Showing papers on "Spectrum of a matrix published in 2009"
TL;DR: A new (non-Hamiltonian) half-size singularity test matrix is derived for use with admittance parameter state-space models, which gives a computational speedup by a factor of eight; it is applicable to both symmetric and unsymmetrical models; and it produces noiseless eigenvalues for reliable passivity assessment.
Abstract: One major difficulty in the rational modeling of linear systems is that the obtained model can be nonpassive, thereby leading to unstable simulations. The model's passivity properties are usually assessed by computing the eigenvalues of a Hamiltonian matrix, which is derived from the model parameters. The purely imaginary eigenvalues represent crossover frequencies where the model's conductance matrix is singular, allowing to pinpoint frequency intervals of passivity violations. Unfortunately, the eigenvalue computation time can be excessive for large models. Also, the test applies only to symmetrical models, and the testing is made difficult by numerical noise in the extracted eigenvalues. In this paper a new (non-Hamiltonian) half-size singularity test matrix is derived for use with admittance parameter state-space models, which overcomes these shortcomings. It gives a computational speedup by a factor of eight; it is applicable to both symmetric and unsymmetrical models; and it produces noiseless eigenvalues for reliable passivity assessment.
117 citations
TL;DR: In this article, the authors prove existence of eigenvalues for the scalar Helmholtz equation (isotropic and anisotropic cases) and Maxwell's equations under the condition that the contrast of the scattering medium is large enough.
Abstract: The investigation of the far field operator and the Factorization Method in
inverse scattering theory leads naturally to the study of corresponding interior
transmission eigenvalue problems. In contrast to the classical Dirichlet- or
Neumann eigenvalue problem for $-\Delta$ in bounded domains these interior
transmiision eigenvalue problem fail to be selfadjoint. In general, existence of
eigenvalues is an open problem. In this paper we prove existence of eigenvalues
for the scalar Helmholtz equation (isotropic and anisotropic cases) and
Maxwell's equations under the condition that the contrast of the scattering
medium is large enough.
111 citations
TL;DR: In this article, the problem of assigning eigenvalues of a linear vibratory system by state feedback control in the presence of time delay is considered and a method of a posteriori analysis is proposed to identify the primary eigenvalue and to ensure that they have been properly assigned.
58 citations
TL;DR: In this paper, an approximate non-state-space based approach is proposed for the calculation of eigenvalues of single and multiple-degree-of-freedom linear viscoelastic systems.
57 citations
TL;DR: In this paper, the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitian matrix model was described and the behavior of these newborn zeroes (eigenvalues) appearing in the new band was analyzed and connected with the location of the zeros of certain Freud polynomials.
Abstract: We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitian matrix model. The method is rigorously based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials. We provide an analysis with an error term of order N
−2γ
where 1/γ=2ν+2 is the exponent of non-regularity of the effective potential, thus improving even in the usual case the analysis of the pertinent literature. The behavior of the first finite number of zeroes (eigenvalues) appearing in the new band is analyzed and connected with the location of the zeroes of certain Freud polynomials. In general, all these newborn zeroes approach the point of nonregularity at the rate N
−γ
, whereas one (a stray zero) lags behind at a slower rate of approach. The kernels for the correlator functions in the scaling coordinate near the emerging band are provided together with the subleading term. In particular, the transition between K and K+1 eigenvalues is analyzed in detail.
48 citations
TL;DR: In this article, the eigenvalues of Laplacian with any order on a bounded domain in an n-dimensional Euclidean space were studied and the Yang-type inequalities were obtained.
Abstract: In this paper, we study eigenvalues of Laplacian with any order on a bounded domain in an n-dimensional Euclidean space and obtain estimates for eigenvalues, which are the Yang-type inequalities. In particular, the sharper result of Yang is included here. Furthermore, for lower order eigenvalues, we obtain two sharper inequalities. As a consequence, a proof of results announced by Ashbaugh [1] is also given.
39 citations
TL;DR: In this paper, the authors studied the structure of the Fisher-Hartwig matrices with real α and β and 0 <α <|β| < 1, where the behavior is particularly simple.
