Distribution of Subdominant Eigenvalues of Random Matrices

We mainly investigate the behavior of the subdominant eigenvalue of matrices B= (b _i,j)^n,n whose entries are independent random variables with an expectation Eb _i,j=1/n and with a variance n c/n ² for some constant c 0. For such matrices we show that for large n, the subdominant eigenvalue is, with great probability, in a small neighborhood of 0. We also show that for large n, the spectral radius of such matrices is, with great probability, in a small neighborhood of 1.     

利用矩阵的初等变换求方阵的特征值

高阶方阵的特征值的求得,需求解一元高次方程,这往往有一定的难度.本文依据矩阵的初等变换的一些良好性质,介绍两种利用矩阵的初等变换化简方阵的特征值的计算的方法.     

Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices

Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian. The deviations from Gaussian behavior can be interpreted as arising from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one considers real symmetric palindromic Toeplitz matrices, matrices where the first row is a palindrome. A similar result was previously proved for a related circulant ensemble through an analysis of the explicit formulas for eigenvalues. By Cauchy’s interlacing property and the rank inequality, this ensemble has the same limiting spectral distribution as the palindromic Toeplitz matrices; a consequence of combining the two approaches is a version of the almost sure Central Limit Theorem. Thus our analysis of these Diophantine equations provides an alternate technique for proving limiting spectral measures for certain ensembles of circulant matrices. A. Massey’s current address: Department of Mathematics, UCLA, Los Angeles, CA 90095, USA. e-mail: amassey3102@math.ucla.edu.     

Cluster robust error estimates for the Rayleigh-Ritz approximation II: Estimates for eigenvalues

This is the second part of a paper that deals with error estimates for the Rayleigh-Ritz approximations of the spectrum and invariant subspaces of a bounded Hermitian operator in a Hilbert or Euclidean space. This part addresses the approximation of eigenvalues. Two kinds of estimates are considered: (i) estimates for the eigenvalue errors via the best approximation errors for the corresponding invariant subspaces, and (ii) estimates for the same via the corresponding residuals. Estimates of these two kinds are needed for, respectively, the a priori and a posteriory error analysis of numerical methods for computing eigenvalues. The paper’s major concern is to ensure that the estimates in question are accurate and ‘cluster robust’, i.e. are not adversely affected by the presence of clustered, i.e. closely situated eigenvalues among those of interest. The paper’s main new results introduce estimates for clustered eigenvalues whereby not only the distances between eigenvalues in the cluster are not present but also the distances between the cluster and the rest of the spectrum appear in asymptotically insignificant terms only.     

关于厄米特矩阵乘积特征值的讨论

讨论厄米特矩阵乘积的特征值 ,推广了文 [1 ]的结果 .指出了文 [2 ]中的一个错误 ,给出了关于迹的一个不等式 .     

利用矩阵的初等变换求方阵的特征值

高阶方阵的特征多项式以及特征值的求得,在计算上往往有一定的难度.本文首先从理论上分析了存在一个上三角矩阵或者下三角矩阵与一个方阵相似;接着,提出了相似变换的概念,分析了相似变换中初等矩阵的选择方法;然后指出了利用相似变换在求方阵的特征多项式以及特征值时的方法,并列举若干实例给予了说明.     

Distribution of Eigenvalues for the Ensemble of Real Symmetric Toeplitz Matrices

Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d. random variable from a fixed probability distributionpof mean 0,variance 1, and finite moments of all order. The limiting spectral measure (the density of normalized eigenvalues) converges weakly to a new universal distribution with unbounded support, independent of pThis distribution’s moments are almost those of the Gaussian’s, and the deficit may be interpreted in terms of obstructions to Diophantine equations; the unbounded support follows from a nice application of the Central Limit Theorem. With a little more work, we obtain almost sure convergence. An investigation of spacings between adjacent normalized eigenvalues looks Poissonian, and not GOE. A related ensemble (real symmetric palindromic Toeplitz matrices) appears to have no Diophantine obstructions, and the limiting spectral measure’s first nine moments can be shown to agree with those of the Gaussian; this will be considered in greater detail in a future paper.     

The Eigenvalues of Very Sparse Random Symmetric Matrices

Let W _n be an n × n random symmetric sparse matrix with independent identically distributed entries such that the values 1 and 0 are taken with probabilities p/n and 1-p/n, respectively; here is independent of n. We show that the limit of the expected spectral distribution functions of W _n has a discrete part. Moreover, the set of positive probability points is dense in (- +). In particular, the points , and 0 belong to this set.     

Cluster robust error estimates for the Rayleigh-Ritz approximation I: Estimates for invariant subspaces

This is the first part of a paper that deals with error estimates for the Rayleigh-Ritz approximations to the spectrum and invariant subspaces of a bounded Hermitian operator in a Hilbert or Euclidean space. This part addresses estimates for the angles between the invariant subspaces and their approximations via the corresponding best approximation errors and residuals and, for invariant subspaces corresponding to parts of the discrete spectrum, via eigenvalue errors. The paper’s major concern is to ensure that the estimates in question are accurate and ‘cluster robust’, i.e. are not adversely affected by the presence of clustered, i.e. closely situated eigenvalues in the spectrum. Available estimates of such kind are reviewed and new estimates are derived. The paper’s main new results introduce estimates for invariant subspaces in which the operator may have clustered eigenvalues whereby not only the distances between eigenvalues in the cluster are not present but also the distances between the cluster and the rest of the spectrum appear in asymptotically insignificant terms only.     

