矩阵特征值的几个扰动定理

1 引言 设A∈C~(n×m),B∈C~(m×m)(m≤n),它们的特征值分别为{λ_k}_(k=1)~n和{μ_k}_(k=1)~m.令 R=AQ-QB (1)这里Q∈C~(n×m)为列满秩矩阵.Kahan研究了矩阵A在C~(n×m)上的Rayleigh商的性质,证明了下列定理:设A为Hermite矩阵,Q为列正交矩阵,即Q~HQ=I,而B=Q~HAQ,则存在 1,2,… ,n的某个排列π,使得 {sum from j=1 to m │μ_j-λ_(π(j))│~2}~(1/2)≤2~(1/2)‖R‖_F (2)其中R如(1)所示,‖·‖_F为矩阵的Frobenius范数.刘新国在[2]中将此定理推广到B为可对角化矩阵的情形,并且还建立了较为一般的扰动定理:设A为正规矩阵,B为可对角化矩阵;存在非奇异矩阵G,使得G~(-1)BG为对角阵,则存在1,2,…,n的某个排列π,使得 │μ_j-λ_(π(j))│≤2(2~(1/2))nK(G)_(σ_m~(-1))‖R‖_F,j=1,2,…,m. (3)     

关于Kahan定理的一个注记

设A∈C~(n×n),B∈C~(k×k)均为Hermite矩阵,它们的特征值分别为{λ_j}_(j=1)~n和{μ_j}_(j=1)~k(k≤n);Q∈~(n×k)为列满秩矩阵.令 (1) 则存在A的k个特征值λ_(j_2),λ_(j_2),…,λ_(j_k),使得 (2) 其中σ_k为Q的最小奇异值,||·||_2表示矩阵的谱范数.这是著名的Kahan定理·1996年曹志浩等在[2]中将(2)加强为 (3) 这是Kahan的猜想.在本文中,我们讨论将Kahan定理中“B为k阶Hermite矩阵”改为B为k阶(任意)方阵后,特征值的扰动估计,有以下结果. 定理 设A∈C~(n×n)为Hermite矩阵,其特征值为{λ_j}_(j=1)~n,B∈C~(k×k)的特征值为{μ_j}_(j=1)~k,而Q∈C~(n×k)为列满秩矩阵.则存在A的k个特征值λ_(j_1),λ_(j_2),…,λ_(j_k),使得     

求解多项式特征值问题的部分正交投影方法及其变形

正1引言为表述方便,用C~(m×n)表示m×n复矩阵的全体,C~m=C~(m×1).‖·‖表示向量或矩阵的2-范数.对A∈C~(m×n),v∈C~m及正整数m,K[A,v,m]=[v,Av,A~2v,...,A~(m-1)v]称为Krylov矩阵,span(K[A,v,m])就是由A和v生成的Krylov子空间.e_j是适当阶单位矩阵的第j列.设A_i∈C~(m×n)(i=0,1,…,d)是给定的矩阵,记     

伪谱的边界曲线及其跟踪算法的步长控制

1 引言 1990年L.N.Trefethen为度量矩阵的非正规性引入了矩阵的伪谱,对于任意的ε≥0,矩阵A∈C~(n×n)的ε-伪谱定义为 ∧_ε(A)={z:z∈σ(A △A),||△A||≤ε},其中σ(·)表示矩阵的特征值集合(谱集合),||·||表示矩阵的2-范数。利用预解式(zI—A)~(-1),     

求解Sylvester方程的广义非对称PMHSS算法

<正>1引言考虑如下Sylvester方程:AX+XB=F(1)这里A∈C~(m×m),B∈C~(n×n),F∈C~(m×n)是复数矩阵.令A=W+iT,B=U+iV,Q,T∈R~(m×m),U,V∈R~(n×n)都是实对称矩阵,且W,U是不定的,T,V是正定的.我们假定-TW≤T,-VU≤V.对于任意矩阵W和T,WT(W≤T)意味着T-W是     

