n阶矩阵m次方幂的一般求法探讨

文[1]、文[2]给出了全部特征值相等及全部不同特征值为两个,并满足一定条件的n阶矩阵m次方幂的求法。本文对一般的n阶矩阵A的m次方幂A~m的求法进行探讨。本文要点: 1.提出将A~m化为次数低于n的A的多项式r(A)的一个比较简单的途径,即本文(3)式。2.对矩阵λE—A进行λ矩阵的初等变换,     

关于正定厄米特矩阵乘积特征值估计的改进

设 A、B 都是 n×n 阶厄米特矩阵,其中有一个是半正定的,本文不仅给出了矩阵乘积 AB 的最大、最小特征值的一个最优估计,并且对 AB 的每一个“中间”特征值也给出了估计,大大改进并推广了文[3]的结果.     

特征值全是实数的实二重随机矩阵的存在性

<正> P.M.Gibson 在[1]中已证明了下面的性质:对于每一整数 n≥7,存在一个 n 阶不可约化的二重随机矩阵 A,使得 A 的特征值全是实数.李健生在[2]中进一步得到下面的结果:对于每一个自然数 n,存在一个 n 阶不可约化的二重随机矩阵 A,使得 A 的特征值     

关于正定厄米特矩阵乘积的特征值的最优估计

设A,B是两个n×n阶正定厄米特矩阵,本文指出了关于矩阵乘积AB的特征值的一类最优估计,它大大改进了文[1]、[2]的结论。     

对称正交反对称矩阵反问题解存在的条件

矩阵反问题和矩阵特征值反问题在科学和工程技术中具有广泛的应用,有关它们的研究已取得了许多进展[1,2].[3]和[4]分别研究了反对称矩阵反问题和双反对称矩阵特征值反问题等.本文研究一类更广泛的对称正交反对称矩阵反问题.用Rn×m（Cn×m）表示n×m实（复）矩阵的全体,ASRn×n表示n阶反对称矩阵的全体,ABSRn×n表示n阶双反对称矩阵的全体,ORn×n表示n阶正交矩阵的全体.A+表示矩阵A的Moore-Penrose广义逆.In表示n阶单位矩阵.ei表示n阶单位矩阵的第i列,Sn=[en,en-1,     

迹为4的n—可扩张的TC S—矩阵的一个刻划

在文[1]中,给出了迹为0和2的n阶n-可扩张的TC S-矩阵的一个刻划。本文将给出迹为4的这类矩阵的一个刻划。     

关于两个厄米特矩阵乘积的特征值的估计问题

设A,B是两个任意的n阶厄米特矩阵(不假定A,B正定)。本文利用A,B的特征值给出了乘积矩阵AB的特征值的取值范围,基本上解决了对两个n阶厄米特矩阵乘积的特征值的估计,当A,B都是正定阵时,我们的结果大大地改进了[3]的结果。     

p.n.p.矩阵的一些性质

一个n阶实方阵若其各阶主子式皆非正,则称为部分非正阵,简写作p.n.p.矩阵.特别地,各阶主子式皆负的p.n.p.矩阵称为部分负矩阵,简写为p.n.矩阵。文[1]、[5]讨论了p.n.p.矩阵的谱性质。本文在[5]的基础上讨论了p.n.p.矩阵的若干性质,并给出p.n.p.矩阵特征值的某些估计式。 引理1 设A=(A_(ij)_n×n为一p.n.p.矩阵,则A的特征值之实部不全为负(n≥2)。 证 设λ_1,λ_2,…,λ_n为A的全部特征值。假定A的每一特征值之实部皆为负。分两种情     

关于矩阵方程X+A*X-1A=P的解及其扰动分析

考虑非线性矩阵方程X＋A^＊（X^-1）A=P其中A是n阶非奇异复矩阵,P是n阶Hermite正定矩阵．本文给出了Hermite正定解和最大解的存在性以及获得最大解的一阶扰动界,改进了文[5,6]中的部分结论．     

关于矩阵对角化的一种判别方法

阐述判别矩阵对角化的一种方法．如果复数域C上n阶矩阵A的对应于不同特征值的线性无关特征向量的个数都恰好等于该特征值的重数,则A相似于对角矩阵．     

Graphs with four distinct Laplacian eigenvalues

In this paper, we investigate connected nonregular graphs with four distinct Laplacian eigenvalues. We characterize all such graphs which are bipartite or have exactly one multiple Laplacian eigenvalue. Other examples of interest are also presented.     

Principal Submatrices of a Hermitian Matrix

Suppose k 1 ,<, k m and n are positive integers such that k 1 + … + k m h n . We characterize those k i × k i Hermitian matrices A i , i = 1, < , m that can appear as diagonal blocks of an n × n Hermitian matrix C with prescribed eigenvalues. The characterization will be given in terms of the eigenvalues of C and A i , i = 1, <, m . Our results extend those of Thompson and Freede, Horn, Fan and Pall.     

