共查询到20条相似文献,搜索用时 15 毫秒
1.
Yunkai Zhou 《Applied mathematics and computation》2011,217(24):10267-10270
We study the eigenvalues of a matrix A perturbed by a few special low-rank matrices. The perturbation is constructed from certain basis vectors of an invariant subspace of A, such as eigenvectors, Jordan vectors, or Schur vectors. We show that most of the eigenvalues of the low-rank perturbed matrix stayed unchanged from the eigenvalues of A; the perturbation can only change the eigenvalues of A that are related to the invariant subspace. Existing results mostly studied using eigenvectors with full column rank for perturbations, we generalize the results to more general settings. Applications of our results to a few interesting problems including the Google’s second eigenvalue problem are presented. 相似文献
2.
We consider the eigenvalues and eigenvectors of finite, low rank perturbations of random matrices. Specifically, we prove almost sure convergence of the extreme eigenvalues and appropriate projections of the corresponding eigenvectors of the perturbed matrix for additive and multiplicative perturbation models.The limiting non-random value is shown to depend explicitly on the limiting eigenvalue distribution of the unperturbed random matrix and the assumed perturbation model via integral transforms that correspond to very well-known objects in free probability theory that linearize non-commutative free additive and multiplicative convolution. Furthermore, we uncover a phase transition phenomenon whereby the large matrix limit of the extreme eigenvalues of the perturbed matrix differs from that of the original matrix if and only if the eigenvalues of the perturbing matrix are above a certain critical threshold. Square root decay of the eigenvalue density at the edge is sufficient to ensure that this threshold is finite. This critical threshold is intimately related to the same aforementioned integral transforms and our proof techniques bring this connection and the origin of the phase transition into focus. Consequently, our results extend the class of ‘spiked’ random matrix models about which such predictions (called the BBP phase transition) can be made well beyond the Wigner, Wishart and Jacobi random ensembles found in the literature. We examine the impact of this eigenvalue phase transition on the associated eigenvectors and observe an analogous phase transition in the eigenvectors. Various extensions of our results to the problem of non-extreme eigenvalues are discussed. 相似文献
3.
Alexei A. Mailybaev 《Numerical Linear Algebra with Applications》2006,13(5):419-436
The paper develops Newton's method of finding multiple eigenvalues with one Jordan block and corresponding generalized eigenvectors for matrices dependent on parameters. It computes the nearest value of a parameter vector with a matrix having a multiple eigenvalue of given multiplicity. The method also works in the whole matrix space (in the absence of parameters). The approach is based on the versal deformation theory for matrices. Numerical examples are given. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
4.
This paper deals with the asymptotic distribution of Wishart matrix and its application to the estimation of the population matrix parameter when the population eigenvalues are block-wise infinitely dispersed. We show that the appropriately normalized eigenvectors and eigenvalues asymptotically generate two Wishart matrices and one normally distributed random matrix, which are mutually independent. For a family of orthogonally equivariant estimators, we calculate the asymptotic risks with respect to the entropy or the quadratic loss function and derive the asymptotically best estimator among the family. We numerically show (1) the convergence in both the distributions and the risks are quick enough for a practical use, (2) the asymptotically best estimator is robust against the deviation of the population eigenvalues from the block-wise infinite dispersion. 相似文献
5.
The tracking of eigenvalues and eigenvectors for parameterized matrices is of major importance in optimization and stability problems. In the present paper, we consider a one-parameter family of matrices with distinct eigenvalues. A complete system of differential equations is developed for both the eigenvalues and the right and left eigenvectors. The computational feasibility of the differential system is demonstrated by means of a numerical example.The work of R. Kalaba and L. Tesfatsion was partially supported by the National Science Foundation under Grant No. ENG-77-28432 and by the National Institutes of Health under Grant No. GM-23732-03. 相似文献
6.
We consider the asymptotic joint distribution of the eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues become infinitely dispersed. We show that the normalized sample eigenvalues and the relevant elements of the sample eigenvectors are asymptotically all mutually independently distributed. The limiting distributions of the normalized sample eigenvalues are chi-squared distributions with varying degrees of freedom and the distribution of the relevant elements of the eigenvectors is the standard normal distribution. As an application of this result, we investigate tail minimaxity in the estimation of the population covariance matrix of Wishart distribution with respect to Stein's loss function and the quadratic loss function. Under mild regularity conditions, we show that the behavior of a broad class of tail minimax estimators is identical when the sample eigenvalues become infinitely dispersed. 相似文献
7.
Jerome Eisenfeld 《Linear algebra and its applications》1976,15(3):205-215
This paper deals with block diagonalization of partitioned (not necessarily square) matrices. The process is shown to be analogous to calculating eigenvalues and eigenvectors. Computer techniques and examples are provided. Several various types of applications are discussed including application to liver disease. 相似文献
8.
Tetsuro Yamamoto 《Numerische Mathematik》1980,34(2):189-199
Summary On the basis of an existence theorem for solutions of nonlinear systems, a method is given for finding rigorous error bounds for computed eigenvalues and eigenvectors of real matrices. It does not require the usual assumption that the true eigenvectors span the whole space. Further, a priori error estimates for eigenpairs corrected by an iterative method are given. Finally the results are illustrated with numerical examples.Dedicated to Professor Yoshikazu Nakai on his sixtieth birthday 相似文献
9.
10.
A. G. Mazko 《Ukrainian Mathematical Journal》2011,62(8):1234-1250
We consider the problem of localization of eigenvalues of polynomial matrices. We propose sufficient conditions for the spectrum
of a regular matrix polynomial to belong to a broad class of domains bounded by algebraic curves. These conditions generalize
the known method for the localization of the spectrum of polynomial matrices based on the solution of linear matrix inequalities.
