic F-Ritz vectors are presented. As an application, assuming that A is diagonalizable, we show that the global harmonic Arnoldi method is able to solve multiple eigenvalue problems both in theory and in practice. To be practical, we develop an implicitly restarted global harmonic Arnoldi algorithm with certain harmonic F-shifts suggested. In particular, this algorithm can be adaptively used to solve multiple eigenvalue problems. Numerical experiments show that the algorithm is efficient for the eigenproblem and is reliable for quite ill-conditioned multiple eigenproblems.     

基于信息压缩矩阵特征值映射的时变系统UD分解辨识算法

在系统辨识领域遗忘因子UD分解算法(一种通过对系统数据矩阵进行UD分解的在线辨识算法)具有对时变系统阶次和参数同步估计的优异性能,但传统的遗忘策略不能从根本上解决信息压缩矩阵数据过饱和问题,为了拓展现有UD分解算法在时变系统的适用范围,同时针对数据空间分布不均匀性,提出一种基于信息压缩矩阵特征值映射的UD分解辨识算法.从理论上分析辨识算法跟踪能力与参数估计矩阵有界性的对应关系,从而构造出一种基于信息压缩矩阵特征值映射的有界函数,特征值映射函数能够根据系统数据传递过程中信息量的大小动态调整遗忘因子,解决了参数辨识过程中数据过饱和及数据分布不均匀问题.仿真结果表明,相比于常规时变遗忘因子策略,带有特征值映射的UD分解算法能够更加准确跟踪系统参数的变化,且能够保证系统不是2N阶持续激励信号的情况下,也能对时变系统参数进行跟踪.     

Image segmentation by using the localized subspace iteration algorithm

An image segmentation algorithm called"segmentation based on the localized subspace iterations"(SLSI)is proposed in this paper.The basic idea is to combine the strategies in Ncut algorithm by Shi and Malik in 2000 and the LSI by E,Li and Lu in 2007.The LSI is applied to solve an eigenvalue problem associated with the affinity matrix of an image,which makes the overall algorithm linearly scaled.The choices of the partition number,the supports and weight functions in SLSI are discussed.Numerical experiments for real images show the applicability of the algorithm.     

基于梯度的Sylvester共轭矩阵方程的迭代解

本文提出了一种基于梯度的Sylvester共轭矩阵方程的迭代算法.通过引入一个松弛参数和采用递阶辨识原理,构造一个迭代算法求解Sylvester矩阵方程.通过应用复矩阵的实数表达以及实数表示的一些性质,收敛性分析表明在一定假设条件下,对于任意初始值,迭代方法均收敛到精确解,数值算例也表明了所给方法的有效性.     

Interlacing properties of the eigenvalues of some matrix classes

We establish the eigenvalue interlacing property (i.e. the smallest real eigenvalue of a matrix is less than the smallest real eigenvalue of any of its principal submatrices) for the class of matrices introduced by Kotelyansky (all principal and almost principal minors of these matrices are positive). We show that certain generalizations of Kotelyansky and totally positive matrices possess this property. We also prove some interlacing inequalities for the other eigenvalues of Kotelyansky matrices.     

A new approach for calculating the real stability radius

We present a new fast algorithm to compute the real stability radius with respect to the open left half plane which is an important problem in many engineering applications. The method is based on a well-known formula for the real stability radius and the correspondence of singular values of a transfer function to pure imaginary eigenvalues of a three-parameter Hamiltonian matrix eigenvalue problem. We then apply the implicit determinant method, used previously by the authors to compute the complex stability radius, to find the critical point corresponding to the desired singular value. This corresponds to a two-dimensional Jordan block for a pure imaginary eigenvalue in the parameter dependent Hamiltonian matrix. Numerical results showing quadratic convergence of the algorithm are given.     

不变性在线性代数中应用两例

本文介绍数学中“不变性”思想，讨论了线性代数中某种条件下秩数不变、特征多项式、特征值不变；对称性、半正定、正定性不变；以及度量性不变．以初等变换为首要方法，解决线性代数中一类重要问题．阐述了矩阵或线性方程组线性变换的本质     

The {\cal {D Q R}} algorithm, basic theory, convergence, and conditional stability

Summary. We describe a fast matrix eigenvalue algorithm that uses a matrix factorization and reverse order multiply technique involving three factors and that is based on the symmetric matrix factorization as well as on –orthogonal reduction techniques where is computed from the given matrix . It operates on a similarity reduction of a real matrix to general tridiagonal form and computes all of 's eigenvalues in operations, where the part of the operations is possibly performed over , instead of the 7–8 real flops required by the eigenvalue algorithm. Potential breakdo wn of the algorithm can occur in the reduction to tridiagonal form and in the –orthogonal reductions. Both, however, can be monitored during the computations. The former occurs rather rarely for dimensions and can essentially be bypassed, while the latter is extremely rare and can be bypassed as well in our conditionally stable implementation of the steps. We prove an implicit theorem which allows implicit shifts, give a convergence proof for the algorithm and show that is conditionally stable for general balanced tridiagonal matrices . Received April 25, 1995 / Revised version received February 9, 1996