Abstract: A Toeplitz matrix is one in which the matrix elements are constant along diagonals. The Fisher-Hartwig matrices are much-studied singular matrices in the Toeplitz family. The matrices are defined for all orders, N.T hey are parameterized by two constants, α and β. Their spectrum of eigenvalues has a simple asymptotic form in the limit as N goes to infinity. Here we study the structure of their eigenvalues and eigenvectors in this limiting case. We specialize to the case with real α and β and 0 <α <|β| < 1, where the behavior is particularly simple. The eigenvalues are labeled by an index l which varies from 0 to N − 1. An asymptotic analysis using Wiener-Hopf methods indicates that for large N ,t hejth component of the lth eigenvector varies roughly in the fashion ln ψ l ≈ ip l j +O (1/N ). The lth wavevector, p l ,v aries as
28 citations
01 Nov 2009
TL;DR: In this paper, a Sturm-Liouville expression with indefinite weight of the form sgn (−d^2/dx^2+V ) on R and the non-real eigenvalues of an associated selfadjoint operator in a Krein space was studied.
Abstract: We study a Sturm-Liouville expression with indefinite weight of the form sgn (−d^2/dx^2+V ) on \mathbb{R} and the non-real eigenvalues of an associated selfadjoint operator in a Krein space. For real-valued potentials V with a certain behaviour at \pm \infty we prove that there are no real eigenvalues and the number of non-real eigenvalues (counting multiplicities) coincides with the number of negative eigenvalues of the selfadjoint operator associated to −d^2/dx^2 + V in L^2(\mathbb{R}). The general results are illustrated with examples.
28 citations
TL;DR: In this article, the eigenvalues of the poly-Laplacian with any order on a domain in an n-dimensional unit sphere were studied and the optimal result of Cheng and Yang (Math Ann 331:445-460, 2005) was included in their results.
Abstract: In this paper we study eigenvalues of the poly-Laplacian with any order on a domain in an n-dimensional unit sphere and obtain estimates for eigenvalues. In particular, the optimal result of Cheng and Yang (Math Ann 331:445–460, 2005) is included in our ones. In order to prove our results, we introduce 2(l + 1) functions ai and bi, for i = 0, 1, . . . , l and two operators μ and η. First of all, we study properties of functions ai and bi and the operators μ and η. By making use of these properties and introducing k free constants, we obtain estimates for eigenvalues.
26 citations
TL;DR: The most used approaches to the numerical solution of the Sturm–Liouville problem are discussed: finite differences and variational methods, both leading to a matrix eigenvalue problem; shooting methods using an initial-value solver; and coefficient approximation methods.
24 citations
TL;DR: In this article, a system of three arbitrary quantum particles on a three-dimensional lattice that interact via attractive pair contact potentials is considered and a condition for a gap to appear in the essential spectrum and it is shown that there are infinitely many eigenvalues of the Hamiltonian of the corresponding threeparticle system in this gap.
Abstract: We consider a system of three arbitrary quantum particles on a three-dimensional lattice that interact via attractive pair contact potentials. We find a condition for a gap to appear in the essential spectrum and prove that there are infinitely many eigenvalues of the Hamiltonian of the corresponding three-particle system in this gap.
TL;DR: Several localization techniques for the generalized eigenvalues of a matrix pair, obtained via the famous Gersgorin theorem and its generalizations are introduced.
Abstract: We introduce several localization techniques for the generalized eigenvalues of a matrix pair, obtained via the famous Gersgorin theorem and its generalizations. Specifically, we address the techniques of computing and graphing of the obtained localization sets of a matrix pair. The work that follows involves much about nonnegative matrices, strictly diagonally dominant (SDD) matrices, H- and M-matrices. We show the utility of our results theoretically, as well as with numerical examples and graphs. Copyright © 2009 John Wiley & Sons, Ltd.
02 Nov 2009
TL;DR: The idea is to convert the Hamiltonian matrix to an equivalent sparse form, termed the “extended Hamiltonian pencil”, and solve for its eigenvalues efficiently using a special eigensolver, which demonstrates an 80X speed-up compared with standard direct eIGensolvers.
