Showing papers on "Spectrum of a matrix published in 2005"

Singular values and eigenvalues of tensors: a variational approach

Abstract: We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are particularly useful in generalizing certain areas where the spectral theory of matrices has traditionally played an important role. For illustration, we will discuss a multilinear generalization of the Perron-Frobenius theorem

Eigenvalues, singular values, and Littlewood-Richardson coefficients

Abstract: We characterize the relationship between the singular values of a Hermitian (resp., real symmetric, complex symmetric) matrix and the singular values of its off-diagonal block. We also characterize the eigenvalues of a Hermitian (or real symmetric) matrix C = A + B in terms of the combined list of eigenvalues of A and B . The answers are given by Horn-type linear inequalities. The proofs depend on a new inequality among Littlewood-Richardson coefficients.

Remarks on the spectrum of the Neumann problem with magnetic field in the half-space

Abstract: We consider a Schrodinger operator with a constant magnetic field in a one-half three-dimensional space, with Neumann-type boundary conditions. It is known from the works by Lu–Pan and Helffer–Morame that the lower bound of its spectrum is less than b, the intensity of the magnetic field, provided that the magnetic field is not normal to the boundary. We prove that the spectrum under b is a finite set of eigenvalues (each of infinite multiplicity). In the case when the angle between the magnetic field and the boundary is small, we give a sharp asymptotic expansion of the number of these eigenvalues.

Matrix Models as Non-Local Hidden Variables Theories

Abstract: It is shown that the matrix models which give non-perturbative definitions of string and M theory may be interpreted as non-local hidden variables theories in which the quantum observables are the eigenvalues of the matrices while their entries are the non-local hidden variables. This is shown by studying the bosonic matrix model at finite temperature, with T taken to scale as 1/N, with N the rank of the matrices. For large N the eigenvalues of the matrices undergo Brownian motion due to the interaction of the diagonal elements with the off diagonal elements, giving rise to a diffusion constant that remains finite as N goes to infinity. The resulting probability density and current for the eigenvalues are then found to evolve in agreement with the Schroedinger equation, to leading order in 1/N, with hbar proportional to the thermal diffusion constant for the eigenvalues. The quantum uctuations and uncertainties in the eigenvalues are then consequences of ordinary statistical uctuations in the values of the off-diagonal matrix elements. Furthermore, this formulation of the quantum theory is background independent, as the definition of the thermal ensemble makes no use of a particular classical solution. The derivation relies on Nelson's stochastic formulation of quantum theory, which is expressed in terms of a variational principle.

On graphs whose Laplacian matrices have distinct integer eigenvalues

Abstract: In this paper, we investigate graphs for which the corresponding Laplacian matrix has distinct integer eigenvalues. We define the set Si,n to be the set of all integers from 0 to n, excluding i. If there exists a graph whose Laplacian matrix has this set as its eigenvalues, we say that this set is Laplacian realizable. We investigate the sets Si,n that are Laplacian realizable, and the structures of the graphs whose Laplacian matrix has such a set as its eigenvalues. We characterize those i < n such that Si,n is Laplacian realizable, and show that for certain values of i, the set Si,n is realized by a unique graph. Finally, we conjecture that Sn,n is not Laplacian realizable for n ≥ 2 and show that the conjecture holds for certain values of n. © 2005 Wiley Periodicals, Inc. J Graph Theory

On the eigenvalues of a "dumb-bell with a thin handle"

Abstract: We consider the Neumann boundary-value problem of finding the small-parameter asymptotics of the eigenvalues and eigenfunctions for the Laplace operator in a singularly perturbed domain consisting of two bounded domains joined by a thin "handle". The small parameter is the diameter of the cross-section of the handle. We show that as the small parameter tends to zero these eigenvalues converge either to the eigenvalues corresponding to the domains joined or to the eigenvalues of the Dirichlet problem for the Sturm-Liouville operator on the segment to which the thin handle contracts. The main results of this paper are the complete power small-parameter asymptotics of the eigenvalues and the corresponding eigenfunctions and explicit formulae for the first terms of the asymptotics. We consider critical cases generated by the choice of the place where the thin "handle" is joined to the domains, as well as by the multiplicity of the eigenvalues corresponding to the domains joined.

