关于对广义的正定矩阵进一步研究

通常讨论矩阵的正定性只局限在实对称矩阵范围内(以下我们把全体n阶实对称正定矩阵的集合记为S~+),随着数学本身的发展和其它学科的需要,有不少人开始研究未必对称的较广义的实正定矩阵.李炯生在文[1]中给出了一类较广义的实正定矩阵的定义: 设A是n阶实方阵.如果对于任何非零的n维列向量X都有 X~TAX>0,其中X~T表示X的转置,则把A叫做正定矩阵.全体这类矩阵的集合记为P(I).文[1]证明了A∈P(I)的充分必要条件是A的对称分量是对称正定矩阵(即把A表示为对称矩阵与反对称阵的和的形式,前者称为对称分量,后者称为反对称分量).同时还推得P(I)中矩阵其     

非齐次对称特征值问题

引言 用SR~(n×n)表示所有。n×n实对称矩阵的集合。R~n表示n维线性空间。||·||_2表示向量的Euclid范数或矩阵的谱范数。 本文研究如下问题: 问题ISEP 给定矩阵A∈SR~n×n和向量b∈R~n,求实数λ和向量X∈R~n使得 AX=λX+b, (1) ||X||_2=1. (2) 若b=0,则问题ISEP就是通常的实对称矩阵特征值问题,若b≠0,则问题ISEP称为非齐次对称特征值问题,使(1)和(2)式成立的数λ和向量X分别称为非齐次特征值和相应的非齐     

iljak猜想的反例及进一步结果

R~(n×n)表示 n 阶实矩阵组成的集合,R~n 表示 n 维实向量空间.本文中的矩阵假定都属于 R~(n×n).给定一个矩阵 A∈R~(n×n),A>0(A≥0)表示 A 是一个对称正定(非负定)矩阵;A 称为正(非负)矩阵,如 A 的元素都是正的(非负的).矩阵 A 称为稳定矩阵,如A 的特征值的实部都是负的.     

Siljak 猜想的反例及进一步结果

R~(n×n)表示 n 阶实矩阵组成的集合,R~n 表示 n 维实向量空间.本文中的矩阵假定都属于 R~(n×n).给定一个矩阵 A∈R~(n×n),A>0(A≥0)表示 A 是一个对称正定(非负定)矩阵;A 称为正(非负)矩阵,如 A 的元素都是正的(非负的).矩阵 A 称为稳定矩阵,如A 的特征值的实部都是负的.     

实对称五对角矩阵逆特征值问题

1 引 言 对于n阶实对称矩阵A=(aij),r是一个正整数,且1≤r≤n-1,当|i-j|>r时,aij=0(i,j=1,2,…,n),至少有一个i使得ai,i+r≠0,则称矩阵A是带宽为2r+1的实对称带状矩阵.特别地,当r=1时,称A为实对称三对角矩阵;当r=2时,称A为实对称五对角矩阵. 实对称带状矩阵逆特征值问题应用十分广泛,这类问题不仅来自微分方程逆特征值问     

Householder矩阵的又一特性

给出了Householder矩阵的其它若干性质,利用本文中得到的正交向量组所对应的Householder矩阵的重要性质,解决了形如A=k1H1 k2H2 … knHn(ki∈R,Hi为n阶Householder矩阵,i=1,2,…n)的实对称阵的特性值与特征向量的问题,且任一实对称矩阵A均可表示为上述形式.     

(0,1)实对称矩阵特征值的图论意义

A为元素只取 0 ,1且主对角线元素均为 0的 n阶实对称方阵 ,n维列向量 J=( 1 ,1 ,1 ,… ,1 ) T ,且 AJ=( d1,d2 ,d3,… ,dn) T。若 λi 是 A的特征值 ,试证明 :∑ni=1λ2i =∑ni=1di ( 0 )　　这是一道典型的线性代数中关于实对称矩阵特征值方面的问题。对它的求解如下 :设 n维非零向量 x是 A的对应于特征值λi 的特征向量 ,则有 Ax=λix.两边同时左乘 A,得A2 x =A(λix) =λi( Ax) =λ2ix ( 1 )而上式说明 λ2i 即方阵 A2 的特征值。由 [1 ],对任一 n阶方阵 A=[aij]n× n,若 λi 是 A的特征值 ,则有 ∑ni=1λi=tr( A) =∑ni=1aii 。…     

代数特征值反问题之解的稳定性分析

考虑如下代数特征值反问题: 问题 G(A;{A_k}_1~n;λ).设 A=(a_(ij)),A_k=(a_(ij)~((k))),k=1,…,n是n+1个n×n的实对称矩阵,λ=(λ_1,…,λ_n)是n维实向量且λ_i≠λ_j,i≠j.求n维实向量c=(c_1,…,c_n)~T,使矩阵A(c)=A+sum from k=1 to n (c_kA_k)的特征值是λ_1,…,λ_n. 这一问题是经典加法问题的推广.当A_k-e_ke_k~~T(e_k是n阶单位阵的第k列)时,     

矩阵方程AXB+CYD=E对称最小范数最小二乘解的极小残差法

<正>1引言本文用R~(n×m)表示全体n×m实矩阵集合,用SR~(n×n)表示全体n×n实对称矩阵集合,OR~(n×n)表示全体n×n实正交矩阵集合.用I_n表示n阶单位矩阵,用A*B表示矩阵A与B的Hadamard乘积.对任意矩阵A,B∈R~(n×m),定义内积〈A,B〉=tr(B~T A),其中     

