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Showing papers on "Spectrum of a matrix published in 2010"


Journal ArticleDOI
TL;DR: In this article, it was shown that if the secular equation is real, the Hamiltonian is necessarily PT symmetric, and the set of all operators C that nontrivially obey the two equations [C, H ] = 0 and C 2 = 1] is nonempty and the energy eigenvalues of H are all real only if every such operator commutes with PT.

111 citations


Journal ArticleDOI
TL;DR: In this article, isolated and embedded eigenvalues in the generalized eigenvalue problem defined by two self-adjoint operators with a positive essential spectrum and a finite number of isolated eigen values were studied.
Abstract: We study isolated and embedded eigenvalues in the generalized eigenvalue problem defined by two self-adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues. The generalized eigenvalue problem determines the spectral stability of nonlinear waves in infinite-dimensional Hamiltonian systems. The theory is based on Pontryagin’s invariant subspace theorem and extends beyond the scope of earlier papers of Pontryagin, Krein, Grillakis, and others. Our main results are (i) the number of unstable and potentially unstable eigenvalues equals the number of negative eigenvalues of the self-adjoint operators, (ii) the total number of isolated eigenvalues of the generalized eigenvalue problem is bounded from above by the total number of isolated eigenvalues of the self-adjoint operators, and (iii) the quadratic forms defined by the two self-adjoint operators are strictly positive on the subspace related to the continuous spectrum of the generalized eigenvalue problem. Applicatio...

67 citations


Journal ArticleDOI
TL;DR: In this paper, a stochastic representation of the absorption time of a Markov chain is given in terms of a mixture of sums of independent geometric laws whose parameters are successive Dirichlet eigenvalues (starting from the smallest one).
Abstract: This paper gives a stochastic representation in spectral terms for the absorption time T of a finite Markov chain which is irreducible and reversible outside the absorbing point. This yields quantitative informations on the parameters of a similar representation due to O'Cinneide for general chains admitting real eigenvalues. In the discrete time setting, if the underlying Dirichlet eigenvalues (namely the eigenvalues of the Markov transition operator restricted to the functions vanishing on the absorbing point) are nonnegative, we show that T is distributed as a mixture of sums of independent geometric laws whose parameters are successive Dirichlet eigenvalues (starting from the smallest one). The mixture weights depend on the starting law. This result leads to a probabilistic interpretation of the spectrum, in terms of strong random times and local equilibria through a simple intertwining relation. Next this study is extended to the continuous time framework, where geometric laws have to be replaced by exponential distributions having the (opposite) Dirichlet eigenvalues of the generator as parameters. Returning to the discrete time setting we consider the influence of negative eigenvalues which are given another probabilistic meaning. These results generalize results of Karlin and McGregor and Keilson for birth and death chains.

47 citations


Journal ArticleDOI
TL;DR: In this article, bounds on the variance of a finite universe were derived for the roots of the polynomial equations and bounds for the largest and smallest eigenvalues of a square matrix with real spectrum.
Abstract: We derive bounds on the variance of a finite universe. Some related inequalities for the roots of the polynomial equations and bounds for the largest and smallest eigenvalues of a square matrix with real spectrum are obtained.

45 citations


Journal ArticleDOI
TL;DR: In this article, a detailed procedure based on an analytical transfer matrix method is presented to solve bound-state problems and the energy eigenvalues for an arbitrary one-dimensional potential can be obtained by the method.
Abstract: A detailed procedure based on an analytical transfer matrix method is presented to solve bound-state problems. The derivation is strict and complete. The energy eigenvalues for an arbitrary one-dimensional potential can be obtained by the method. The anharmonic oscillator potential and the rational potential are two important examples. Checked by numerical techniques, the results for the two potentials by the present method are proven to be exact and reliable.