We also develop a method for the localization of eigenvalues of a parametric family of matrix polynomials in the form of a
system of linear matrix inequalities. 相似文献
11.
This article is a continuation of the article [F. Zhang, Ger?gorin type theorems for quaternionic matrices, Linear Algebra Appl. 424 (2007), pp. 139–153] on the study of the eigenvalues of quaternion matrices. Profound differences in the eigenvalue problems for complex and quaternion matrices are discussed. We show that Brauer's theorem for the inclusion of the eigenvalues of complex matrices cannot be extended to the right eigenvalues of quaternion matrices. We also provide necessary and sufficient conditions for a complex square matrix to have infinitely many left eigenvalues, and analyse the roots of the characteristic polynomials for 2?×?2 matrices. We establish a characterisation for the set of left eigenvalues to intersect or be part of the boundary of the quaternion balls of Ger?gorin. 相似文献
12.
Hisao Nagao 《Annals of the Institute of Statistical Mathematics》1988,40(3):477-489
This paper deals with some problems of eigenvalues and eigenvectors of a sample correlation matrix and derives the limiting distributions of their jackknife statistics with some numerical examples. 相似文献
13.
An algorithm has been developed for finding the eigenvalues of a symmetric matrixA in a given interval [a, b] and the corresponding eigenvectors using a modification of the method of simultaneous iteration with the same favorable convergence properties. The technique is most suitable for large sparse matrices and can be effectively implemented on a parallel computer such as the ILLIAC IV. 相似文献
14.
Axel Ruhe 《BIT Numerical Mathematics》1970,10(3):343-354
When a matrix is close to a matrix with a multiple eigenvalue, the arithmetic mean of a group of eigenvalues is a good approximation to this multiple eigenvalue. A theorem of Gershgorin type for means of eigenvalues is proved and applied as a perturbation theorem for a degenerate matrix.For a multiple eigenvalue we derive bounds for computed bases of subspaces of eigenvectors and principal vectors, relating them to the spaces spanned by the last singular vectors of corresponding powers of the matrix. These bounds assure that, provided the dimensionalities are chosen appropriately, the angles of rotation of the subspaces are of the same order of magnitude as the perturbation of the matrix.A numerical example is given. 相似文献
15.
Lennart Bondesson 《Linear and Multilinear Algebra》2013,61(3):239-247
A nonsymmetric N?×?N matrix with elements as certain simple functions of N distinct real or complex numbers r 1, r 2, …, rN is presented. The matrix is special due to its eigenvalues???the consecutive integers 0,1,2, …, N?1. Theorems are given establishing explicit expressions of the right and left eigenvectors and formulas for recursive calculation of the right eigenvectors. A special case of the matrix has appeared in sampling theory where its right eigenvectors, if properly normalized, give the inclusion probabilities of the conditional Poisson sampling design. 相似文献
16.
W. Murray 《Journal of Optimization Theory and Applications》1971,7(3):189-196
Hessian matrices play a key role in optimization. Knowledge of their behavior is useful both in giving insight into optimization problems and in designing algorithms to solve them. In this paper, analytical expressions are obtained for the eigenvalues and eigenvectors at the intermediate minima of barrier and penalty functions. This in turn leads to an analytical expression for the inverse of the Hessian matrix (it is singular) at the solution.The author acknowledges the programming assistance of D. Dennis. This work has been carried out at the National Physical Laboratory, Teddington, Middlesex, England. 相似文献
17.
An algorithm is presented in this paper by which the rth root of real or complex matrices can be found without the computation of the eigenvalues and eigenvectors of the matrix. All required computations are in the real domain. The method is based on the Newton-Raphson algorithm and is capable of finding roots even when the matrix is defective. Computing the root of a matrix from eigenvalues and eigenvectors would be the preferred method if these data were available. 相似文献
18.
Recently the authors proposed a simultaneous iteration algorithm for the computation of the partial derivatives of repeated eigenvalues and the corresponding eigenvectors of matrices depending on several real variables. This paper analyses the properties of that algorithm and extends it in several ways. The previous requirement that the repeated eigenvalue be dominant is relaxed, and the new generalized algorithm given here allows the simultaneous treatment of simple and repeated eigenvalues. Methods for accelerating convergence are examined. Numerical results support our theoretical analysis. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献
19.
矩阵特征值的一类新的包含域 总被引:1,自引:0,他引:1
李华 《纯粹数学与应用数学》2010,26(4):673-678
用盖尔圆盘定理来估计矩阵的特征值是一个经典的方法,这种方法仅利用矩阵的元素来确定特征值的分布区域.本文利用相似矩阵有相同的特征值这一理论,得到了矩阵特征值的一类新的包含域,它们与盖尔圆盘等方法结合起来能提高估计的精确度. 相似文献
20.
We investigate lower bounds for the eigenvalues of perturbations of matrices. In the footsteps of Weyl and Ipsen & Nadler, we develop approximating matrices whose eigenvalues are lower bounds for the eigenvalues of the perturbed matrix. The number of available eigenvalues and eigenvectors of the original matrix determines how close those approximations can be, and, if the perturbation is of low rank, such bounds are relatively inexpensive to obtain. Moreover, because the process need not be restricted to the eigenvalues of perturbed matrices, lower bounds for eigenvalues of bordered diagonal matrices as well as for singular values of rank-k perturbations and other updates of n×m matrices are given. 相似文献