子空间上的一类矩阵反问题

1 引言 设Rn×m为所有n×m实矩阵的集合,ASRn×n为n阶实反对称矩阵的集合,ORn×n 为n阶实正交矩阵的全体. In是n阶单位矩阵,A+,R(A),N(A)分别表示矩阵A的 Moore-Penrose广义逆、值域及零空间,并记EA=I-AA+,FA=I-A+A(I为单位矩 阵,A为任意矩阵).对A=(aij),B=(bij)∈Rn×m,A*B=(aijbij)表示矩阵A与B 的Hadamard积.在Rn×m上定义矩阵A与B的内积为(A,B)=tr(BT A),则由此内积 导出的范数‖A‖=(A,A)~(1/2)是矩阵的Frobenius范数,并且Rn×m构成一个完备的内积 空间.     

判定3,4或5阶对称矩阵偕正性的算法

文献[1]给出了判定阶数不大于5的对称矩阵偕正性的充分必要条件.本文在此基础上,进一步给出了它们严格偕正的条件,并提出了三个算法,它们能够用来有效地判定3,4,5阶对称矩阵严格偕正、偕正或非偕正.     

A copositive Q-matrix which is notR 0

Jeter and Pye gave an example to show that Pang's conjecture, thatL ₁ ?Q ?R ₀, is false while Seetharama Gowda showed that the conjecture is true for symmetric matrices. It is known thatL ₁-symmetric matrices are copositive matrices. Jeter and Pye as well as Seetharama Gowda raised the following question: Is it trueC ₀ ?Q ?R ₀? In this note we present an example of a copositive Q-matrix which is notR ₀. The example is based on the following elementary proposition: LetA be a square matrix of ordern. SupposeR ₁ =R ₂ whereR _i stands for theith row ofA. Further supposeA ₁₁ andA ₂₂ are Q-matrices whereA _ii stands for the principal submatrix omitting theith row andith column fromA. ThenA is a Q-matrix.     

Copositive approximation by rational functions with prescribed numerator degree

The paper proves that, if f（x） ∈ L^p[-1,1],1≤p〈∞ ,changes sign I times in （-1, 1）,then there exists a real rational function r（x） ∈ Rn^（2μ-1）l which is eopositive with f（x）, such that the following Jackson type estimate ｜｜f-r｜｜p≤Cδl^2μωφ（f,1/n）p holds, where μ is a natural number ≥3/2＋1/p, and Cδ is a positive constant depending only on δ.     

On copositive approximation in some classical spaces of sequences

In this paper the author writes a simple characterization for the best copositive approximation in c; the space of convergent sequences, by elements of finite dimensional Chebyshev subspaces, and shows that it is unique.     

Contribution of copositive formulations to the graph partitioning problem

This article provides analysis of several copositive formulations of the graph partitioning problem and semidefinite relaxations based on them. We prove that the copositive formulations based on results from Burer [S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120 (Ser. A) (2009), pp. 479–495] and the author of the paper [J. Povh, Semidefinite approximations for quadratic programs over orthogonal matrices. J. Global Optim. 48 (2010), pp. 447–463] are equivalent and that they both imply semidefinite relaxations which are stronger than the Donath–Hoffman eigenvalue lower bound [W.E. Donath and A.J. Hoffman, Lower bounds for the partitioning of graphs. IBM J. Res. Develop. 17 (1973), pp. 420–425] and the projected semidefinite lower bound from Wolkowicz and Zhao [H. Wolkowicz and Q. Zhao, Semidefinite programming relaxations for the graph partitioning problem. Discrete Appl. Math. 96–97 (1999), pp. 461–479].     

New results on the cp-rank and related properties of co(mpletely )positive matrices

Copositive and completely positive matrices play an increasingly important role in Applied Mathematics, namely as a key concept for approximating NP-hard optimization problems. The cone of copositive matrices of a given order and the cone of completely positive matrices of the same order are dual to each other with respect to the standard scalar product on the space of symmetric matrices. This paper establishes some new relations between orthogonal pairs of such matrices lying on the boundary of either cone. As a consequence, we can establish an improvement on the upper bound of the cp-rank of completely positive matrices of general order and a further improvement for such matrices of order six.     

OnQ-matrices

Recently, Jeter and Pye gave an example to show that Pang's conjecture, thatL ₁ Q , is false. We show in this article that the above conjecture is true for symmetric matrices. Specifically, we show that a symmetric copositive matrix is inQ if and only if it is strictly copositive.     

On Copositive Approximation in Spaces of Continuous Functions Ⅱ:The Uniqueness of Best Copositive Approximation

This paper is part Ⅱ of "On Copositive Approximation in Spaces of Continuous Functions". In this paper, the author shows that if Q is any compact subset of real numbers, and M is any finite dimensional strict Chebyshev subspace of C(Q), then for any admissible function f ∈C(Q)\M, the best copositive approximation to f from M is unique.     

On common linear copositive Lyapunov functions for pairs of stable positive linear systems

We study the common linear copositive Lyapunov functions of positive linear systems. Firstly, we present a theorem on pairs of second order positive linear systems, and give another proof of this theorem by means of properties of geometry. Based on the process of the proof, we extended the results to a finite number of second order positive linear systems. Then we extend this result to third order systems. Finally, for higher order systems, we give some results on common linear copositive Lyapunov functions.