Abstract: Passivity is an important property for a macro-model generated from measured or simulated data. Existence of purely imaginary eigenvalues of a Hamiltonian matrix provides useful information in assessing and correcting the passivity of a system. Since direct computation of eigenvalues is very expensive for large-scale systems, several authors have proposed to solve iteratively for a subset of the eigenvalues based on heuristic sampling along the imaginary axis. However, completeness is not guaranteed in such methods and thus potential risk of missing important eigenvalues is difficult to avoid. In this paper we are aiming at finding all eigenvalues efficiently to avoid both the high cost and the potential risk of missing important eigenvalues. The idea of the proposed method is to convert the Hamiltonian matrix to an equivalent sparse form, termed the "extended Hamiltonian pencil", and solve for its eigenvalues efficiently using a special eigensolver. Experiments on several realistic systems demonstrate an 80X speed-up compared with standard direct eigensolvers.
TL;DR: Asymptotic expansions for negative eigenvalues λ−ke of the Dirichlet problem with the density becoming negatively small in either a subdomain of fixed size, or a small, of diameter O(e 1/m), neighborhood of an interior point are constructed in this article.
Abstract: Asymptotic expansions are constructed for negative eigenvalues λ−ke of the Dirichlet problem with the density becoming negatively small, of order e, in either a subdomain of fixed size, or a small, of diameter O(e1/m), neighborhood of an interior point. Such the eigenvalues lie far away from the coordinate origin and their order with respect to the small parameter is e−1 in the first case and e−m/(m+2) in the second one. The limit problems are formulated and investigated, the eigenvalues of which imply the limits as e → +0 of the quantities −eλ−ke and −em/(m+2)λ−ke, respectively. Asymptotically precise estimates are obtained for the remainders in the expansions of the eigenvalues and eigenfunctions.
TL;DR: In this paper, the authors studied the spectrum of relaxation rates in the electron glass system and showed that the spectrum can be represented by the distribution function of the relaxation rate and the noise in the fluctuating occupancies of the localized electronic sites.
Abstract: Recently we have shown that slow relaxations in the electron glass system can be understood in terms of the spectrum of a matrix describing the relaxation of the system close to a metastable state. The model focused on the electron glass system, but its generality was demonstrated on various other examples. Here, we study the noise spectrum in the same framework. We obtain a remarkable relation between the spectrum of relaxation rates $\lambda$ described by the distribution function $P(\lambda) \sim 1/\lambda$ and the $1/f$ noise in the fluctuating occupancies of the localized electronic sites. This noise can be observed using local capacitance measurements. We confirm our analytic results using numerics, and also show how the Onsager symmetry is fulfilled in the system.
TL;DR: In this paper, the authors studied the spectrum of the relaxation rate of the electron glass system in terms of the spectrum spectrum of a matrix describing the relaxation of the system close to a metastable state.
Abstract: Recently we have shown that slow relaxations in the electron glass system can be understood in terms of the spectrum of a matrix describing the relaxation of the system close to a metastable state. The model focused on the electron glass system, but its generality was demonstrated on various other examples. Here, we study the noise spectrum in the same framework. We obtain a remarkable relation between the spectrum of relaxation rates λ described by the distribution function P (λ) ∼ 1/λ and the 1/f noise in the fluctuating occupancies of the localized electronic sites. This noise can be observed using local capacitance measurements. We confirm our analytic results using numerics, and also show how the Onsager symmetry is fulfilled in the system.
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TL;DR: In this article, the authors extend some existence's results concerning the generalized eigenvalues for fully nonlinear operators singular or degenerate for the radial case and prove the existence of an infinite number of eigen values, simple and isolated.
Abstract: In this paper we extend some existence's results concerning the generalized eigenvalues for fully nonlinear operators singular or degenerate. We consider the radial case and we prove the existence of an infinite number of eigenvalues, simple and isolated. This completes the results obtained by the author with Isabeau Birindelli for the first eigenvalues in the radial case, and the results obtained for the Pucci's operator by Busca Esteban and Quaas and for the $p$-Laplace operator by Del Pino and Manasevich.
TL;DR: In this paper, it was shown that the eigenvalues of a non-self-adjoint differential operator originated from the linearization of some Cauchy problem, and that they are related to the Eigenvalue of Heun's differential equation.
Abstract: Eigenmode solutions are very important in stability analysis of dynamical systems. The set of eigenvalues of a non-self-adjoint differential operator originated from the linearization of some Cauchy problem is investigated. It is shown that the eigenvalues are purely imaginary, and that they are related to the eigenvalues of Heun’s differential equation. These two results are used to derive the asymptotic behavior of the eigenvalues and to compute them numerically.