15 citations


Journal ArticleDOI
TL;DR: In this paper, Dai et al. constructed methods for finding convergent expansions for eigenvectors and eigenvalues of large-toeplitz matrices based on a situation in which the analogous infinite- matrix would be singular.
Abstract: This paper constructs methods for finding convergent expansions for eigenvectors and eigenvalues of large- Toeplitz matrices based on a situation in which the analogous infinite- matrix would be singular. It builds upon work done by Dai, Geary, and Kadanoff [H Dai et al., J. Stat. Mech. P05012 (2009)] on exact eigenfunctions for Toeplitz operators which are infinite-dimension Toeplitz matrices. One expansion for the finite- case is derived from the operator eigenvalue equations obtained by continuing the finite- Toeplitz matrix to plus infinity. A second expansion is obtained by continuing the finite- matrix to minus infinity. The two expansions work together to give an apparently convergent expansion for the finite- eigenvalues and eigenvectors, based upon a solvability condition for determining eigenvalues. The expansions involve an expansion parameter expressed as an inverse power of . A variational principle is developed, which gives an approximate expression for determining eigenvalues. The lowest order asymptotics for eigenvalues and eigenvectors agree with the earlier work [H Dai et al., J. Stat. Mech. P05012 (2009)]. The eigenvalues have a term as their leading finite- correction in the central region of the spectrum. The correction in this region is obtained here for the first time. Received: 19 October 2009; Accepted: 29 September 2010; Edited by: A. G. Green; Reviewed by: T. Ehrhardt, Math. Dept., Univ. California, Santa Cruz, USA; DOI: 10.4279/PIP.020003

11 citations


Journal ArticleDOI
A. M. Nazari1, D. Rajabi1
TL;DR: In this article, the distance from a complex square matrix A to the set of matrices X that have λ 1 and λ 2 as some of their eigenvalues is calculated.

8 citations


Journal ArticleDOI
TL;DR: In this paper, an inclusion region for the eigenvalues of a matrix that can be considered an alternative to the Brauer set is derived, accompanied by non-singularity conditions.
Abstract: We derive an inclusion region for the eigenvalues of a matrix that can be considered an alternative to the Brauer set. It is accompanied by non-singularity conditions.

8 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that there are 25 connected integral graphs with exactly two main eigenvalues and index 3, where the eigenvector of a graph has an eigen vector such that the sum of whose entries is not equal to zero.

8 citations


Journal ArticleDOI
TL;DR: An auto-detection corner based on eigenvalues product of covariance matrices (ADEPCM) of boundary points over multi-region of support is presented, which considers that points corresponding to peaks of eigen values product graph are reported as corners, which avoids human judgment and curvature threshold settings.
Abstract: In this paper we present an auto-detection corner based on eigenvalues product of covariance matrices (ADEPCM) of boundary points over multi-region of support. The algorithm starts with extracting the contour of an object, and then computes the eigenvalues product of covariance matrices of this contour at various regions of support. Finally determine automatically peaks of the graph of eigenvalues product function. We consider that points corresponding to peaks of eigenvalues product graph are reported as corners, which avoids human judgment and curvature threshold settings. Experimental results show that the proposed method has more robustness for noise and various geometrical transform.

7 citations


Posted Content
TL;DR: In this article, the eigenvalues and eigenfunctions of the Hodge Laplacian for generic metrics on a closed 3-manifold were analyzed, and it was shown that the nonzero eigen values are simple.
Abstract: In this paper we analyze the eigenvalues and eigenfunctions of the Hodge Laplacian for generic metrics on a closed 3-manifold $M$. In particular, we show that the nonzero eigenvalues are simple and the zero set of the eigenforms of degree 1 or 2 consists of isolated points for a residual set of $C^r$ metrics on $M$, for any integer $r\geq2$. The proof of this result hinges on a detailed study of the Beltrami (or rotational) operator on co-exact 1-forms.

