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)     

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,     

解广义特征值反问题的同伦方法

1.引言 我们讨论下列广义特征值反问题: (G)已知B是n×n阶对称半正定矩阵,λ=(λ_1,…,λ_(2n-1))~T∈R~(2n-1),且{λ_i}~(n_3),和{λ_i}_(n+1)~(2n-1)严格交错。问题是欲求一个实对称三对角n×n阶矩阵A,使得λ_1…,λ_n是Ax=λBx的特征值,λ_(n+1),…,λ_(2n-1)是A_(n-1)x=λB_(n-1)x的特征值,其中A_(n-1),B_(n-1)分别是矩阵A,B的前n-1阶主子阵。     

可对称化矩阵特征值的扰动界

在[1]中,Kahan证明了如下的定理:设A为n×n Hermite矩阵,B为n×n。可对称化矩阵,即存在非奇异矩阵Q,使得Q~(-1)BQ为实对角矩阵。又设A,B的特征值分别为λ_1     

反特征值问题的最佳逼近的一点注记

对于[1]与[2]中提出的关于矩阵的最佳逼近问题,本文用一个简洁的方法,证明其主要结果. 1.问题及条件的转化 设X∈R~(n×k),A∈R~(n×n),λ_1,…λ_k为A的部分特征值,A=daig(λ_1…λ_k)以及     

由谱数据数值稳定地构造实对称带状矩阵

§1.引言 设r,n是正整数并且0r有a_(ij)=0.     

关于正规矩阵特征值的扰动

设N与A均为n×n正规矩阵,其特征值分别为{v_i}_(i=1)~n与{α_i}_(i=1)~n。Hoffman和Wielandt证明了:存在1,2,…,n的一个排列π(1),π(2),…,π(n),使得|| ||_F表示Frobenius范数。 当N为n×n Hermite矩阵,A为n×n可对称化矩阵,即存在非奇异矩阵Q=I X,使得Q~(-1)AQ为Hermite矩阵时,Stewart证明了:如果N与A的特征值分别     

加法与乘法逆特征值问题的可解性

1.引言 本文讨论如下代数特征值反问题可解的充分条件: 问题A(加法逆特征值问题)。给定一Hermite矩阵A=(a_(ij))_(n×n)及n个实数λ_1,…,λ_n,求一实对角阵D=diag(c_1…,c_n),使得A+D的特征值为λ_1,…,λ_n。 问题M(乘法逆特征值问题)。给定一正定Hermite矩阵A=(a_(ij))_(n×n)和n个正实数     

等式约束加权线性最小二乘问题的解法

1 引言 在实际应用中常会提出解等式约束加权线性最小二乘问题 min||b-Ax||_M,(1.1) x∈C~n s.t.Bx=d, 其中B∈C~(p×n),A∈C~(q×n),d∈C~p,b∈C~q,M∈C~(q×q)为Hermite正定阵. 对于问题(1.1),目前已有多种解法,见文[1—3).本文将利用广义逆矩阵的知识,给出(1.1)的通解及迭代解法.本文中关于矩阵广义逆与投影算子(矩阵)的记号基本上与文[4]的相同.例如,A~+表示A的MP逆,P_L表示到子空间L上的正交投影算子,λ_(max)(MAY)表示矩阵M~(1/2)AY的最大特征值.我们还要用到广义BD逆的概念: 设A∈C~(n×n),L为C~n的子空间,则称A_(L)~(+)=P_L(AP_L+P_L⊥)~+为A关于L的广义BD逆.     

不变子空间上特征值的扰动

1引言设矩阵A∈C~(n×n),B∈C~(m×m),Q∈C~(n×m)为列满秩矩阵,令R=AQ-QB．当R的范数很小的时候,我们分析矩阵B的特征值对A的特征值的逼近性．当A,B都是Hermite阵时,上述问题已经被Kahan解决．近年来,对可对角化矩阵的情形,取得了一些新的成果．[4][5][6]中给出了几个范数不等式,并应用于矩阵特征值     

A REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

CONTENTS OF VOLUME 30

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Instructions for contributions

正Applied Mathematics-A Journal of Chinese Universities,Series B(Appl.Math.J.Chinese Univ.,Ser.B)is a comprehensive applied mathematics journal jointly sponsored by Zhejiang University,China Society for Industrial and Applied Mathematics,and Springer-Verlag.It is a quarterly journal with     

Journal of Mathematical Research with Applications Guide for Authors

正Journal overview:Journal of Mathematical Research with Applications(JMRA),formerly Journal of Mathematical Research and Exposition(JMRE)created in 1981,one of the transactions of China Society for Industrial and Applied Mathematics,is a home for original research papers of the highest quality in all areas of mathematics with applications.The target audience comprises:pure and applied mathematicians,graduate students in broad fields of sciences and technology,scientists and engineers interested in mathematics.     

Staircase transportation problems with superadditive rewards and cumulative capacities

A cumulative-capacitated transportation problem is studied. The supply nodes and demand nodes are each chains. Shipments from a supply node to a demand node are possible only if the pair lies in a sublattice, or equivalently, in a staircase disjoint union of rectangles, of the product of the two chains. There are (lattice) superadditive upper bounds on the cumulative flows in all leading subrectangles of each rectangle. It is shown that there is a greatest cumulative flow formed by the natural generalization of the South-West Corner Rule that respects cumulative-flow capacities; it has maximum reward when the rewards are (lattice) superadditive; it is integer if the supplies, demands and capacities are integer; and it can be calculated myopically in linear time. The result is specialized to earlier work of Hoeffding (1940), Fréchet (1951), Lorentz (1953), Hoffman (1963) and Barnes and Hoffman (1985). Applications are given to extreme constrained bivariate distributions, optimal distribution with limited one-way product substitution and, generalizing results of Derman and Klein (1958), optimal sales with age-dependent rewards and capacities.To our friend, Philip Wolfe, with admiration and affection, on the occasion of his 65th birthday.Research was supported respectively by the IBM T.J. Watson and IBM Almaden Research Centers and is a minor revision of the IBM Research Report [6].