Showing papers on "Spectrum of a matrix published in 1999"
TL;DR: In this article, the general parameter dependent distribution between two sets of eigenvalues with two different parameter values can be expressed as a quaternion determinant, and a connection formula relating the distributions at the hard and soft edge is obtained, a universal asymptotic behaviour of the two point correlation is identified.
175 citations
TL;DR: A simple formula is presented for the approximate (limiting Frechet) subdifferential of an arbitrary function of the eigenvalues of an symmetric matrix nonsmoothly, subsuming earlier results on convex and Clarke subgradients.
Abstract: The eigenvalues of a symmetric matrix depend on the matrix nonsmoothly. This paper describes the nonsmooth analysis of these eigenvalues. In particular, I present a simple formula for the approximate (limiting Frechet) subdifferential of an arbitrary function of the eigenvalues, subsuming earlier results on convex and Clarke subgradients. As an example I compute the subdifferential of the k'th largest eigenvalue.
120 citations
TL;DR: In this paper, an unfolding procedure for complex eigenvalues was introduced and applied to data from lattice QCD at finite chemical potential to construct the nearest-neighbor spacing distribution of adjacent eigen values in the complex plane.
Abstract: In quantum chromodynamics (QCD) at nonzero chemical potential, the eigenvalues of the Dirac operator are scattered in the complex plane. Can the fluctuation properties of the Dirac spectrum be described by universal predictions of non-Hermitian random matrix theory? We introduce an unfolding procedure for complex eigenvalues and apply it to data from lattice QCD at finite chemical potential $\mu$ to construct the nearest-neighbor spacing distribution of adjacent eigenvalues in the complex plane. For intermediate values of $\mu$, we find agreement with predictions of the Ginibre ensemble of random matrix theory, both in the confinement and in the deconfinement phase.
97 citations
TL;DR: In this paper, the spectrum of matrix-valued functions depending analytically on two parameters is studied and generalizations of the Rellich theorem on analytic dependence of the spectrum and complete regular splitting of multiple eigenvalues are established.
Abstract: In this paper behaviour of the spectrum of matrix-valued functions depending analytically on two parameters is studied. Generalizations of the Rellich theorem on analytic dependence of the spectrum and complete regular splitting of multiple eigenvalues are established.
59 citations
TL;DR: This paper computed the spectrum of the scattering integral operator for a sphere and the eigenvalues of the coefficient matrices that arise from the discretization of the integral equation, and tried to use this information to predict the performance of iterative methods.
Abstract: The volume integral equation of electromagnetic scattering can be used to compute the scattering by inhomogeneous or anisotropic scatterers. In this paper we compute the spectrum of the scattering integral operator for a sphere and the eigenvalues of the coefficient matrices that arise from the discretization of the integral equation. For the case of a spherical scatterer, the eigenvalues lie mostly on a line in the complex plane, with some eigenvalues lying below the line. We show how the spectrum of the integral operator can be related to the well-posedness of a modified scattering problem. The eigenvalues lying below the line segment arise from resonances in the analytical series solution of scattering by a sphere. The eigenvalues on the line are due to the branch cut of the square root in the definition of the refractive index. We try to use this information to predict the performance of iterative methods. For a normal matrix the initial guess and the eigenvalues of the coefficient matrix determine the rate of convergence of iterative solvers. We show that when the scatterer is a small sphere, the convergence rate for the nonnormal coefficient matrices can be estimated but this estimate is no longer valid for large spheres.
51 citations
TL;DR: In this paper, an efficient method is used to compute the eigenvalues required in a discrete-ordinates solution to a special class of radiative-transfer problems, which is relevant to a non-gray model with scattering that allows complete frequency redistribution.
Abstract: An efficient method is used to compute the eigenvalues required in a discrete-ordinates solution to a special class of radiative-transfer problems The basis for this computation is an algorithm for finding eigenvalues of a matrix that consists of the sum of a diagonal matrix and a rank-one matrix, a form that can arise in a discrete-ordinates solution of some basic transport problems To illustrate the efficiency of the approach, a radiative-transfer problem relevant to a non-gray model with scattering that allows complete frequency redistribution is discussed
30 citations
TL;DR: By means of computer checking, all simple graphs with at most two nonnegative eigenvalues are determined, and all minimalsimple graphs with exactly two (respectively, three) non negative eigen values are determined.
Abstract: In this paper, by means of computer checking, all simple graphs with at most two nonnegative eigenvalues, and all minimal simple graphs with exactly two (respectively, three) nonnegative eigenvalues are determined.