Journal ArticleDOI
TL;DR: In this paper, a reduced-order method to find critical eigenvalues of ultra large-scale power system was proposed, where the numerical solution of matrix exponential is computed by precise time-step integration.
Abstract: This study presents a novel reduced-order method to find critical eigenvalues of ultra large-scale power system. First, the numerical solution of matrix exponential is computed by precise time-step integration. Secondly, based on the numerical solution, the numerical curve of the trace of matrix exponential is formed. Thirdly, the candidates of critical eigenvalues are extracted from the numerical curve of the trace by Prony analysis, and finally, a set of weight coefficients is calculated to confirm critical eigenvalues from candidate eigenvalues. Since the trace contains all eigenvalues, no critical eigenvalue can be lost in analysing the numerical curve of the trace. In the later period of time integration, the effect of all non-critical eigenvalues to the trace is decayed and oppositely the effect of all critical eigenvalues is amplified. Thus, several rightmost eigenvalues can only be extracted from the numerical curve of the trace. Case studies for 16 and 9004 order system have validated that the proposed method can be used to find all critical eigenvalues of ultra large-scale power system.

Journal ArticleDOI
TL;DR: In this article, it was shown that only the numbers - 3, - 2, - 1, 0, 1, 2 d can be integer eigenvalues of a circuit distance power.

Journal ArticleDOI
TL;DR: In this paper, the authors prove the first of these conjectures by generalizing the original Markov matrix by introducing extra parameters, explicitly calculating its eigenvalues, and showing that the new matrix reduces to the original one by a suitable specialization.
Abstract: In recent work on nonequilibrium statistical physics, a certain Markovian exclusion model called an asymmetric annihilation process was studied by Ayyer and Mallick. In it they gave a precise conjecture for the eigenvalues (along with the multiplicities) of the transition matrix. They further conjectured that to each eigenvalue, there corresponds only one eigenvector. We prove the first of these conjectures by generalizing the original Markov matrix by introducing extra parameters, explicitly calculating its eigenvalues, and showing that the new matrix reduces to the original one by a suitable specialization. In addition, we outline a derivation of the partition function in the generalized model, which also reduces to the one obtained by Ayyer and Mallick in the original model.

Journal Article
TL;DR: Two sequences are presented which converge to the minimal and the maximal modulus of eigenvalues, respectively, and it is pointed out that the two sequences are not recommendable for practical use for finding the minimal
Abstract: This paper is concerned with bounds of eigenvalues of a complex matrix. Both lower and upper bounds of modulus of eigenvalues are given by the Stein equation. Furthermore, two sequences are presented which converge to the minimal and the maximal modulus of eigenvalues, respectively. We have to point out that the two sequences are not recommendable for practical use for finding the minimal and the maximal modulus of eigenvalues.

Book ChapterDOI
TL;DR: In this paper, the essential spectrum of a weighted Sturm-Liouville operator is studied under the assumption that the weight function has one turning point, and an abstract approach to the problem is given via a functional model for indefinite Sturm Liouville operators.
Abstract: Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues are obtained. Also, operators with finite singular critical points are considered.

Journal ArticleDOI
TL;DR: In this paper, the existence of an infinite number of eigenvalues for a model "three-particle" Schrodinger operator H was studied and the necessary and sufficient conditions for its existence below the lower boundary of its essential spectrum were proved.
Abstract: We study the existence of an infinite number of eigenvalues for a model “three-particle” Schrodinger operator H. We prove a theorem on the necessary and sufficient conditions for the existence of an infinite number of eigenvalues of the model operator H below the lower boundary of its essential spectrum.

Posted Content
TL;DR: A notion of central eigenvalues permits to describe criterium of diagonalization of square matrices whose coefficients are intervals and a notion of Exponential mapping is defined.
Abstract: In a previous paper, we have given an algebraic model to the set of intervals. Here, we apply this model in a linear frame. We define a notion of diagonalization of square matrices whose coefficients are intervals. But in this case, with respect to the real case, a matrix of order $n$ could have more than $n$ eigenvalues (the set of intervals is not factorial). We consider a notion of central eigenvalues permits to describe criterium of diagonalization. As application, we define a notion of Exponential mapping.