矩阵不等式约束下矩阵方程最小二乘问题的增广Lagrangian方法北大核心CSCD

称X∈R^(m×n)为实(R,S)对称矩阵,若满足X=RXS,其中R∈R^(m×m)和S∈R^(n×n)为非平凡实对合矩阵,即R=R^(-1)≠±I_m,S=S^(-1)≠±I_n.该文将优化理论中求凸集上光滑函数最小值的增广Lagrangian方法应用于求解矩阵不等式约束下实(R,S)对称矩阵最小二乘问题,即给定正整数m,n,p,t,q和矩阵A_i∈R^(m×m),B_i∈R^(n×n)(i=1,2,…,q),C∈R^(m×m),E∈R^(p×m),F∈R^(n×t)和D∈R^(p×t),求实(R,S)对称矩阵X∈R^(m×m)且在满足相容矩阵不等式EXF≥D约束下极小化‖∑_(i=1)~qA_iXB_i-C‖,其中EXF≥D表示矩阵EXF-D非负,‖·‖为Frobenius范数.该文给出求解问题的矩阵形式增广Lagrangian方法的迭代格式,并用数值算例验证该方法是可行且高效的.     

幂等矩阵下Yang-Baxter-like方程解的新的显式表达式

正1引言设C~(m×n)表示m×n阶复矩阵的集合,I_n表示n阶单位矩阵.对于矩阵A∈C~(m×n),A~*表示它的共轭转置矩阵.设矩阵A∈C~(n×n),如果A~2=A,则称矩阵A为幂等矩阵;如果A~2=A=A~*,则称矩阵A为正交投影矩阵.设A∈C~(n×n)本文主要研究下面的二次矩阵方程AXA=XAX,(1.1)称之为Yang-Baxter-like方程,因为其与统计物理中分别由Yang[1]和Baxter[2]独立得到的经典Yang-Baxter方程相似.     

关于阶数为奇数的布尔矩阵行空间的基数

设Bn表示所有的n阶布尔矩阵的集合，R(A)表示A∈Bn的行空间，|R(A)|表示R(A)的基数．设m，n为正整数，本文证明了(1)Vm∈[1，46]，[1，78]，分别存在A∈B7，A∈B8，使得|R(A)|=m．(1)当n≥9为奇数时，则V m∈[1．2^(n 3)/2 2^(n 1)/2 … 2^3]．存在A∈Bm，使得|R(A)|=m．     

布尔矩阵行空间基数的存在点

设Bn表示所有的n阶布尔矩阵的集合，R(A)表示A∈Bn的行空间．|R(A)|表示R（A）的基数。设m，n,k为正整数，本证明了当n≥9,[n 5/2]≤k≤n-3时,对任意的m、2^k≤m≤2^k 2^n-k 2 2^n-k 1 … 2^3，存在A∈B．使得|R（A）|=m．     

对称正交矩阵反问题及其最佳逼近

本文主要讨论下面两个问题：问题Ⅰ：给定矩阵X,B∈R~(m×n),求对称正交矩阵A∈SOR~(m×m),使得AX=B．问题Ⅱ：给定矩阵(?)∈R~(m×m),求矩阵A~*∈S_E使得(?)这里S_E问题Ⅰ的解集合,‖·‖指Frobenius范数．本文首先讨论具有k阶对称主子阵的n(n＞k)阶正交矩阵的C-S分解,利用这个结果,得到了问题Ⅰ有解的充要条件和通解的一般形式．然后,对给定矩阵(?)∈R~(m×m),讨论了矩阵(?)在问题Ⅰ的解集合S_E中的最佳逼近,得到了最佳逼近解的表达式．     

FC-空间中的不动点定理定理及应用

在FC-空间中建立了一个不动点定理,推广了近期文献的一个相关结论利用这个结果,我们建立了一些广义型定理,推广了著名的截口定理到FC-空间中借助这个截口定理,在FC-空间得到一些新的广义向量均衡问题解的存在性定理.     