11 citations
TL;DR: In this paper, two generalizations of the Perron-Frobenius theory to the matrix pencil Ax = λBx are discussed, and their relationships are studied.
9 citations
01 Jan 1999
TL;DR: In this article, the authors proposed a matrix inverse problem for the determination of the nonhomogeneity of vibrating elastic membranes, which is a classical problem in Mathematical Physics and arises in a wide variety of physical applications.
Abstract: The vibrating elastic membrane is a classical problem in Mathematical Physics which arises in a wide variety of physical applications. Since the geometry of the membrane is usually well defined for a particular problem, determination of the nature of any nonhomogeneity is critical. The eigenvalues of particular membranes are often quite accessible experimentally and so a method for the determination of the nonhomogeneity based on the available eigenvalues is of practical importance. Projection of the boundary value problem and its coefficients onto appropriate vector spaces leads to a matrix inverse problem. Although the matrix inverse problem is of nonstandard form, it can be solved by a fix ed-point iterative method. Convergence of the method for a rectangular membrane is discussed and numerical evidence of the success of the method is presented.
8 citations
TL;DR: In this paper, the Wielandt inequality is used to obtain interesting inequalities about covariance matrix and various correlation coefficients including the canonical correlation, multiple and simple correlation, where the canonical covariance is a generalized inverse of the simple correlation.
Abstract: Suppose thatA is ann ×n positive definite Hemitain matrix. LetX andY ben ×p andn ×q matrices (p + q≤ n), such thatX* Y = 0. The following inequality is proved
$$X^* AY(Y^* AY)^ - Y^* AX \leqslant \left( {\frac{{\lambda _1 - \lambda _n }}{{\lambda _1 + \lambda _n }}} \right)^2 X^* AX,$$
where λ1, and λn, are respectively the largest and smallest eigenvalues ofA, andM- stands for a generalized inverse ofM. This inequality is an extension of the well-known Wielandt inequality in which bothX andY are vectors. The inequality is utilized to obtain some interesting inequalities about covariance matrix and various correlation coefficients including the canonical correlation, multiple and simple correlation.
7 citations
TL;DR: In this paper, a computer-assisted proof of eigenvalues below the essential spectrum of the Sturm-Liouville problem on the half-line is presented, using methods of functional analysis and interval analysis.
Abstract: This paper is concerned with a computer–assisted proof of eigenvalues below the essential spectrum of the Sturm–Liouville problem on the half–line. It uses methods of functional analysis and interval analysis to derive algorithms that may be used to prove the existence of such eigenvalues.
TL;DR: In this article, the spectral radius of a non-normal matrix A under linear perturbations is used to analyze the quality of the approximation of a projection method and to justify the presence of unexpected approximate eigenvalues.
Abstract: We are interested in some aspects of the perturbation effects in the spectrum of a real nonnormal matrix A under linear perturbations. We discuss some known results and we use them to justify some recent experimental observations. Moreover, we demonstrate that the qualitative behavior of the eigenvalues of A under linear perturbations may be predicted by inspecting the spectral radius of a related matrix. Then, we show how this information can be used to analyze the quality of the approximation of a projection method and to justify the presence of unexpected approximate eigenvalues.
Patent•
04 Jan 1999TL;DR: In this paper, a system relationship function is defined which describes the external response of a set of coupled linear components in an electrical circuit, and an error is indicated for the eigenvalues which is caused by ignoring those eigen values which have not been determined within the homogeneous differential equation system.
Abstract: A system relationship function is defined which describes the “external response” of a set of coupled linear components in an electrical circuit. A differential equation system is defined for the set of coupled linear components, and a predetermined number of eigenvalues are determined for the homogeneous differential equation system. An error is indicated for the eigenvalues which is caused by ignoring those eigenvalues which have not been determined within the homogeneous differential equation system. If the error is less than a given setpoint error, then the system relationship function is produced from the differential equations which are described by the determined eigenvalues. If the error is greater than the limit, then further eigenvalues are determined, for which an error is in turn determined. This method is continued iteratively until the error is less than the limit.
07 Dec 1999
TL;DR: In this article, the authors study operator Lyapunov equations in which the infinitesimal generator is not necessarily stable, but it satisfies a spectrum decomposition assumption and it has at most finitely many unstable eigenvalues.