Alternative method of calculating the eigenvalues of the transfer matrix of the τ2 model for N = 2

Abstract: It has been shown that the τ2 (Baxter-Bazhanov-Stroganov) model for N = 2 with arbitrary parameters is a particular case of the generalized Ising model. The model satisfies the free-fermion condition, which enables one to solve it by the method of the auxiliary Grassmann field. Explicit expressions have been derived for the partition function on a finite-size lattice and eigenvalues of the transfer matrix. In this approach, in contrast to the functional relation method, there is no problem with the multiplicities of the eigenvalues of transfer matrix.

Exclusion and Inclusion Intervals for the Real Eigenvalues of Positive Matrices

Abstract: Given a real matrix, we analyze an open interval, called a row exclusion interval, such that the real eigenvalues do not belong to it. We characterize when the row exclusion interval is nonempty. In addition to the exclusion interval, inclusion intervals for the real eigenvalues, alternative to those provided by the Gerschgorin disks, are also considered for matrices whose off-diagonal entries present a restricted dispersion. The results are applied to obtain a sharp upper bound for the real eigenvalues different from 1 of a positive stochastic matrix and a sufficient condition for the stability of a negative matrix, among other applications.

Neutron transport with anisotropic scattering: theory and applications

Abstract: This thesis is a blend of neutron transport theory and numerical analysis. We start with the study of the problem of the Mika/Case eigenexpansion used in the solution process of the homogeneous one-speed Boltzmann neutron transport equation with anisotropic scattering for plane symmetry. The anisotropic scattering is expressed as a finite Legendre series in which the coefficients are the ``scattering coefficients'. This eigenexpansion consists of a discrete spectrum of eigenvalues with its corresponding eigenfunctions and the continuous spectrum [-1,+1] with its corresponding eigendistributions. In the general case where the anisotropic scattering can be of any (finite) order, multiple discrete eigenvalues exist and these have to be located to have the complete spectrum. We have devised a stable and robust method that locates all these discrete eigenvalues. The method is a two-step process: first the number of discrete eigenvalues is calculated and this is followed by the calculation of the discrete eigenvalues themselves, now being able to count them down and make sure none are forgotten. During our numerical experiments, we came across what we called near-singular eigenvalues: discrete eigenvalues that are located extremely close to the continuum and hence lead to near-singular behaviour in the eigenfunction. Our solution method has been adapted and allows for the automatic detection of such a near-singular eigenvalue. For the elements of the continuous spectrum [-1,+1], there is no non-zero function satisfying the associated eigenequation but there is a non-zero distribution that does satisfy it. It is not feasible to compute a distribution as such but one can evaluate integrals in which this distribution appears. The continuum part of the eigenexpansion can hence only be characterised by its (angular) moments. Accurate and fast numerical quadrature is needed to evaluate these integrals. Several quadrature methods have been evaluated on a representative test function. The eigenexpansion was proved to be orthogonal and complete and hence can be used to represent the infinite medium Green's function. The latter is the building block of the Boundary Sources Method, an integral solution method for the neutron transport equation. Using angular and angular/spatial moments of the Green's function, it is possible to solve with high accuracy slab problems. We have written a one-dimensional slab code implementing this Boundary Sources Method allowing for media with arbitrary order anisotropic scattering. Our results are very good and the code can be considered as a benchmark code for others. As a final application, we have used our code to study the discrete spectrum of a well-known scattering kernel in radiative transfer, the Henyey-Greenstein kernel. This kernel has one free parameter which is used to fit the kernel to experimental data. Since the kernel is a continuous function, a finite Legendre approximation needs to be adopted. Depending on the free parameter, the approximation order and the number of secondaries per collision, the number of discrete eigenvalues ranges from two to thirty and even more. Bounds for the minimum approximation order are derived for different requirements on the approximation: non-negativity, an absolute and relative error tolerance.

