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.     

一种三对角矩阵的特征值及其应用

给出了一种三对角矩阵的特征值和特征向量的算法,利用矩阵方法和对称多项式证明了一些与Lucas数以及第一类Chebyshev多项式有关的三角恒等式.     

三对角矩阵的特征值及其应用

给出了计算一种三对角矩阵的特征值和特征向量的公式.利用矩阵的特征值理论证明了一些三角恒等式,特别是一些与Fibonacci数和第二类Chebyshev多项式有关的三角恒等式.     

求非亏损矩阵特征值的一种数值计算方法

借助相似变换将非亏损矩阵转为Hessenberg矩阵,通过获得确定Hessenberg矩阵特征多项式系数的方法,利用特征值与特征多项式系数间的关系,给出求非亏损矩阵特征值的一种数值算法。     

关于方阵多项式的特征问题

对方阵及其矩阵多项式,给出了它们特征值、特征向量之间关系的刻画.     

特征值问题的预变换方法(I): 杨辉三角阵变换与二阶PDE 特征多项式

本文提出一类求解特征值问题的下三角预变换方法, 目标是通过相似变换后矩阵下三角元素平方和明显减少、且变换后的特征值及其特征向量较易求解, 使变换后的对角线可作为全体特征值很好的一组初值, 其作用如同对于解方程组找到好的预条件子, 加速迭代收敛. 以二阶PDE 数值计算为例,对于以Laplace 方程为代表的特征波向量组及正交多项式组有广泛的应用前景.
杨辉三角是我国古代数学家的一项重要成就. 本文引入杨辉三角矩阵作为预变换子, 给出一般矩阵用杨辉三角矩阵作为左、右预变换子时变为上三角矩阵的充要条件, 给出了元素为行指标二次多项式的两个矩阵类(三对角线阵与五对角线阵) 中特征值何时保持二次多项式的充要条件, 并应用于构造新的二元PDE 正交多项式.     

图的Laplacian特征值

本文研究了连通图的Laplacian特征值,利用图的Laplacian矩阵的特征多项式的行列式表示式,对存在两个不同顶点,但有相同邻集的一类图,得到了一个Laplacian特征值,并给出了它的应用.     

利用半群代数中良序基构造解稀疏多项式方程组特征值方法

本文利用半群代数k[A]中良序基,构造了求稀疏多项式方程组解的特征值矩阵,并给出了可以构造方阵的条件.     

M-矩阵最小特征值下界的新不等式

非奇异M-矩阵B的最小特征值τ(B)的下界是矩阵论中重要的研究课题.利用特征值定位定理,首先给出非负矩阵与M-矩阵的逆矩阵Hadamard积的谱半径上界,进而给出M-矩阵最小特征值下界的新不等式.新不等式只与矩阵的元素有关,易于计算.理论分析和数值例子表明所给结果改进了现有结果.     

一类Hermitian鞍点矩阵的特征值估计

本文研究了一类大型稀疏Hermitian鞍点线性系统Az=(B E E* 0)(x y)=(f g)=b系数矩阵的特征值,其中B∈C~(p×p)是Hermitian正定阵矩阵,E∈C~(p×q)是列降秩.本文分别给出了该系数矩阵正特征值与负特征值界的一个估计式,同时通过数值算例验证本文所给出的特征值界的估计是合理且有效的.     

Bounds for Eigenvalues of Matrix Polynomials Over Quaternion Division Algebra

Localization theorems are discussed for the left and right eigenvalues of block quaternionic matrices. Basic definitions of the left and right eigenvalues of quaternionic matrices are extended to quaternionic matrix polynomials. Furthermore, bounds on the absolute values of the left and right eigenvalues of quaternionic matrix polynomials are devised and illustrated for the matrix p norm, where \({p = 1, 2, \infty, F}\). The above generalizes the bounds on the absolute values of the eigenvalues of complex matrix polynomials, which give sharper bounds to the bounds developed in [LAA, 358, pp. 5–22 2003] for the case of 1, 2, and \({\infty}\) matrix norms.     

Bounds and Majorization Relations for the Zeros of a Class of Fibonacci-Type Polynomials

We apply certain matrix inequalities involving eigenvalues, the numerical radius, and the spectral radius to obtain new bounds and majorization relations for the zeros of a class of Fibonacci-type polynomials. Our results improve upon some earlier bounds for the zeros of these polynomials.     

The spread of matrices and polynomials

The spread of a matrix (or polynomial) is the maximum distance between any two of its eigenvalues (or its zeros). E. Deutsch has recently given upper bounds for the spread of matrices and polynomials. We obtain sharper, simpler upper bounds and observe that they are also upper bounds for the sum of the absolute values of the two largest eigenvalues (or zeros).     

On the spread of matrices and polynomials

The spread of a matrix (polynomial) is defined as the maximum distance between two of its eigenvalues (zeros). In this note upper bounds are found for the spread of matrices and polynomials.     

Estimates for the real and imaginary parts of the eigenvalues of matrices and applications

This paper is a continuation of our recent work on the localization of the eigenvalues of matrices. We give new bounds for the real and imaginary parts of the eigenvalues of matrices. Applications to the localization of the zeros of polynomials are also given.     

Computable eigenvalue bounds for rank-k perturbations

We investigate lower bounds for the eigenvalues of perturbations of matrices. In the footsteps of Weyl and Ipsen & Nadler, we develop approximating matrices whose eigenvalues are lower bounds for the eigenvalues of the perturbed matrix. The number of available eigenvalues and eigenvectors of the original matrix determines how close those approximations can be, and, if the perturbation is of low rank, such bounds are relatively inexpensive to obtain. Moreover, because the process need not be restricted to the eigenvalues of perturbed matrices, lower bounds for eigenvalues of bordered diagonal matrices as well as for singular values of rank-k perturbations and other updates of n×m matrices are given.     

矩阵特征值的几个不等式

研究矩阵特征值的上、下界以及特征值的实部、虚部的不等式 ,给出了特征值一些新的上界和下界     

Bounds for the signless Laplacian energy

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. The Laplacian (respectively, the signless Laplacian) energy of G is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the vertex degrees of the graph. In this paper, among some results which relate these energies, we point out some bounds to them using the energy of the line graph of G. Most of these bounds are valid for both energies, Laplacian and signless Laplacian. However, we present two new upper bounds on the signless Laplacian which are not upper bounds for the Laplacian energy.     

Instability indices for matrix polynomials

There is a well-established instability index theory for linear and quadratic matrix polynomials for which the coefficient matrices are Hermitian and skew-Hermitian. This theory relates the number of negative directions for the matrix coefficients which are Hermitian to the total number of unstable eigenvalues for the polynomial. Herein we extend the theory to ?-even matrix polynomials of any finite degree. In particular, unlike previously known cases we show that the instability index depends upon the size of the matrices when the degree of the polynomial is greater than two. We also consider Hermitian matrix polynomials, and derive an index which counts the number of eigenvalues with nonpositive imaginary part. The results are refined if we consider the Hermitian matrix polynomial to be a perturbation of a ?-even polynomials; however, this refinement requires additional assumptions on the matrix coefficients.     

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.