幂零矩阵的一个性质

通过对2009年全国硕士研究生入学考试数学试卷中的一道线性代数试题的推广,得到幂零矩阵的一个性质,从而揭示出该试题的命题思路与背景.     

关于四元数自共轭矩阵乘积迹和特征值的几个定理

给出四元数自共轭矩阵乘积迹的几个定理及特征值之界的几个新估计，在四元数体上改进和推广了文［１－１２］的相应结果     

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.     

Asymptotics of Eigenvalues for a Class of Jacobi Matrices with Limiting Point Spectrum

In this paper, we study a class of Jacobi matrices with very rapidly decreasing weights. It is shown that the Weyl function (the matrix element of the resolvent of the operator) for the class under study can be expressed as the ratio of two entire transcendental functions of order zero. It is shown that the coefficients in the expansion of these functions in Taylor series are proportional to the generating functions of the number of integral solutions defined by certain Diophantine equations. An asymptotic estimate for the eigenvalues is obtained.     

Commutators, Spectral Trace Identities, and Universal Estimates for Eigenvalues

Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the eigenvalue gaps. As particular examples, we present simple proofs of the classical universal estimates for eigenvalues of the Dirichlet Laplacian, as well as of some known and new results for other differential operators and systems. We also suggest an extension of the methods to the case of non-self-adjoint operators.     

The Convexity Estimates for the Solutions of Two Elliptic Equations

In this paper, for the solutions of two elliptic equations we find the auxiliary curvature functions which attain respective minimum on the boundary. These results are the generalization of the classical ones in Makar-Limanov [17 Makar-Limanov , L.G. ( 1971 ). Solution of Dirichlet's problem for the equation Δu = ?1 on a convex region . Math. Notes Acad. Sci. USSR 9 : 52 – 53 .[Crossref] , [Google Scholar]] for the torsion equation and Acker et al. [1 Acker , A. , Payne , L.E. , Philippin , G. ( 1981 ). On the convexity of level lines of the fundamental mode in the clamped membrane problem, and the existence of convex solutions in a related free boundary problem . Z. Angew. Math. Phys. 32 : 683 – 694 .[Crossref], [Web of Science ®] , [Google Scholar]] for the first eigenfunction of the Laplacian in convex domains of dimension 2. Then we get the new proof of the specific convexity of the solutions of the above two elliptic equations. As a consequence, for the elliptic equation vΔv = ? (1 + |?v|²) in a smooth, bounded and strictly convex domain Ω in ? ⁿ with homogeneous Dirichlet boundary value condition, we also get a sharply lower bound estimate of the Gaussian curvature for the solution surface by the curvature of the boundary of the domain.     

亏损特征值的灵敏度分析

研究了广义特征问题中特征值和不变特征子空间对参数的导数,利用隐函数定理证明了亏损广义特征值问题的平均特征值对参数的解析性,并利用标准特征值的灵敏度分析得到了可约化广义亏损特征值的平均值和相应的不变子空间对参数的导数.这一结果在结构优化、模型修正、以及故障诊断等领域中有着重要应用,为工程计算提供了理论依据.     

Augmentation and Modification Problems for Hermitian Matrices

We obtain necessary and sufficient conditions for the solvability of the augmentation and modification problems of order for Hermitian matrices. The augmentation problem consists in the construction of a Hermitian -matrix with a given -block in block -representation and with the prescribed eigenvalues. The modification problem consists in the construction of a Hermitian -matrix of rank not greater than so that the obtained matrix, being added to a given Hermitian -matrix , will have the required spectrum. We give an estimate for the minimal number of different eigenvalues of the solutions to these problems.     

实对称矩阵的一类逆特征值问题

利用矩阵的奇异值分解及广义逆，给出了矩阵约束下矩阵反问题AX=B有实对称解的充分必要条件及其通解的表达式．此外，给出了在矩阵方程的解集合中与给定矩阵的最佳逼近解的表达式．     

Preconditioners for Ill-Conditioned Toeplitz Matrices

This paper is concerned with the solution of systems of linear equations A _N x = b, where denotes a sequence of positive definite Hermitian ill-conditioned Toeplitz matrices arising from a (real-valued) nonnegative generating function f C ₂ with zeros. We construct positive definite Hermitian preconditioners M _N such that the eigenvalues of M _N ^–1 A _N are clustered at 1 and the corresponding PCG-method requires only O(N log N) arithmetical operations to achieve a prescribed precision. We sketch how our preconditioning technique can be extended to symmetric Toeplitz systems, doubly symmetric block Toeplitz systems with Toeplitz blocks and non-Hermitian Toeplitz systems. Numerical tests confirm the theoretical expectations.     

论利用相似矩阵的性质来确定矩阵中未知元素方法的完整性

讨论了利用相似矩阵的性质来确定矩阵中未知元素方法的完整性.