Showing papers on "Spectrum of a matrix published in 1983"
TL;DR: In this paper, the stability of Parisi's solution for the long-range spin-glass was studied, and it was shown that it is marginally stable in terms of the number of zero modes.
Abstract: We study, near ${T}_{c}$, the stability of Parisi's solution for the long-range spin-glass. In addition to the discrete, "longitudinal" spectrum found by Thouless, de Almeida, and Kosterlitz, we find "transverse" bands depending on one or two continuous parameters, and a host of zero modes occupying most of the parameter space. All eigenvalues are non-negative, proving that Parisi's solution is marginally stable.
107 citations
TL;DR: In this paper, the sign matrices uniquely associated with the matrices (M − ζ j I ) 2, where the corners of a rectangle oriented at π /4 to the axes of a Cartesian coordinate system, were used to compute the number of eigenvalues of the arbitrarily chosen matrix M which lie within the rectangle, and to determine the left and right invariant subspaces of M associated with these eigen values.
44 citations
TL;DR: In this article, a simple explicit convergence criterion is given, as well as the algorithm and two numerical examples for those eigenvalues of a λ-matrix whose elements are functions of a parameter λ.
Abstract: The matrix N(λ) whose elements are functions of a parameter λ is called the λ-matrix. Those values of λ that make the matrix singular are of great interest in many applied fields. An efficient method for those eigenvalues of a λ-matrix is presented. A simple explicit convergence criterion is given, as well as the algorithm and two numerical examples.
41 citations
TL;DR: A robust Lanczos algorithm is presented which is fast, easy to understand, uses about 30 words of extra storage, and has a fairly short program.
41 citations
TL;DR: The eigenvalues of a non-singular conservative ergodic transformation of a separable measure space form a Borel subgroup of the circle of measure zero.
Abstract: The eigenvalues of a non-singular conservative ergodic transformation of a separable measure space form a Borel subgroup of the circle of measure zero. We show that this is the only metric restriction on their size. However, the larger the eigenvalue group of the transformation, the “less recurrent” it is.
16 citations
TL;DR: In this article, it was shown that complex time eigenvalues do actually exist for the one-speed, isotropic scattering neutron transport equation for a homogeneous sphere with vacuum boundary conditions.
Abstract: The time eigenvalue spectrum of the one-speed, isotropic scattering neutron transport equation has been studied for a homogeneous sphere with vacuum boundary conditions. There is a close relationship between the time eigenvalue problem and the criticality problem of the time independent equation for the same model. It is shown that this relation holds even when the time eigenvalues are complex. Using Carlvik's method to solve the criticality problem, it is shown that complex time eigenvalues do actually exist for this model problem. Thus the real eigenvalues found by van Norton(1) do not form the complete spectrum.
8 citations
TL;DR: In this article, the N-points Friedrichs (N>1) model is considered and it is shown that the non-existence of point eigenvalues embedded in the continuous spectrum is incorrect.
Abstract: Considers the N-points Friedrichs (N>1) model (1948) and shows, by means of a counterexample, that a theorem of Marchand (1967) about the non-existence of point eigenvalues embedded in the continuous spectrum is incorrect.
4 citations
TL;DR: In this article, the authors define disc regions centred on the eigenvalues of the diagonal blocks of a general complex matrix, which contain the corresponding eigen values of the matrix matrix.
Abstract: This note presents a new result which defines disc regions, centred on the eigenvalues of the diagonal blocks of a general complex matrix, which contain the eigenvalues of the matrix. A result of this type is useful in the application of a generalized Nyquist type stability criterion to interconnected systems
4 citations
01 Dec 1983
TL;DR: In this article, the authors present a method based on an earlier algorithm (constructed by A. Manitius, G. Payre and R. Roy) which solves directly the characteristic equation of the closed loop system, and compare it with a direct computation of eigenvalues of a symplectic hamiltonian matrix arising from a finite dimensional approximation of a functional differential equation.