TL;DR: New exclusion intervals of the real eigenvalues of a real matrix are presented, which are further applied to localize the real Eigenvalues different from 1 of a positive stochastic matrix.
Abstract: A real square matrix with positive row sums and all its off-diagonal elements bounded below by the corresponding row means is called a $C$-matrix, which is introduced by Pena [Exclusion and inclusion intervals for the real eigenvalues of positive matrices, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 908-917]. In this paper, a new class of nonsingular matrices—$MC$-matrices containing $C$-matrices—is first defined. By properties of a subclass of $MC$-matrices, we present new exclusion intervals of the real eigenvalues of a real matrix, which are further applied to localize the real eigenvalues different from 1 of a positive stochastic matrix. Secondly, an inclusion interval for the real parts of eigenvalues of a real matrix is established. Finally, for real matrices with nonnegative off-diagonal elements, lower and upper bounds of real eigenvalues are obtained. Furthermore, sufficient conditions are derived to indicate that the real inclusion intervals provided by Pena [On an alternative to Gerschgorin circles and ovals of Cassini, Numer. Math., 95 (2003), pp. 337-345] are subsets of those provided by the ovals of Cassini.
01 Jan 2009
TL;DR: In this paper, the eigenvalues of a compression of a monic quadratic weakly hyperbolic polynomial to an (n − 1)-dimensional subspace of ℂ n block-interlace were proved.
Abstract: Let L be a monic quadratic weakly hyperbolic or hyperbolic n × n matrix polynomial. We solve some direct spectral problems: We prove that the eigenvalues of a compression of L to an (n − 1)-dimensional subspace of ℂ n block-interlace and that the eigenvalues of a one-dimensional perturbation of L (−,+)-interlace the eigenvalues of L. We also solve an inverse spectral problem: We identify two given block-interlacing sets of real numbers as the sets of eigenvalues of L and its compression.
TL;DR: In this paper, it was shown that the antisymmetric component HAS of the original Hamiltonian is an outer projection of K with respect to the ant-isymmetrizer O, so that HAS = OKO.
Abstract: The expectation value of the Hamiltonian H for a many-electron system may be expressed in terms of the reduced second-order density matrix and the “reduced” Hamiltonian K for a two-electron system. It may be shown that the antisymmetric component HAS of the original Hamiltonian is an “outer projection” of K with respect to the antisymmetrizer O, so that HAS = OKO. This result implies that the eigenvalues of K are, in order, lower bounds to the eigenvalues of HAS, i.e. to the eigenvalues of H associated with the antisymmetry requirement. The expectation values of K are hence often below the ground state energy of H, and the eigenvalues may be used for the calculation of lower bounds. Some numerical applications to atoms serve as an illustration, and it is shown that the lower bounds become worse as the number of electrons increases. The implications for the “representability problem” are discussed.
TL;DR: This direct method borrows stable methods from numerical linear algebra to compute a large number of eigenvalues with high precision from a differential operator by an infinite matrix.
Abstract: We are concerned with the computation of eigenvalues of singular non- selfadjoint Sturm — Liouville problems by the method of determinants. The represen- tation of a differential operator by an infinite matrix allows the use of Lidskii's theorem to define its determinant. The finite section is then used to compute eigenvalues in a simple way. This direct method borrows stable methods from numerical linear algebra to compute a large number of eigenvalues with high precision. Numerical examples with nondifferentiable and complex valued coefficients are treated at the end. 2000 Mathematics Subject Classification: 34L16, 47A58.
TL;DR: In this paper, the authors give necessary and sufficient conditions for the occurrence of Q-spectral integral variation only in two places, as the first case never occurs, while the second case always occurs.
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TL;DR: In this article, a new class of sign-symmetric matrices is introduced, called J--signsymmetric and irreducible matrices, which have complex eigenvalues on the largest spectral circle.