On a matrix with integer eigenvalues and its relation to conditional Poisson sampling

Abstract: A special non-symmetric N ×N matrix with eigenvalues 0, 1, 2, . . . , N − 1 is presented. The matrix appears in sampling theory. Its right eigenvectors, if properly normalized, give the inclusion probabilities of the Conditional Poisson design (for all different fixed sample sizes). The explicit expressions for the right eigenvectors become complicated for N large. Nevertheless, the left eigenvectors have a simple analytic form. An inversion of the left eigenvector matrix produces the right eigenvectors − the inclusion probabilities. Finally, a more general matrix with similar properties is defined and expressions for its left and right eigenvectors are derived.

Threshold properties of matrix-valued Schrödinger operators

Abstract: We present some results on the perturbation of eigenvalues embedded at a threshold for a two-channel Hamiltonian with three-dimensional Schrodinger operators as entries and with a small off-diagonal perturbation. In particular, we show how the threshold eigenvalue gives rise to discrete eigenvalues below the threshold and, moreover, we establish a criterion on existence of half-bound states associated with embedded pseudo eigenvalues.

Density profiles of small Dirac operator eigenvalues for two color QCD at nonzero chemical potential compared to matrix models

Abstract: We investigate the eigenvalue spectrum of the stagerred Dirac matrix in two color QCD at finite chemical potential. The profiles of complex eigenvalues close to the origin are compared to a complex generalization of the chiral Gaussian Symplectic Ensemble, confirming its predictions for weak and strong non-Hermiticity. They differ from the QCD symmetry class with three colors by a level repulsion from both the real and imaginary axis.

Calculation of energy eigenvalues for two-dimensional anharmonic oscillators

Abstract: We combine a variational method with a normal ordering technique to express the Hamiltonian of two-dimensional anharmonic oscillators, in terms of Wick's ordered-product of the functions of the generalized creation and annihilation operators. An optimal orthonormal basis is constructed from the tensor product of two optimal generalized number state bases, and it is used to set up the Hamiltonian matrix in a simple manner. The eigenvalues of the Hamiltonian matrix are obtained by standard methods.

States with v1=lambda, v2=-lambda and reciprocal equations in the six-vertex model

Abstract: The eigenvalues of the transfer matrix in a six-vertex model (with periodic boundary conditions) can be written in terms of n constants v1,,vn, the zeros of the function Q(v) A peculiar class of eigenvalues are those in which two of the constants v1, v2 are equal to lambda, -lambda, with Delta=-cosh lambda and Delta related to the Boltzmann weights of the six-vertex model by the usual combination Delta=(a^2+b^2-c^2)/2 a b The eigenvectors associated to these eigenvalues are Bethe states (although they seem not) We count the number of such states (eigenvectors) for n=2,3,4,5 when N, the columns in a row of a square lattice, is arbitrary The number obtained is independent of the value of Delta, but depends on N We give the explicit expression of the eigenvalues in terms of a,b,c (when possible) or in terms of the roots of a certain reciprocal polynomial, being very simple to reproduce numerically these special eigenvalues for arbitrary N in the blocks n considered For real a,b,c such eigenvalues are real

Strong and weak coupling of eigenvalues of complex matrices

Abstract: The paper presents a general theory of coupling of eigenvalues of complex matrices of arbitrary dimension smoothly depending on real parameters. The cases of weak and strong coupling are distinguished and their geometric interpretation in two and three-dimensional spaces is given. General asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios. Two numerical exam- ples from crystal optics illustrate effectiveness and accuracy of the presented theory. I. INTRODUCTION Behavior of eigenvalues of matrices dependent on param- eters is a problem of general interest having many important applications in natural and engineering sciences. In modern physics, e.g. quantum mechanics, crystal optics, physical chemistry, acoustics and mechanics, multiple eigenvalues in matrix spectra associated with specific effects attract great interest of researchers since the papers (1), (2). In recent papers, see e.g. (3)-(6), two important cases are distinguished: the diabolic points (DPs) and the exceptional points (EPs). From mathematical point of view DP is a point where the eigenvalues coalesce, while corresponding eigenvectors remain different; and EP is a point where both eigenvalues and eigenvectors merge forming a Jordan block. Both the DP and EP cases are interesting in applications and were observed in experiments (6), (7). In this paper we present a general theory of coupling of eigenvalues of complex matrices of arbitrary dimen- sion smoothly depending on multiple real parameters. Two essential cases of weak and strong coupling based on a Jordan form of the system matrix are distinguished. These two cases correspond to diabolic and exceptional points, respectively. We derive general formulae describing coupling and decoupling of eigenvalues, crossing and avoided crossing of eigenvalue surfaces. It is emphasized that the presented theory of coupling of eigenvalues of complex matrices gives not only qualitative, but also quantitative results on behavior of eigenvalues based only on the information taken at the singular points. The paper is based on the author's previous research on interaction of eigenvalues for matrices and differential operators depending on multiple parameters (8)- (11); for more references see the recent book (12).

