Showing papers on "Spectrum of a matrix published in 2012"

Transmission Eigenvalues in Inverse Scattering Theory

Abstract: This survey aims to present the state of the art of research on the transmission eigenvalue problem focussing on three main topics, namely the discreteness of transmission eigenvalues, the existence of trans- mission eigenvalues and estimates on transmission eigenvalues, in particular, Faber-Krahn type inequalities.

Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators

Abstract: Transmission eigenvalues are points in the spectrum of the interior transmission operator, a coupled $2 \times 2$ system of elliptic partial differential equations, where one unknown function must satisfy two boundary conditions and the other must satisfy none. We show that the interior transmission eigenvalues are discrete and depend continuously on the contrast by proving that the interior transmission operator has upper triangular compact resolvent, and that the spectrum of these operators share many of the properties of operators with compact resolvent. In particular, the spectrum is discrete and the generalized eigenspaces are finite-dimensional. Our main hypothesis is a coercivity condition on the contrast that must hold only in a neighborhood of the boundary.

Finite Element Methods for Maxwell's Transmission Eigenvalues

Abstract: The transmission eigenvalue problem plays a critical role in the theory of qualitative methods for inhomogeneous media in inverse scattering theory. Efficient computational tools for transmission eigenvalues are needed to motivate improvements to theory, and, more importantly, are parts of inverse algorithms for estimating material properties. In this paper, we propose two finite element methods to compute a few lowest Maxwell's transmission eigenvalues which are of interest in applications. Since the discrete matrix eigenvalue problem is large, sparse, and, in particular, non-Hermitian due to the fact that the problem is neither elliptic nor self-adjoint, we devise an adaptive method which combines the Arnoldi iteration and estimation of transmission eigenvalues. Exact transmission eigenvalues for balls are derived and used as a benchmark. Numerical examples are provided to show the viability of the proposed methods and to test the accuracy of recently derived inequalities for transmission eigenvalues.

Antiperiodic dynamical 6-vertex model I: Complete spectrum by SOV, matrix elements of the identity on separate states and connections to the periodic 8-vertex model

Abstract: The spin-1/2 highest weight representations of the dynamical 6-vertex and the standard 8-vertex Yang-Baxter algebra on a finite chain are considered in this paper. For the antiperiodic dynamical 6-vertex transfer matrix defined on chains with an odd number of sites, we adapt the Sklyanin's quantum separation of variable (SOV) method and explicitly construct SOV representations from the original space of representations. We provide the complete characterization of eigenvalues and eigenstates proving also the simplicity of its spectrum. Moreover, we characterize the matrix elements of the identity on separated states by determinant formulae. The matrices entering in these determinants have elements given by sums over the SOV spectrum of the product of the coefficients of separate states. This SOV analysis is not reduced to the case of the elliptic roots of unit and the results here derived define the required setup to extend to the dynamical 6-vertex model the approach recently developed in [1]-[5] to compute the form factors of the local operators in the SOV framework, these results will be presented in a future publication. For the periodic 8-vertex transfer matrix, we prove that its eigenvalues have to satisfy a fixed system of equations. In the case of a chain with an odd number of sites, this system of equations is the same entering in the SOV characterization of the antiperiodic dynamical 6-vertex transfer matrix spectrum. This implies that the set of the periodic 8-vertex eigenvalues is contained in the set of the antiperiodic dynamical 6-vertex eigenvalues. A criterion is introduced to find simultaneous eigenvalues of these two transfer matrices and associate to any of such eigenvalues one nonzero eigenstate of the periodic 8-vertex transfer matrix by using the SOV results. Moreover, a preliminary discussion on the degeneracy of the periodic 8-vertex spectrum is also presented.

An invariant subspace‐based approach to the random eigenvalue problem of systems with clustered spectrum

Abstract: The repeated or closely spaced eigenvalues and corresponding eigenvectors of a matrix are usually very sensitive to a perturbation of the matrix, which makes capturing the behavior of these eigenpairs very difficult. Similar difficulty is encountered in solving the random eigenvalue problem when a matrix with random elements has a set of clustered eigenvalues in its mean. In addition, the methods to solve the random eigenvalue problem often differ in characterizing the problem, which leads to different interpretations of the solution. Thus, the solutions obtained from different methods become mathematically incomparable. These two issues, the difficulty of solving and the non-unique characterization, are addressed here. A different approach is used where instead of tracking a few individual eigenpairs, the corresponding invariant subspace is tracked. The spectral stochastic finite element method is used for analysis, where the polynomial chaos expansion is used to represent the random eigenvalues and eigenvectors. However, the main concept of tracking the invariant subspace remains mostly independent of any such representation. The approach is successfully implemented in response prediction of a system with repeated natural frequencies. It is found that tracking only an invariant subspace could be sufficient to build a modal-based reduced-order model of the system. Copyright (C) 2012 John Wiley & Sons, Ltd.