Abstract: A new class of sign-symmetric matrices is introduced in this paper Such matrices are named J--sign-symmetric The spectrum of a J--sign-symmetric irreducible matrix is studied under assumptions that its second compound matrix is also J--sign-symmetric and irreducible The conditions, when such matrices have complex eigenvalues on the largest spectral circle, are given The existence of two positive simple eigenvalues $\lambda_1 > \lambda_2 > 0$ of a J--sign-symmetric irreducible matrix A is proved under some additional conditions The question, when the approximation of a J--sign-symmetric matrix with a J--sign-symmetric second compound matrix by strictly J--sign-symmetric matrices with strictly J--sign-symmetric compound matrices is possible, is also studied in this paper
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TL;DR: In this article, the existence of point eigenvalues in gaps in the essential spectrum was studied and the problem of counting the number of such eigen values in a gap was studied.
Abstract: In this paper we consider a Schrodinger eigenvalue problem with a potential consisting of a periodic part together with a compactly supported defect potential. Such problems arise as models in condensed matter to describe color in crystals as well as in engineering to describe optical photonic structures. We are interested in studying the existence of point eigenvalues in gaps in the essential spectrum, and in particular in counting the number of such eigenvalues. We use a homotopy argument in the width of the potential to count the eigenvalues as they are created. As a consequence of this we prove the following significant generalization of Zheludev's theorem: the number of point eigenvalues in a gap in the essential spectrum is exactly one for sufficiently large gap number unless a certain Diophantine approximation problem has solutions, in which case there exists a subsequence of gaps containing 0,1 or 2 eigenvalues. We state some conditions under which the solvability of the Diophantine approximation problem can be established.
01 Jan 2009
TL;DR: In this paper, the authors showed that a line distance matrix of size n > 1, associated with biological sequences, has one positive and n − 1 negative eigenvalues, and the spread of the spectrum of line distance matrices is considered.
Abstract: In [1], the authors showed that a line distance matrix of size n > 1, associated with biological sequences, has one positive and n − 1 negative eigenvalues. The energy E(G) of a graph G is defined as the sum of the absolute values of the eigenvalues of G in [2]. Similarly, we obtain bounds on the energy of line distance matrix. The spread of the spectrum of line distance matrix is considered.
Journal Article•
TL;DR: Lower bounds for the rank and the estimation for eigenvalues of matrices are discussed in this paper, where the authors prove that all the eigen values of any complex matrix are located in one disk.
Abstract: Lower bounds for the rank and the estimation for eigenvalues of matrices are discussed.Two lower bounds for the rank and estimation for the real part and imaginary part of eigenvalues are obtained.Also,we prove that all the eigenvalues of any complex matrix are located in one disk.Some numerical examples show the effectiveness of our results.
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TL;DR: In this paper, the existence of eigenvalues and eigenfunctions of 1-homogeneous fully nonlinear operators has been studied in the framework of viscosity solutions.
Abstract: In this paper we present an elementary theory about the existence of eigenvalues for fully nonlinear radially symmetric 1-homogeneous operators. A general theory for first eigenvalues and eigenfunctions of 1-homogeneous fully nonlinear operators exists in the framework of viscosity solutions. Here we want to show that for the radially symmetric operators (and one dimensional) a much simpler theory can be established, and that the complete set of eigenvalues and eigenfuctions characterized by the number of zeroes can be obtained.
TL;DR: In this article, the spectrum of a Schroedinger operator in a multi-dimensional cylinder perturbed by a shrinking potential was considered and the phenomenon of a new eigenvalue emerging from the threshold of the essential spectrum was studied.
Abstract: We consider the spectrum of a Schroedinger operator in a multi-dimensional cylinder perturbed by a shrinking potential. We study the phenomenon of a new eigenvalue emerging from the threshold of the essential spectrum and give the sufficient conditions for such eigenvalues to emerge. If such eigenvalues exist, we construct their asymptotic expansions.
01 Jan 2009
TL;DR: A new algorithm, which is called weighted harmonic projection algorithm for computing the eigenvalues of a nonsymmetric matrix, is presented, which converges fast and works with high accuracy.
Abstract: The harmonic projection method can be used to find interior eigenpairs of large matrices. Given a target point or shift t to which the needed interior eigenvalues are close, the desired interior eigenpairs are the eigenvalues nearest t and the associated eigenvectors. In this paper, we present a new algorithm, which is called weighted harmonic projection algorithm for computing the eigenvalues of a nonsymmetric matrix. The implementation of the algorithm has been tested by numerical examples, the results show that the algorithm converges fast and works with high accuracy