Cutoff-effects in the spectrum of dynamical Wilson fermions

Abstract: We investigate the low–lying eigenvalues of the improved Wilson–Dirac operator in the Schrodinger functional with two dynamical quark flavors. At a lattice spacing of approximately 0.1 fm we find more very small eigenvalues than in the quenched case. These cause problems with HMC–type algorithms and in the evaluation of fermionic correlation functions. Through a simulation at a finer lattice spacing we are able to establish their nature as cutoff–effects.

Some results on graph spectra, with applications to geographic spatial modelling

Abstract: Exact Gaussian maximum likelihood estimation for a spatial process requires evaluation of the determinant and inverse of the covariance matrix. In geographic modelling, it is common to specify the inverse matrix in terms of I - bW, for some known matrix W, but the evaluation of the determinant can still be slow, even when expressed as a function of the eigenvalues of W. This paper considers the eigenvalues of two versions of W related to some easily specified graphs, and some approximations to the determinant of I - bW.

Tracing of critical eigenvalues for power system analysis via continuation of invariant subspaces and projected Arnoldi method

Abstract: The continuation of invariant subspaces algorithm combined with projected Arnoldi method is presented to trace the critical (rightmost or least damping ratio) eigenvalues for the power system stability analysis. A predictor-corrector method is applied to calculate the critical eigenvalues trajectories as power system parameters change. The method can handle eigenvalues with any multiplicity during the trace processes. The subspace dimension inflation and deflation is applied to deal with eigenvalue overlap. And the subspace update is proposed by an efficient projected Arnoldi method which can utilize known information from the traced eigenvalues and subspaces.

A proof of Horn's result on a complex matrix with prescribed singular values and eigenvalues

Abstract: We give another proof of a result of Horn on the existence of a matrix with prescribed singular values and eigenvalues.

On higher order eigenvalues of the spherical laplacian operator

Abstract: In this paper, we use the boundary measurements of normalized eigenfunctions to estimate the variation of the corresponding eigenvalues. With this form, we can show that some eigenvalues, as functions of domain, possess monotonicity as the domain varies according some constraints.

Stochastic eigenvalues in system with matrix functions

Abstract: In this paper, a numerical method for the structural stochastic eigenvalue problem using 2D matrix functions (vector-valued and matrix-valued functions) is presented. With the Kronecker algebra and matrix calculus, expressions for the eigenvalue and eigenvector sensitivities are obtained. By using these two sensitivities together with the perturbation technique, the stochastic moments of the eigenvalues in structural vibrations are obtained. When the mean values, variances and covariances of structural random parameters are known, it is easy to get the first and second order moments of the eigenvalues. The results derived are easily amenable to computational procedures.

A fully parallel method for the singular eigenvalue problem

Abstract: In this paper, a fully parallel method for finding some or all finite eigenvalues of a real symmetric matrix pencil (A, B) is presented, where A is a symmetric tridiagonal matrix and B is a diagonal matrix with b"1 > 0 and b"i >= 0, i = 2,3,...,n. The method is based on the homotopy continuation with rank 2 perturbation. It is shown that there are exactly m disjoint, smooth homotopy paths connecting the trivial eigenvalues to the desired eigenvalues, where m is the number of finite eigenvalues of (A, B). It is also shown that the homotopy curves are monotonic and easy to follow.