Showing papers on "Spectrum of a matrix published in 2000"

On the isolated spectrum of the Perron-Frobenius operator

Abstract: We discuss the existence of large isolated (non-unit) eigenvalues of the Perron-Frobenius operator for expanding interval maps. Corresponding to these eigenvalues (or `resonances') are distributions which approach the invariant density (or equilibrium distribution) at a rate slower than that prescribed by the minimal expansion rate. We consider the transitional behaviour of the eigenfunctions as the eigenvalues cross this `minimal expansion rate' threshold, and suggest dynamical implications of the existence and form of these eigenfunctions. A systematic means of constructing maps which possess such isolated eigenvalues is presented.

A Fast Recursive Algorithm for Constructing Matrices with Prescribed Eigenvalues and Singular Values

Abstract: The Weyl--Horn theorem characterizes a relationship between the eigenvalues and the singular values of an arbitrary matrix. Based on that characterization, a fast recursive algorithm is developed to construct numerically a matrix with prescribed eigenvalues and singular values. Besides being of theoretical interest, the technique could be employed to create test matrices with desired spectral features. Numerical experiment shows this algorithm to be quite efficient and robust.

Algorithm 800: Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices. I: the square-reduced method

Abstract: This article describes LAPACK-based Fortran 77 subroutines for the reduction of a Hamiltonian matrix to square-reduced form and the approximation of all its eigenvalues using the implicit version of Van Loan's method. The transformation of the Hamiltonian matrix to a square-reduced form transforms a Hamiltonian eigenvalue problem of order 2n to a Hessenberg eigenvalue problem of order n. The eigenvalues of the Hamiltonian matrix are the square roots of those of the Hessenberg matrix. Symplectic scaling and norm scaling are provided, which, in some cases, improve the accuracy of the computed eigenvalues. We demonstrate the performance of the subroutines for several examples and show how they can be used to solve some control-theoretic problems.

Bounds for sums of eigenvalues and applications

Abstract: Let A be a matrix of order n × n with real spectrum λ1 ≥ λ2 ≥ ⋯ ≥ λn. Let 1 ≤ k ≤ n − 2. If λn or λ1 is known, then we find an upper bound (respectively, lower bound) for the sum of the k-largest (respectively, k-smallest) remaining eigenvalues of A. Then, we obtain a majorization vector for (λ1, λ2,…, λn−1) when λn is known and a majorization vector for (λ2, λ3,…, λn) when λ1 is known. We apply these results to the eigenvalues of the Laplacian matrix of a graph and, in particular, a sufficient condition for a graph to be connected is given. Also, we derive an upper bound for the coefficient of ergodicity of a nonnegative matrix with real spectrum.

Application of random matrix theory to study cross-correlations of stock prices

Abstract: We address the question of how to precisely identify correlated behavior between different firms in the economy by applying methods of random matrix theory (RMT) Specifically, we use methods of random matrix theory to analyze the cross-correlation matrix of price changes of the largest 1000 US stocks for the 2-year period 1994–1995 We find that the statistics of most of the eigenvalues in the spectrum of agree with the predictions of random matrix theory, but there are deviations for a few of the largest eigenvalues To prove that the rest of the eigenvalues are genuinely random, we test for universal properties such as eigenvalue spacings and eigenvalue correlations We demonstrate that shares universal properties with the Gaussian orthogonal ensemble of random matrices In addition, we quantify the number of significant participants, that is companies, of the eigenvectors using the inverse participation ratio, and find eigenvectors with large inverse participation ratios at both edges of the eigenvalue spectrum — a situation reminiscent of results in localization theory

Calculating Derivatives of Repeated and Nonrepeated Eigenvalues Without Explicit Use of Eigenvectors

Abstract: The analysis of inverse problems in parametric model updating often require the sensitivities of eigenvalues. The calculation of these sensitivities is mathematically related to the derivatives of the eigenvalues with respect to the model parameters. A common method to calculate these derivatives is the Nelson method, which requires the eigenvectors. The method introduced in this paper is derived from the characteristic equation of the underlying general eigenvalue problem and allows the derivatives of eigenvalues with respect to the model parameters to be calculated without explicit use of the eigenvectors. The method is extended for the case of repeated eigenvalues, which leads to restrictions on the parameterization. For repeated eigenvalues of multiplicity two, these restrictions are formulated expicitly. Applications and limitations of the method are demonstrated by examples.

Eigenvalues of a λ-rational Sturm–Liouville problem

Abstract: The Sturm-Liouville problem (1.1), (1.2) is considered, which depends rationally on the eigenvalue parameter. Estimates for the eigenvalues and especially for the embedded eigenvalues are proved. Moreover, conditions for the non-existence of embedded eigenvalues are given.

Method of determining an intrinsic spectrum from a measured spectrum

Abstract: In determining an intrinsic spectrum from a measured spectrum using the Maximum Entropy Algorithm, it is hardly or even not at all practical to determine the eigenvalues of an NxN matrix of large dimensions (N of the order of from 10?4 to 105?). According to the invention such a large matrix is subdivided into a large number of much smaller partial matrices that are located on the diagonal or trace of the large matrix. The set of eigenvalues to be determined then consists of all eigenvalues of the partial matrices which can be determined much faster. Because of the Toeplitz-like character of the partial matrices, their eigenvalues can be determined very fast by Fourier transformation of a single row of such a matrix. Using the set of eigenvalues thus obtained, the intrinsic spectrum is determined by means of a minimizing algorithm. The convergence rate of the minimizing algorithm can be highly enhanced by adding a random noise value to the variables of the minimizing process and by decreasing that noise value to zero in a number of iteration steps.