Journal ArticleDOI
TL;DR: In this paper, the eigenvalues of weighted composition operators uCφ on the Bloch space are discussed and an example of a non-compact operator Cφ whose eigen values can be determined precisely is given.
Abstract: This note discusses eigenvalues of weighted composition operators uCφ on the Bloch space. The main result provides a class of uCφ for which computation of eigenvalues is possible. We also construct an example of a non-compact operator Cφ whose eigenvalues can be determined precisely.

Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of estimating the eigenvalues of a self-adjoint operator defined by a Jacobi matrix in the Hilbert space l 2 (ℕ) by eigen values of principal finite submatrices of an infinite Jacobi matrices that defines this operator.
Abstract: We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l 2(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].

01 Jan 2010
TL;DR: In this paper, the spectral properties of the interior transmission problem have been investigated in inverse scattering theory and lower and upper bounds for the first transmission eigenvalue in terms of the geometry and physical properties of a scattering object were provided.
Abstract: The interior transmission problem arises in inverse scattering theory for inhomogeneous media. It is a boundary value problem for a set of equations defined in a bounded domain coinciding with the support of the scattering object. Of particular interest is the spectrum associated with this boundary value problem, more specifically the existence of eigenvalues known as transmission eigenvalues. Indeed, on one hand, in the context of sampling methods for reconstructing the support of the scatterer one needs to avoid those frequencies that correspond to transmission eigenvalues, and hence it is important to know that the transmission eigenvalues form a discrete set. On the other hand, one can use transmission eigenvalues to obtain information about physical properties of the scattering medium [1], [3] and therefore it is important to know whether they exist and to understand their connection with the index of refraction. The latter application is based on the recent results in [2] which justify the numerical observation that the transmission eigenvalues can be computed from the far field data. Either way, the investigation of the spectral properties of the interior transmission problem has become an interesting mathematical question in inverse scattering theory. We present here the most recent developments on transmission eigenvalues, in particular we show the existence of infinitely many transmission eigenvalues and provide lower and upper bounds for the first transmission eigenvalue in terms of the geometry and physical properties of the scattering object [4], [5]. Then, we show how to use these bounds to obtain information on the index of refraction of a general anisotropic scattering medium, as well on the presence of defects inside the scattering medium.

Journal ArticleDOI
TL;DR: In this article, the spectral problem associated with the Klein-Gordon equation for unbounded electric potentials is considered and the spectrum of this problem is contained in two disjoint real intervals and the two inner boundary points are eigenvalues.
Abstract: We consider the spectral problem associated with the Klein-Gordon equation for unbounded electric potentials. If the spectrum of this problem is contained in two disjoint real intervals and the two inner boundary points are eigenvalues, we show that these extremal eigenvalues are simple and possess strictly positive eigenfunctions. Examples of electric potentials satisfying these assumptions are given.

Posted Content
TL;DR: This paper proposes a general framework that covers the existing schemes and reveals that the matrix mismatch leads to a threshold effect caused by "steering vector competition", and finds that, if there are generalized eigenvalues that are infinite, the threshold will increase unboundedly with the interference power.
Abstract: Matrix pair beamformer (MPB) is a blind beamformer. It exploits the temporal structure of the signal of interest (SOI) and applies generalized eigen-decomposition to a covariance matrix pair. Unlike other blind algorithms, it only uses the second order statistics. A key assumption in the previous work is that the two matrices have the same interference statistics. However, this assumption may be invalid in the presence of multipath propagations or certain "smart" jammers, and we call it as matrix mismatch. This paper analyzes the performance of MPB with matrix mismatch. First, we propose a general framework that covers the existing schemes. Then, we derive its normalized output SINR. It reveals that the matrix mismatch leads to a threshold effect caused by "steering vector competition". Second, using matrix perturbation theory, we find that, if there are generalized eigenvalues that are infinite, the threshold will increase unboundedly with the interference power. This is highly probable when there are multiple periodical interferers. Finally, we present simulation results to verify our analysis.

