Showing papers on "Spectrum of a matrix published in 1998"
TL;DR: In this article, the authors developed a theory which describes the behavior of eigenvalues of a class of one-dimensional random non-Hermitian operators introduced by Hatano and Nelson.
Abstract: We develop a theory which describes the behavior of eigenvalues of a class of one-dimensional random non-Hermitian operators introduced recently by Hatano and Nelson. We prove that the eigenvalues are distributed along a curve in the complex plane. An equation for the curve is derived, and the density of complex eigenvalues is found in terms of spectral characteristics of a ``reference'' Hermitian disordered system. The generic properties of the eigenvalue distribution are discussed.
141 citations
TL;DR: In this article, it was shown that small eigenvalues (singular values) are determined to high relative accuracy by the data much more accurately than the classical perturbation theory would indicate.
Abstract: The classical perturbation theory for Hermitian matrix eigenvalue and singular value problems provides bounds on the absolute differences between approximate eigenvalues (singular values) and the true eigenvalues (singular values) of a matrix These bounds may be bad news for small eigenvalues (singular values), which thereby suffer worse relative uncertainty than large ones However, there are situations where even small eigenvalues are determined to high relative accuracy by the data much more accurately than the classical perturbation theory would indicate In this paper, we study how eigenvalues of a Hermitian matrix A change when it is perturbed to $\wtd A=D^*AD$, where D is close to a unitary matrix, and how singular values of a (nonsquare) matrix B change when it is perturbed to $\wtd B=D_1^*BD_2$, where D1 and D2 are nearly unitary It is proved that under these kinds of perturbations small eigenvalues (singular values) suffer relative changes no worse than large eigenvalues (singular values) Many well-known perturbation theorems, including the Hoffman--Wielandt and Weyl--Lidskii theorems, are extended
139 citations
TL;DR: In this article, the authors present a collection of relative perturbation results which have emerged during the past ten years and show that the derivation of many relative bounds can be based on absolute bounds.
Abstract: It used to be good enough to bound absolute of matrix eigenvalues and singular values. Not any more. Now it is fashionable to bound relative errors. We present a collection of relative perturbation results which have emerged during the past ten years.No need to throw away all those absolute error bound, though. Deep down, the derivation of many relative bounds can be based on absolute bounds. This means that relative bounds are not always better. They may just be better sometimes – and exactly when depends on the perturbation.
91 citations
TL;DR: In this article, the authors proposed a method to solve the problem of the problem: without abstracts, without abstractions, and without abstracting abstracts. But without abstract.
Abstract: Abstract. ((Without abstract))
41 citations
TL;DR: In this article, the glueball masses as eigenvalues for a supergravity wave equation in a black hole geometry were derived for 3-dimensional lattice QCDs, and the first states with masses m 2 = 11.58766, 34.52698, 68.33171, 241.236607, 321.626549, etc.
41 citations
TL;DR: The Chebyshev?tau spectral method may produce spurious eigenvalues with large positive real parts, even when all true eigen values of the problem are known to have negative real parts as discussed by the authors.
35 citations
TL;DR: In this article, the eigenvalues and eigenvectors of a square matrix A over a complete and completely distributive lattice were characterized and also the roots of the characteristic equation of A.
30 citations
TL;DR: In this article, the spectral properties of the Wilson-Dirac operator in topologically non-trivial gauge field configurations were studied and the role of eigenvectors with real eigenvalues as lattice equivalents of the continuum zero-modes was discussed.
19 citations
TL;DR: In this paper, the problem of simultaneous asymptotic estimation of eigenvalues of covariance matrix of Wishart matrix is considered under a weighted quadratic loss function.
Abstract: The problem of simultaneous asymptotic estimation of eigenvalues of covariance matrix of Wishart matrix is considered under a weighted quadratic loss function. James-Stein type of estimators are obtained which dominate the sample eigenvalues. The relative merits of the proposed estimators are compared to the sample eigenvalues using asymptotic quadratic distributional risk under loal alternatives. It is shown that the proposed estimators are asymptotically superior to the sample eigenvalues. Further, it is demonstrated that the James-Stein type estimator is dominated by its truncated part.
13 citations
TL;DR: In this article, the location of singular values of a matrix A is described in terms of its deleted absolute row sums and column sums, which is similar to Brauer's and Brualdi's theorems for eigenvalues.
10 citations
TL;DR: In this paper, the bound-state eigenfunctions and eigenvalues of a Schrodinger Hamiltonian are determined as functions of the strength of the potential and the method is able to determine the bound state energies for arbitrarily weak strengths of the possible potential.
Abstract: We present a simply applied numerical technique that allows the accurate determination of the bound-state eigenfunctions and eigenvalues of a differential operator such as the one-particle Schrodinger Hamiltonian. The method applies for potentials that asymptotically vanish. The eigenvalues and eigenfunctions are determined as functions of the strength of the potential and the method is able to determine the bound-state energies for arbitrarily weak strengths of the potential. At no point is a matrix diagonalized thus the method may be applied to problems with space dimension greater than unity.
TL;DR: In this article, the maximal dimension of a subspace of a matrix over a set of n n matrices over a field F such that each A 2 L has at most k distinct eigenvalues (in the algebraic closure of F ).
Abstract: Abstract. The following problem, originally proposed by Omladi c and Semrl [Linear Algebra Appl., 249:29{46 (1996)], is considered. Let k and n be positive integers such that k < n. Let L be a subspace of Mn(F ), the space of n n matrices over a eld F , such that each A 2 L has at most k distinct eigenvalues (in the algebraic closure of F ). Then, what is the maximal dimension of L. Omladi c and Semrl assumed that F = C and solved the problem for k = 1, k = 2 and n odd, and k = n 1 (under a mild assumption). In this paper, their results for k = 1 and k = n 1 are extended to any F such that char(F ) = 0, and a solution for k = 2 and any n, and for k = 3 is given.