G-凸空间中的KKM定理，匹配定理和截口定理

在G-凸空间中证明了一些新的KKM型定理．作为应用,在G-凸空间中得到了一些新的匹配定理和截口定理,所得结果改进和推广了[2,3,7]中的相关结果．     

Some comparison theorems for nonnegative splittings of matrices

Summary In a recent paper the author has proposed some theorems on the comparison of the asymptotic rates of convergence of two nonnegative splittings. They extended the corresponding result of Miller and Neumann and implied the earlier theorems of Varga, Beauwens, Csordas and Varga. An open question by Miller and Neumann, which additional and appropriate conditions should be imposed to obtain strict inequality, was also answered. This article continues to investigate the comparison theorems for nonnegative splittings. The new results extend and imply the known theorems by the author, Miller and Neumann.The Project Supported by the Natural Science Foundation of Jiangsu Province Education Commission     

Inductive tools for connected delta-matroids and multimatroids

We prove a splitter theorem for tight multimatroids, generalizing the corresponding result for matroids, obtained independently by Brylawski and Seymour. Further corollaries give splitter theorems for delta-matroids and ribbon graphs.     

关于半直积同构的讨论

这篇文章从半直积的定义出发,给出了当群H或当从群H到N的自同构群Aut(N)的两个同态φ,ψ满足一定条件时半直积N∝φH与N∝ψH同构的两个定理,而其中一个定理是一个已知结果的推广.     

Strong convergence of an iterative algorithm for an infinite family of strict pseudo-contractions in Banach spaces

In this paper, we introduce a new W-mapping and present an iterative algorithm for an infinite family of strict pseudo-contractions. Strong convergence theorems are proved in Banach spaces. Our results improve and extend the corresponding result announced by Cai and Hu [G. Cai, C. Hu, Strong convergence theorems of modified Ishikawa iterative process with errors for an infinite family of strict pseudo-contractions, Nonlinear Anal. 71(12) (2009) 6044-6053].     

带有确定生成元的倒向随机微分方程的共单调定理

对倒向随机微分方程(简记BSDE)的解(y,z),利用Malliavin微分的方法进行了研究．给出了某些关于比较z的方法,在此基础上继续研究(y,z)的某些重要性质,同时推广了Chen Zengjing等人文章中相应的结论．     

Instability of closed invariant sets of semidynamical systems: Method of sign-constant Lyapunov functions

In the paper, a reduction principle for the instability property of a closed positively invariant set M for semidynamical systems is proved. The fact that the result is not traditional is stressed by the assumption on the existence of a closed positively invariant set with respect to which the set M has the attraction property. The corresponding instability theorem of the method of sign-constant Lyapunov functions is presented. The assertion thus obtained generalizes the well-known Chetaev and Krasovskii theorems for systems of ordinary differential equations, theorems on the instability with respect to some of the variables, and also the Shimanov and Hale theorems for systems with retarded argument. Illustrating examples are presented.     

Existence theorems of nash equilibria for non-cooperative n-person games

In this paper, we first obtain existence theorems of Nash equilibria for non-cooperative n-person games which generalize a corresponding result of Nikaido and Isoda (1955). As applications, we give two new existence theorems of -equilibrium points which generalize that of Tijs (1981). Finally, a saddle point theorem of Komiya (1986) is deduced from one of our existence theorems of -equilibrium points.     

Convergence and stability of ishikawa iterative methods with errors for quasi-contractive mappings in q-uniformly smooth banach spaces

In this paper, We show convergence and stability theorems of the Ishikawa iteration methods with errors for quasi-Contractive mappings in any nonempty closed bounded convex subsets of q-uniformly smooth Banach spaces.A related result deals with stability of the Ishikawa iteration method forquasi-contractive the corresponding results of chidume,Chidume-Osilike Osilike and others.