Eigenvalue spectrum of transition matrix of dual Sierpinski gaskets and its applications

Abstract: The eigenvalue spectrum of the transition matrix of a network contains much information about its structural properties and is related to the behavior of various dynamical processes performed on it. In this paper, we study the eigenvalues of the transition matrix of the dual Sierpinski gaskets embedded in d-dimensional Euclidean spaces. We obtain all the eigenvalues, as well as their corresponding degeneracies, by making use of the spectral decimation technique. We then apply the obtained eigenvalues to determine the exact number of spanning trees in the studied fractals and derive an explicit formula of the eigentime identity for random walks taking place on them.

Exact eigenvalue spectrum of a class of fractal scale-free networks

Abstract: The eigenvalue spectrum of the transition matrix of a network encodes important information about its structural and dynamical properties. We study the transition matrix of a family of fractal scale-free networks and analytically determine all the eigenvalues and their degeneracies. We then use these eigenvalues to evaluate the closed-form solution to the eigentime for random walks on the networks under consideration. Through the connection between the spectrum of transition matrix and the number of spanning trees, we corroborate the obtained eigenvalues and their multiplicities.

Eigenvalues of the laplacian on riemannian manifolds

Abstract: For a bounded domain Ω with a piecewise smooth boundary in a complete Riemannian manifold M, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L2(Ω) in place of the Rayleigh–Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, for lower order eigenvalues, our results extend the results of Chen and Cheng [D. Chen and Q.-M. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan 60 (2008) 325–339].

On Eigenvalues of the Sum of Two Random Projections

Abstract: We study the behavior of eigenvalues of matrix P N +Q N where P N and Q N are two N-by-N random orthogonal projections. We relate the joint eigenvalue distribution of this matrix to the Jacobi matrix ensemble and establish the universal behavior of eigenvalues for large N. The limiting local behavior of eigenvalues is governed by the sine kernel in the bulk and by either the Bessel or the Airy kernel at the edge depending on parameters. We also study an exceptional case when the local behavior of eigenvalues of P N +Q N is not universal in the usual sense.

Spectral properties of Sturm-Liouville operators with discontinuities at finite points

Abstract: In this paper, we investigate a class of Sturm-Liouville operators with eigenparameter-dependent boundary conditions and transmission conditions at finite interior points. By modifying the inner product in a suitable Krein space K associated with the problem, we generate a new self-adjoint operator A such that the eigenvalues of such a problem coincide with those of A. We construct its fundamental solutions, get the asymptotic formulae for its eigenvalues and fundamental solutions, discuss some properties of its spectrum, and obtain its Green function and the resolvent operator. Three important conclusions can be drawn: (1) the new operator A is self-adjoint in the Krein space K; (2) if , and ρ j > 0, j = 1,2, then, the eigenvalues of the problem (Equations 1 to 5) are analytically simple; (3) the residual spectrum of the operator A is empty, i.e., σ r (A) = ∅.

Structure of trajectories of complex-matrix eigenvalues in the Hermitian-non-Hermitian transition.

Abstract: The statistical properties of trajectories of eigenvalues of Gaussian complex matrices whose Hermitian condition is progressively broken are investigated. It is shown how the ordering on the real axis of the real eigenvalues is reflected in the structure of the trajectories and also in the final distribution of the eigenvalues in the complex plane.

An envelope for the spectrum of a matrix

Abstract: We introduce and study an envelope-type region ɛ(A) in the complex plane that contains the eigenvalues of a given n×n complex matrix A. ɛ(A) is the intersection of an infinite number of regions defined by cubic curves. The notion and method of construction of ɛ(A) extend the notion of the numerical range of A, F(A), which is known to be an intersection of an infinite number of half-planes; as a consequence, ɛ(A) is contained in F(A) and represents an improvement in localizing the spectrum of A.

A Matrix Method for Determining Eigenvalues and Stability of Singular Neutral Delay-Differential Systems

Abstract: The eigenvalues and the stability of a singular neutral differential system with single delay are considered. Firstly, by applying the matrix pencil and the linear operator methods, new algebraic criteria for the imaginary axis eigenvalue are derived. Second, practical checkable criteria for the asymptotic stability are introduced.

Source Enumeration in Large Arrays Based on Moments of Eigenvalues in Sample Starved Conditions

Abstract: This paper presents a scheme to enumerate the incident waves impinging on a high dimensional uniform linear array using relatively few samples. The approach is based on Minimum Description Length (MDL) criteria and statistical properties of eigenvalues of the Sample Covariance Matrix (SCM). We assume that several models, with each model representing a certain number of sources, will compete and MDL criterion will select the best model with the minimum model complexity and maximum model decision. Statistics of noise eigenvalue of SCM can be approximated by the distributional properties of the eigenvalues given by Marcenko-Pastur distribution in the signal-free SCM. In this paper we use random matrix theory to determine the statistical properties of the moments of noise eigenvalues of SCM to separate noise and signal eigenvalues. Numerical simulations are used to demonstrate the performance of proposed estimator compared with some other enumerators in sample starved regime.