Posted Content
TL;DR: In this article, the interior transmission problem and transmission eigenvalues for multiplicative perturbations of a linear partial differential operator with constant real coefficients were studied under suitable growth conditions on the symbol of the operator and the perturbation.
Abstract: In this paper we study the interior transmission problem and transmission eigenvalues for multiplicative perturbations of linear partial differential operator of order $\ge 2$ with constant real coefficients. Under suitable growth conditions on the symbol of the operator and the perturbation, we show the discreteness of the set of transmission eigenvalues and derive sufficient conditions on the existence of transmission eigenvalues. We apply these techniques to the case of the biharmonic operator and the Dirac system. In the hypoelliptic case we present a connection to scattering theory.

Journal ArticleDOI
TL;DR: The relationship between the eigenvalues associated with the matrices of the minimum dimension time-domain and frequency-domain approaches used for reconstructing missing uniform samples and the weighted Toeplitz matrix is derived.
Abstract: In this correspondence, we derive the relationship between the eigenvalues associated with the matrices of the minimum dimension time-domain and frequency-domain approaches used for reconstructing missing uniform samples. The dependency of the eigenvalues of the weighted Toeplitz matrix on positive weights are explored. Simple bounds for the maximum and minimum eigenvalues of the weighted Toeplitz matrix are also presented. Alternative matrices possessing the same nonzero eigenvalues as that of the weighted Toeplitz matrix are provided. We verify the theory by the examples presented.

Journal ArticleDOI
TL;DR: In this paper, a new proof of a classical result of A Horn on the existence of a matrix with prescribed singular values and eigenvalues was given, which was later used to prove a new result of the same authors.
Abstract: We give a new proof of a classical result of A Horn on the existence of a matrix with prescribed singular values and eigenvalues

Journal ArticleDOI
TL;DR: In this paper, the eigenvalues of stochastic matrices are estimated for the tensor product of two matrices, and the distribution for the generalized eigenvectors of these matrices is obtained.
Abstract: The purpose of this paper is to locate and estimate the eigenvalues of stochastic matrices. We present several estimation theorems about the eigenvalues of stochastic matrices. Meanwhile, we obtain the distribution theorem for the eigenvalues of tensor product of two stochastic matrices. We will conclude the paper with the distribution for the eigenvalues of generalized stochastic matrices.

01 Jan 2010
TL;DR: In this article, the authors propose a method of disassembling a set of disassembly points, called DISSERTATION, which is based on disassemblage-of-dispersal.
Abstract: OF DISSERTATION

Journal ArticleDOI
TL;DR: The main part of this paper is devoted to the description of how the three step procedure to calculate all discrete eigenvalues was developed, which did not wish to resort to high-precision fixed point arithmetic but would solely rely on IEEE 754 double precision arithmetic.

Book ChapterDOI
01 Jan 2010
TL;DR: In this paper, the Rayleigh quotient is used to recover the eigenvalue of a scalar λ by finding a non-null vector x such that x is associated with λ.
Abstract: Given a square matrix \( {\rm A} \in \mathbb{C}^{{n \times n}} \), the eigenvalue problem consists in finding a scalar λ (real or complex) and a nonnull vector x such that $${\rm Ax} = \lambda{\rm x}$$ (6.1) Any such λ is called an eigenvalue of A, while x is the associated eigenvector. The latter is not unique; indeed all its multiples αx with α≠ 0, real or complex, are also eigenvectors associated with λ. Should x be known, λ can be recovered by using the Rayleigh quotient \( {\rm x}^H {\rm Ax/}\left\| {\rm X} \right\|^2, {\rm x}^H = {\rm \bar{x}}^{\rm T} \) being the vector whose i-th component is equal to \( \bar{x}_{i} \).