TL;DR: In this paper, the spectrum of time eigenvalues has been studied for one-speed neutrons in isotropically scattering spheres with a reflexion coefficient R in the interval −1 ≤ R ≤ 1.
Abstract: The spectrum of time eigenvalues has been studied for one-speed neutrons in isotropically scattering spheres with a reflexion coefficient R in the interval −1 ≤ R ≤ 1. It is proved that a continuous spectrum exists in the Hilbert space. This continuum disappears if we suitably enlarge the function space. It is also proved that for R ≤ 0 there are only discrete eigenvalues on the real axis. However, for R > 0 there is a continuum of eigenvalues beyond the Corngold limit. An expression is given for the lower boundary of the continuum, and it is supported through numerical calculations.
TL;DR: In this article, Li et al. studied some basic properties of generalized eigenvalues of a definite Hermitian matrix pair and proved an interlacing theorem and a minimax theorem.
TL;DR: In this article, an elementary proof is given that some well-known formulae for derivatives of eigenvalues of matrix-valued functions hold under weaker hypotheses than are required by the usual proofs.
TL;DR: The possible usefulness of eigenvectors and eigenvalues of Karle–Hauptman matrices and the use of the sum of the square of the negative eigen values (SSNE) as a goodness-of-fit criterion is examined.
Abstract: The possible usefulness of eigenvectors and eigenvalues of Karle–Hauptman matrices is examined. The eigenvalue spectra of structures with calculated and measured |U|'s are discussed and the result of several attempts at phase refinement are reported. The use of the sum of the square of the negative eigenvalues (SSNE) as a goodness-of-fit criterion is examined. The possible use of a priori information is investigated using the approximation of orthogonal electron densities corresponding to eigenvectors with eigenvalues greater than zero.
Journal Article•
TL;DR: This paper combines the characteristic polynomial method and QR method to provide a new method, which can obtain the multiplex eigenvalues of a matrix, which is better than MATLAB or MATHEMATICA.
Abstract: There are two kinds of method to calculate the eigenvalues of a matrix:characteristic polynomial method and QR method. In this paper,we combine with these two methods to provide a new method,which can obtain the multiplex eigenvalues. The programming,written by Borland C++,is given. The numerical example shows, if the matrix has multiplex eigenvalues, our method is better than MATLAB or MATHEMATICA.
TL;DR: In this paper, the authors considered self-adjoint boundary value problems with discrete spectrum and coefficients periodic in a certain coordinate and established upper bounds for eigenvalues in terms of the eigen values of the corresponding problem with averaged coefficients.
Abstract: We consider self-adjoint boundary-value problems with discrete spectrum and coefficients periodic in a certain coordinate. We establish upper bounds for eigenvalues in terms of the eigenvalues of the corresponding problem with averaged coefficients. We illustrate the results obtained in the case of the Hill vector equation and for circular and rectangular plates with periodic coefficients.
TL;DR: In this paper, the root locus method is used to calculate the eigenvalues step by step from low order systems to high order systems, and the property of Schwartz matrix is also used to determine the factors of the characteristic equation by the searching method.
Abstract: Two methods for calculating eigenvalues of linear systems are proposed. One is to use the root locus method to calculate the eigenvalues step by step from low order systems to high order systems. The other is to use the property of Schwartz matrix to determine the factors of the characteristic equation by the searching method. A method for determining the number of eigenvalues in each half complex plane is also presented.
TL;DR: For a bicontractive operator T on a Kreĭ space, the connections between its eigen values and eigenstructure and the eigenvalues and eigstructure of its minimal unitary dilation U are studied in this paper.
Abstract: For a bicontractive operator T on a Kreĭ space the connections between its eigenvalues and eigenstructure and the eigenvalues and eigenstructure of its minimal unitary dilation U are studied. For eigenvalues on the unit circle of T in general only part of the eigenspace of T will return its an eigenspace of U and the corresponding eigenvalue will be a singular critical point of U.
TL;DR: For estimating error bound of computed eigenvalues of a matrix, there needs more practical perturbation theory of eigen values of matrices and two problems are put forward in such direction.
TL;DR: In this paper, the robustness of a spectral assignment method applied to the system of a flexible beam is studied and it is shown that the intersection between any admissible set of closed-loop eigenvalues of the nominal system and a corresponding set of a perturbed system can only have a finite number of elements.
Abstract: In this paper the robustness of a spectral assignment method applied to the system of a flexible beam is studied. It is shown that the intersection between any admissible set of closed-loop eigenvalues of the nominal system and a corresponding set of a perturbed system can only have a finite number of elements. Then, the impact on the eigenvalues of the perturbed system of a control law that affects only a finite number of eigenvalues of the nominal system is investigated. It is proved that only a finite number of eigenvalues of the perturbed system is moved. In addition, the polynomial whose roots are these closed-loop eigenvalues of the perturbed system is determined. Finally, using a new parameterization of uncertainty, the nonlinearity of the coefficients of this polynomial with respect to the uncertain parameters is simplified, turning it into a multivariate polynomial form whose stability robustness can be studied using several known methods.
TL;DR: In this paper, the eigenvalue spectrum of the staggered Dirac matrix on a 63 × 4 lattice is analyzed in full QCD at finite temperature and in the presence of a chemical potential.