Eigenvalues of a statistical mechanics matrix

Abstract: We consider the problem of finding the spectrum of an n × n matrix which arises in the study of a certain model of long-range interactions in a one-dimensional statistical mechanics system. Our analysis exhibits a curious resemblance of the suitably normalized distribution of eigenvalues to the Marcenko–Pastur law in the limit n → ∞.

Method of solving the generalized problem on eigenvalues of multidimensional circuits, i

Abstract: This work is devoted to the development of a new method for solving the complete generalized problem of large-scale electrical circuit eigenvalues. We prove that the determinant of the conductance node matrix of an arbitrary k-loop LC-circuit is the Weinstein function for the loop impendance matrix of this circuit, and vice versa. A recurrent method of imposing constraints on the basic problem was defined, which permitted us to separate roots of characteristic polynomials in the recurrent process. Also a condition for conservativity of multiple eigenvalues was defined. A solution algorithm for the problem of defining a full range of eigenvalues of an oscillatory system with a finite number of degrees of freedom was suggested.

On Laplacian Energy of Certain Mesh Derived Networks

Abstract: of a graph are the eigenvalues of its adjacency matrix. The multiset of eigenvalues is called its spectrum. There are many properties which can be explained using the spectrum like energy, connectedness, vertex connectivity, chromatic number, perfect matching etc. Laplacian spectrum is the multiset of eigenvalues of Laplacian matrix. The Laplacian energy of a graph is the sum of the absolute values of its Laplacian eigenvalues. In this paper we calculate the Laplacian energy of some grid based networks

Asymptotics of large eigenvalues for a class of band matrices

Abstract: We investigate the asymptotic behaviour of large eigenvalues for a class of finite difference self-adjoint operators with compact resolvent in $l^2$.

Inequalities for Eigenvalues and Singular Values

Abstract: Extremal representations for the maximum and minimum eigenvalues of a symmetric matrix are proved. Singular values are defined and the Singular Value Decomposition is obtained. Courant–Fischer Minimax Theorem, Cauchy Interlacing Principle and majorization of diagonal elements by eigenvalues of a symmetric matrix are proved. The volume of a matrix is defined as the positive square root of the product of the nonzero singular values. Some basic properties of volume are proved. Minimality properties of the Moore–Penrose inverse involving singular values are established.

On the fine spectrum of the operator defined by a lambda matrix over the sequence space c0 and c

Abstract: The main purpose of this paper is to determine the fine spectrum with respect to Goldberg's classification of the operator defined by a lambda matrix over the sequence spaces c0 and c. As a new development, we give the approximate point spectrum, defect spectrum and compression spectrum of the matrix operator Lambda on the sequence spaces c0 and c.

An estimator for the eigenvalues of the system matrix of a periodic-reference LMS algorithm

Abstract: The convergence analysis of the Least Mean Square (LMS) algorithm has been conventionally based on stochastic signals and describes thus only the average behavior of the algorithm. It has been shown previously that a periodic-reference LMS system can be regarded as a linear time-periodic system whose stability can be determined from the monodromy matrix. Generally, the monodromy matrix can only be solved numerically and does not thus reveal the actual factors behind the dynamics of the system. This paper derives an estimator for the eigenvalues of the monodromy matrix. The estimator is easy to calculate, and it also reveals the underlying reason for the bad convergence of the LMS algorithm in some special cases. The estimator is confirmed by comparing it to the precise eigenvalues of the monodromy matrix. The estimator is found to be accurate for the eigenvalues close to unity.

Comparison of Quadrature Rules for the Wielandt-Nyström Method with Statistical Applications

Abstract: Many statistics used to test that a sample of n points in the unit interval [0,1] comes from a known distribution can be studied using the theory of degenerate U - and V-statistics. In this theory, a special role is played by the eigenvalues of an integral operator. The aim of the present paper is to compare several versions of the Wielandt-Nystrom method for the approximation of the eigenvalues of this integral operator. We apply it to compute the eigenvalues of the Anderson-Darling statistic.

The Eigenvalues and The Optimal Potential Functions of Sturm-Liouville Operators

Abstract: In this article, we study the eigenvalues and potential functions of Sturm-Liouville operators with ordinary separated-type boundary condition. When the gap of the first two eigenvalues reaches minimum, we give the specific form of potential function. Meanwhile, for step potential function, we establish an one-to-one relationship between the eigenvalues and the nonnegative real roots of a class of algebraic equation, which provide an effective method for the approximate calculation of eigenvalues.