Linearly dependent subspaces and the eigenvalue spectrum of the one-particle reduced density matrix

Abstract: The structure of the one-particle reduced density matrix when expressed in a Cartesian Gaussian basis set is investigated. A set of exact linear dependency conditions between products of basis functions, which result from the angular behaviour of the basis functions, is discovered. Some of these exact linear dependencies hold simultaneously in both position and momentum spaces making it possible to alter the one matrix while keeping both the position and momentum densities fixed. The magnitude of this space is easily predicted for the Pople and Dunning-Hay basis sets commonly used in quantum chemical calculations, and we give simple rules for their enumeration. It is further shown that alteration of the one-matrix component in this space alters the eigenvalue structure of the one-matrix and therefore has consequences for N-representability. Using the one-matrix corresponding to a wavefunction as a starting point, the eigenvalue change is always in the same direction, that is small eigenvalues get more negative while large ones become more positive. For independent particle model wavefunctions, which are already extreme in their eigenvalues, no change is possible without breaking the N-representability conditions.

Eigenvalues assignment robustness in specified regions for uncertain multi-time-scale systems

Abstract: From the viewpoint of the eigenvalues locations of the multi-time-scale systems, we realize that the eigenvalues positions of the fast subsystems are far from the slow subsystem ones, and the uncertainty or parameter variation will move the eigenvalues of a nominal system away from the desired ones. Therefore, it is significant to guarantee that the eigenvalues of the perturbed system lie inside our desired regions as those of the nominal system. The multi-time-scale system under structured perturbation is considered in this paper. We present some sufficient conditions for the robustness measures of eigenvalues assignment in the specified regions. A design algorithm by the method of constraint optimization to design a robust decentralized controller to assign the eigenvalues of whole multi-time-scale system is also provided in this research. An illustration example is presented to show the applicabilityof our proposed theorems.

Some results, concerning the variation of eigenvalues, when these depend on a real parameter

Abstract: In this article we deal with a Hamiltonial of the form H(v) = Ho + A(v) where Ho is a self-adjoint bounded or unbounded operator on a Hilbert space and A(v) is a bounded self-adjoint perturbation depending on a real parameter v. In quantum mechanics a variety of results has been obtained by taking formally the derivative of the eigenvectors and eigenvalues of H(v).The differentiability of the eigenvectors and eigenvalues has been rigorously proved under several assumptions. Among these assumptions is the assumption that the eigenvalues are simple and the assumption that the perturbation A(v) is a uniformly bounded self-adjoint operator. A part of this article is dealing with examples, which show that these two assumptions are essential. The rest of this article is devoted to different applications concerning asymptotic relations of eigenvalues and a result for the solutions of the equation dy/dt= M(t)y in an abstract infinite dimensional Hilbert space, where iM(t)(12=-1) is self-adjoint for every t in an ...

Comparing lattice Dirac operators with Random Matrix Theory

Abstract: We study the eigenvalue spectrum of different lattice Dirac operators (staggered, fixed point, overlap) and discuss their dependence on the topological sectors. Although the model is 2D (the Schwinger model with massless fermions) our observations indicate possible problems in 4D applications. In particular misidentification of the smallest eigenvalues due to non-identification of the topological sector may hinder successful comparison with Random Matrix Theory (RMT).

The influence of isolated largest eigenvalues on the numerical convergence of the CG method

Abstract: This paper considers the dependence of the convergence history of the CG method on the largest eigenvalues of a symmetric positive-definite matrix. It is demonstrated that, in solving ill-conditioned linear systems, the reproduction of largest eigenvalues can be so intensive that they cannot be treated as isolated. On the other hand, from the moment the smallest isolated eigenvalues start to govern the numerical convergence of the CG method, the convergence is mainly influenced by the smallest Ritz values. Bibliography: 2 titles.

Perturbations of isolated eigenvalues

Abstract: In this paper, we generalize the result given by A. A. Dezin in [1] to the case of operators defined on a Banach space, under slightly different hypothesis. In the first part we consider perturbations of isolated simple eigenvalues, and in the second part we deal with perturbations of isolated multiple eigenvalues.

Accelerated spectral refinement Part II: Cluster of eigenvalues

Abstract: The framework for accelerated spectral refinement for a simple eigenvalue developed in Part I of this paper is employed to treat the general case of a cluster of eigenvalues whose total algebraic multiplicity is finite. Numerical examples concerning the largest and the second largest multiple eigenvalues of an integral operator are given.

Numerical investigation of eigenoscillations near the system of two strips forming a cross in the channel

Abstract: The existence of eigenoscillations near the system mentioned in the title system is proven. The number of oscillation modes is determined. A classification by groups of possible symmetry is carry out. An infinite matrix equation for the coefficients of corresponding expansion is obtained. This equation is investigated numerically. The plots of eigenvalues versus the length of the cross are obtained. An approximate formula for the eigenvalues is found and investigated. The theory of the self-adjoint operators, the "Dirichlet-Neumenn bracket" and variational methods are used.

Supersymmetric and Shape-Invariant Generalization for Non-resonant Jaynes-Cummings Systems

Abstract: A class of shape-invariant bound-state problems which represent transitions in a two-level system introduced earlier are generalized to include arbitrary energy splittings between the two levels. We show that the coupled-channel Hamiltonians obtained correspond to the generalization of the non-resonant Jaynes-Cummings Hamiltonian, widely used in quantized theories of laser. In this general context, we determine the eigenstates, eigenvalues, the time evolution matrix and the population inversion matrix factor.