Abstract: We study operator Lyapunov equations in which the infinitesimal generator is not necessarily stable, but it satisfies a spectrum decomposition assumption and it has at most finitely many unstable eigenvalues. Under mild conditions, these have unique self-adjoint solutions. We give conditions under which the number of negative eigenvalues of this solution equals the number of unstable eigenvalues of the generator. An application to the bounded real lemma is treated.
Posted Content•
TL;DR: In this article, the use of the eigenvalues of the Dirac operator on a curved spacetime as dynamical variables was discussed, with the main motivation coming from their invariance under the action of diffeomorphisms.
Abstract: We review some work done with C. Rovelli on the use of the eigenvalues of the Dirac operator on a curved spacetime as dynamical variables, the main motivation coming from their invariance under the action of diffeomorphisms. The eigenvalues constitute an infinite set of ``observables'' for general relativity and can be taken as variables for an invariant description of the gravitational field dynamics.
01 Jan 1999
TL;DR: This chapter discusses the development of the new learning rules for eigenpairs, and their applications in parallel-pipeline computing and eigenvectors.
Abstract: 111 ACKNOWLEDGEMENTS . IV LIST OF TABLES ■ Vll LIST OF FIGURES Vlll LIST OF GRAPHS . . . . . . . . . . . A ,, . . ix CHAPTER ONE Introduction • L . . . . . .. . • •1 Computing,eigenpairs: background . .^ . . . ■ 2, Using neural networks to compute eigenpairs : 7 Review of previous work . . . . . . . . . . . .11 Thesis preview . . . . . .. . . . . . . . . . 16 CHAPTER TWO The new learning rules, and, algorithms 18 The modified learning rule . . . . . . . . . 19 Derivation . . . . . . . • • . . . . . . 20 Finding all eigenpairs . . . . . : . . . . ,. 22 , 23. Serial deflation . . . . . . . . . . . . . Serial-pipelined'deflation. .. . . . . . . . • 24 Parallel-pipeline rule . . . . . . . . . ... . 26 Derivation of parallel-pipeline rule 27 Relating parallel-pipeline and Danger's; 30 rules . . . . . . . . . .. . . . .. . . . CHAPTER THREE Implementation . . . . . . . . . . . . . . 31 Finding extreme eigenvalues and eigenvectors .. . . 32
01 Jan 1999
TL;DR: In this article, the authors apply eigenvalue techniques for cut evaluation to produce relations between the weight and order of induced subgraphs, and apply these results to bound the stability number.
Abstract: We apply eigenvalue techniques for cut evaluation to produce relations between the weight and order of induced subgraphs, and apply these results to bound the stability number.
01 Jan 1999
TL;DR: In this article, it was shown that all eigenvalues of A are contained in the boundary of the boundary in the sense that all the eigenvectors of A can be found in Ω =属于≠a，i≠j{z∈C: |z-aij| |x-haij| ≤rirj}.
Abstract: Let A= (aij)∈C^n×n and r,. =∑j≠i |aij|. Suppose that for each row of A there is at least one nonzero off-diagonal entry. It is proved that all eigenvalues of A are contained in Ω=属于≠a，i≠j{z∈C: |z-aij| |x-aij| ≤rirj}. The result reduces the number of ovals in original Brauer's theorem in many cases. Eigenvalues (and associated eigenvectors) that locate in the boundary of Ω are discussed.
TL;DR: In this article, general formulae for the lower bound of the first two Dirichlet eigenvalues on a regular domain were presented, and the positivity of the top spectrum and the gap between these two Diricheleteigenvalues were studied.
Abstract: This note presents general formulae for the lower bound of
the rst two Dirichlet eigenvalues on a regular domain As applications,
the positivity of top spectrum and the gap between these two Dirichlet
eigenvalues are studied
TL;DR: In this article, a non-local sensitivity analysis of generalized eigensystem (eigenvectors and eigen values) of an n × n matrix pencil A - λB with pairwise distinct eigenvalues is made.
TL;DR: For specific choice of parameters the spectrum of the discrete quantum pendulum-integral contains the eigenvalues of a finite matrix which depends analytically on the flux as discussed by the authors, which may be obtained by the algebraic Bethe ansatz.
Abstract: For specific choice of parameters the spectrum of the discrete quantum pendulum-integral contains the eigenvalues of a finite matrix which depends analytically on the flux. Under natural continuity assumptions these eigenvalues include the spectral values which may be obtained by the algebraic Bethe ansatz.
TL;DR: In this article, the authors studied the behavior of simple well-separated complex eigenvalues, which are simple in theory, but in practice become defective using finite precision, and provided a sharp estimate of the distance of T n to the nearest defective matrix, when n increases.