On the General Algebraic Inverse Eigenvalue Problems

A number of new results on sufficient conditions for the solvability and numerical algorithms of the following general algebraic inverse eigenvalue problem are obtained: Given $n+1$ real $n\times n$ matrices $A=(a_{ij}),A_k=(a_{ij}^{(k)})(k=1,2,\cdots,n)$ and $n$ distinct real numbers $\lambda_1,\lambda_2,\cdots,\lambda_n,$ find $n$ real number $c_1,c_2,\cdots,c_n$ such that the matrix $A(c)=A+\sum\limits_{k=1}^{n}c_k A_k$ has eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_n.$     

具有确定非零对角元个数的布尔矩阵广义幂敛指数的极矩阵

设A是周期为P的n阶布尔矩阵,1≤i≤n,A的广义幂敛指数k(A,i)是使得Ak和Ak+p有i行对应相等的最小非负整数k.本文刻画了恰含d(1≤d≤n)个非零对角元的n阶布尔矩阵的广义幂敛指数的极矩阵.     

Graphs with three distinct eigenvalues and largest eigenvalue less than 8

In this paper we consider graphs with three distinct eigenvalues and, we characterize those with the largest eigenvalue less than 8. We also prove a simple result which gives an upper bound on the number of vertices of graphs with a given number of distinct eigenvalues in terms of the largest eigenvalue.     

A note on “Methods for constructing distance matrices and the inverse eigenvalue problem”

In this paper, a symmetric nonnegative matrix with zero diagonal and given spectrum, where exactly one of the eigenvalues is positive, is constructed. This solves the symmetric nonnegative eigenvalue problem (SNIEP) for such a spectrum. The construction is based on the idea from the paper Hayden, Reams, Wells, “Methods for constructing distance matrices and the inverse eigenvalue problem”. Some results of this paper are enhanced. The construction is applied for the solution of the inverse eigenvalue problem for Euclidean distance matrices, under some assumptions on the eigenvalues.     

Numerical enclosure for multiple eigenvalues of an Hermitian matrix whose graph is a tree

In this paper we consider a numerical enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. If an Hermitian matrix A whose graph is a tree has multiple eigenvalues, it has the property that matrices which are associated with some branches in the undirected graph of A have the same eigenvalues. By using this property and interlacing inequalities for Hermitian matrices, we show an enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. Since we do not generally know whether a given matrix has exactly a multiple eigenvalue from approximate computations, we use the property of interlacing inequalities to enclose some eigenvalues including multiplicities.In this process, we only use the enclosure of simple eigenvalues to enclose a multiple eigenvalue by using a computer and interval arithmetic.     

单位球面上仿Blaschke张量的特征值为常数的超曲面

设x:M~n→S~(n+1)是(n+1)-维单位球面上不含脐点的超曲面,在S~(n+1)的Moebius变换群下浸入x的四个基本不变量是:一个黎曼度量g称为Moebius度量;一个1-形式Φ称为Moebius形式;一个对称的(0,2)张量A称为Blaschke张量和一个对称的(0,2)张量B称为Moebius第二基本形式.对称的(0,2)张量D=A+λB也是Moebius不变量,称为浸入x的仿Blaschke张量,其中λ是常数,仿Blaschke张量的特征值称为仿Blaschke特征值.李海中和王长平(2003)研究了满足如下条件的超曲面:(i)Φ=0;(ii)存在可微函数λ和μ,使A+λg+μB=0.他们证明了λ和μ都是常数,并且给出了这类超曲面的分类,也就是D的特征值全相等的超曲面的分类.本文对满足如下条件的超曲面进行了分类:(i)Φ=0,(ii)对某一个常数λ,D具有两个互异的常数特征值.     

Tournament matrices and their generalizations,I.

If M is any complex matrix with rank (M + M ^* + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 for its real part. This generalizes a recent finding of de Caen and Hoffman: the rank of any n × n tournament matrix is at least n ? 1. We extend several spectral properties of tournament matrices to this and related types of matrices. For example, we characterize the singular real matrices M with 0 diagonal for which rank (M + M^T + I) = 1 and we characterize the vectors that can be in the kernels of such matrices. We show that singular, irreducible n × n tournament matrices exist if and only n? {2,3,4,5} and exhibit many infinite families of such matrices. Connections with signed digraphs are explored and several open problems are presented.     

TheAB-algorithm and its properties

One presents a new algorithm, called the -algorithm, for solving the generalized eigenvalue problem Ax=λBx, where det (A—λB) ≠ 0, relative to A. The algorithm is iterative, it is based on the application of plane rotations and allows us to pass from the initial problem to the solving of a similar problem having simpler matrices, whose eigenvalues can be easily computed and coincide with the eigenvalues of the initial problem. Thus, if all the eigenvalues of the initial problem are distinct, then the application of the -algorithm leads to the computation of the eigenvalues of a pencil with triangular matrices. In the case of an arbitrary initial pencil A—λB, the problem reduces to solving the eigenvalue problem for a pencil of quasitriangular form. One proves the convergence of the algorithm. One establishes its properties which in many respects are similar with the properties of the known algorithms QR and QZ, the first of which solves the usual eigenvalue problem while the second one solves the generalized problem of the above-mentioned form.