Abstract: A solution of the linear quadratic control problem involving functional differential equations gives a linear feedback control law which modifies the original system dynamics. Under certain assumptions, the eigenvalues of the modified linear system constitute a stable part of a spectrum of a hamiltonian operator associated with the optimization problem. These eigenvalues can be computed without solving the infinite dimensional Riccati equation. In this paper we present a method based on an earlier algorithm (constructed by A. Manitius, G. Payre and R. Roy) which solves directly the characteristic equation of the closed loop system, and compare it with a direct computation of eigenvalues of a symplectic hamiltonian matrix arising from a finite dimensional approximation of a functional differential equation.
2 citations
2 citations
01 Jan 1983
TL;DR: In this article, the sensitivity of the eigenvalues of a defective matrix under small perturbations was studied and a generalization of the special results of Wilkinson, Stewart,Bauer and Fike was presented.
Abstract: This paper is concerned with the sensitivity of the eigenvalues of a defective matrix under small perturbations.The given estimate generalizes all special results of Wilkinson, Stewart,Bauer and Fike,when the eigenvalue is simple and when the matrix is nondefective, and interpretes the phenomenon indicated by Golub and Wilkinson for Multiple eigenvalues
TL;DR: In this paper, the effect of a compact linear feedback control on the eigenvalues of a Hilbert space oscillator was considered and conditions were derived under which a sequence of complex numbers can be obtained as eigen values using such a feedback control.
Abstract: Consider the effect of a compact linear feedback control on the eigenvalues of a Hilbert space oscillator. It is shown that the kth eigenvalue can be perturbed a distance of only if a sequence is summable. Conditions are also derived under which a sequence of complex numbers can be obtained as eigenvalues using such a feedback control. The analysis gives an explicit form for the control in terms of the desired eigenvalues. A simple application to the stabilization problem for water waves in a finite tank is given.
TL;DR: The method is the counterpart of the Rayleigh-Ritz method in the sense that the results obtained from both methods will improve, i.e. the eigenvalues can be bracketed into a small region, and the lower bounds to all eigen values can be obtained from the solution of one transcendental equation.
TL;DR: In this paper, a Koehler-type method to obtain lower bounds for the eigenvalues of a certain class of operators is presented, and general properties required of a problem for the technique to work are discussed and the connection with other classical methods is analyzed.
Abstract: A Koehler-type method to obtain lower bounds for the eigenvalues of a certain class of operators is presented. The general properties required of a problem for the technique to work are discussed and the connection with other classical methods is analyzed.
TL;DR: One presents the ALGOL procedures which implement the algorithm for the determination of the group of smallest (greatest) eigenvalues and their corresponding eigenvectors for a matrix pencil where A and B are real square matrices of simple structure.
Abstract: One presents the ALGOL procedures which implement the algorithm for the determination of the group of smallest (greatest) eigenvalues and their corresponding eigenvectors for a matrix pencil
where A and B are real square matrices of simple structure From the initial pencil one constructs a matrix C, whose eigenvalues are taken as the initial approximations to the eigenvalues from the group of the smallest (greatest) eigenvalues of the pencil The refinement of the eigen-values is performed on the basis of the theory of perturbations Then one determines the eigen-vectors and one computes the infinite norm of the residual One gives ALGOL programs and test examples
22 Jun 1983
TL;DR: In this paper, a method for computing the control gain matrix of a controllable multi-input, linear a system, where the poles (eigenvalues) are placed at arbitrary locations, is presented.
Abstract: A method is developed that computes the control gain matrix, u=-C x, of a controllable multi-input, linear aystem, x = A x + B u, required to place the poles (eigenvalues) at arbitrary locations. The derivation of the method is based upon defining a quadratic performance index that is identically zero when the closed loop system has the desired eigenvalues. The numerical computations involve solving a set of n(n-m) linear algebraic equations where n is the number of states and m is the number of controls. For m ≫ 1, the C matrix is not unique and alternative gain matrices yaielding the desired n eigenvalues can be computed. The method is illustrated by both simDle analyt4cal and higher order numerical eramples. A robust computational program called POLESYS is described and applied to several examples.
TL;DR: In this article, a Hill's matrix Lϱp of odd order corresponding to the theory of Hill's equations is considered and necessary and sufficient conditions on the coefficients of such a matrix are established in order that the matrices L1 and L−1 have all double eigenvalues except one.