Showing papers on "Spectrum of a matrix published in 2016"
TL;DR: This paper proposes a new order-determination procedure that helps to pinpoint the rank of a matrix more precisely than the previous methods by exploiting both patterns: when the eigenvalues of a random matrix are close together, their eigenvectors tend to vary greatly; when they are far apart, their variability tends to be small.
Abstract: In applying statistical methods such as principal component analysis, canonical correlation analysis, and sufficient dimension reduction, we need to determine how many eigenvectors of a random matrix are important for estimation. This problem is known as order determination, and amounts to estimating the rank of a matrix. Previous order-determination procedures rely either on the decreasing pattern, or elbow, of the eigenvalues, or on the increasing pattern of the variability in the directions of the eigenvectors. In this paper we propose a new order-determination procedure by exploiting both patterns: when the eigenvalues of a random matrix are close together, their eigenvectors tend to vary greatly; when the eigenvalues are far apart, their variability tends to be small. The combination of both helps to pinpoint the rank of a matrix more precisely than the previous methods. We establish the consistency of the new order-determination procedure, and compare it with other such procedures by simulation and in an applied setting.
108 citations
TL;DR: In this article, the eigenvalues of Schrodinger operators with complex potentials in odd space dimensions were studied and bounds on the total number of eigen values in the case where V decays exponentially at infinity.
Abstract: We study the eigenvalues of Schrodinger operators with complex potentials in odd space dimensions. We obtain bounds on the total number of eigenvalues in the case where V decays exponentially at infinity.
50 citations
TL;DR: A new approach for selecting the optimal value of r, which is based on the distribution of the eigenvalues of a scaled Hankel matrix is introduced, and the results indicate that the proposed approach has potential in selecting thevalue of r for signal extraction.
Abstract: Singular spectrum analysis (SSA) is a reliable technique for separating an arbitrary signal from a noisy time series (signal+noise). The SSA technique is based upon two main selections: window length, L , and the number of the eigenvalues, r . These values play an important role for the reconstruction stage. In this paper, we introduce a new approach for selecting the optimal value of r , which is based on the distribution of the eigenvalues of a scaled Hankel matrix. The proposed approach is applied to a number of simulated and real data with different structures. The results indicate that the proposed approach has potential in selecting the value of r for signal extraction.
33 citations
TL;DR: The major technical tool is a transformation of the eigenvalue problem for initial fractional Schrödinger equation into that for Fredholm integral equation with hypersingular kernel, solved by means of expansion over the complete set of orthogonal functions in the domain D.
Abstract: We study Levy flights with arbitrary index 0<μ≤2 inside a potential well of infinite depth. Such a problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in disordered quantum systems. The major technical tool is a transformation of the eigenvalue problem for initial fractional Schrodinger equation into that for Fredholm integral equation with hypersingular kernel. The latter equation is then solved by means of expansion over the complete set of orthogonal functions in the domain D, reducing the problem to the spectrum of a matrix of infinite dimensions. The eigenvalues and eigenfunctions are then obtained numerically with some analytical results regarding the structure of the spectrum.
24 citations
TL;DR: In this article, a novel solution strategy of Saint-Venant's problem based on Hamilton's formalism is proposed, which relies upon the determination of the subspace of the system's Hamiltonian matrix associated with its null and pure imaginary eigenvalues.
Abstract: This paper proposes a novel solution strategy of Saint-Venant’s problem based on Hamilton’s formalism. Saint-Venant’s problem focuses on helicoidal beams and its solution hinges upon the determination of the subspace of the system’s Hamiltonian matrix associated with its null and pure imaginary eigenvalues. A projection approach is proposed that reduces the system Hamiltonian matrix to a matrix of size 12 × 12, whose eigenvalues are identical to the null and purely imaginary eigenvalues of the original system, with the same Jordan structure. A fundamental theoretical result is established: Saint-Venant’s solutions exist because rigid-body motions create no strains. Indeed, the solvability conditions for the governing equations of the problem are satisfied because a matrix identity holds, which expresses the fact that rigid-body motions create no strains. Because it avoids the identification of the Jordan structure of the original system, the implementation of the proposed projection for large, realistic problems is straightforward. Closed form solutions of the reduced problem are found and three-dimensional stress and strain fields can be recovered from the closed form solution. Numerical examples are presented to demonstrate the capabilities of the analysis. Predictions are compared to exact solutions of three-dimensional elasticity and three-dimensional FEM analysis.
21 citations
TL;DR: In this paper, the authors give an asymptotic limit theory for the eigenvalues of the covariance matrix of a multivariate time series and derive limit theories for a host of continuous functionals of the Eigenvalues.
18 citations
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TL;DR: This work proposes a theoretically optimal and computationally efficient algorithm for recovering the moments of the eigenvalues of the population covariance matrix and provides finite--sample bounds on the expected error of the recovered eigen Values, which imply that the estimator is asymptotically consistent as the dimensionality of the distribution and sample size tend towards infinity.
Abstract: We consider the problem of approximating the set of eigenvalues of the covariance matrix of a multivariate distribution (equivalently, the problem of approximating the "population spectrum"), given access to samples drawn from the distribution. The eigenvalues of the covariance of a distribution contain basic information about the distribution, including the presence or lack of structure in the distribution, the effective dimensionality of the distribution, and the applicability of higher-level machine learning and multivariate statistical tools. We consider this fundamental recovery problem in the regime where the number of samples is comparable, or even sublinear in the dimensionality of the distribution in question. First, we propose a theoretically optimal and computationally efficient algorithm for recovering the moments of the eigenvalues of the population covariance matrix. We then leverage this accurate moment recovery, via a Wasserstein distance argument, to show that the vector of eigenvalues can be accurately recovered. We provide finite--sample bounds on the expected error of the recovered eigenvalues, which imply that our estimator is asymptotically consistent as the dimensionality of the distribution and sample size tend towards infinity, even in the sublinear sample regime where the ratio of the sample size to the dimensionality tends to zero. In addition to our theoretical results, we show that our approach performs well in practice for a broad range of distributions and sample sizes.
12 citations
TL;DR: In this article, the leading exponents of the eigenvalues of matrices or matrix polynomials over the field of Puiseux series with the tropical analogues of eigen values are derived.
12 citations
TL;DR: In this article, the authors consider the product of independent spherical ensembles and show that the empirical spectral distribution of the product converges, with probability one, to a non-random distribution.
Abstract: We consider the product of independent spherical ensembles. By the special structure of eigenvalues as a rotation-invariant determinant point process, we show that the empirical spectral distribution of the product converges, with probability one, to a non-random distribution. And the limiting eigenvalue distribution is a power of spherical law. We also present an interesting correspondence between the eigenvalues of three classes of random matrix ensembles and zeros of random polynomials.
12 citations
TL;DR: Simulation results illustrate that the generalized algorithm is able to not only maintain the excellent properties of TSA-SPA in terms of convergence and robustness, but also consider various parameter changes effectively in large-scale power systems.
Abstract: The recently-developed Two-sided Arnoldi and Sensitive Pole Algorithm (TSA-SPA) is effective and robust in computing the most sensitive eigenvalues with respect to control parameter changes in large-scale power systems. This paper extends the TSA-SPA to handle different system parameters, including control, system operating and network parameters. The proposed algorithm makes use of perturbation in reduced matrix obtained from Arnoldi/TSA method through linearization and successfully avoids the need for TSA-SPA to formulate the whole state matrix of the system and to explicitly calculate the elements' variations in system state matrix. A new deflation method is also proposed and adopted in the generalized algorithm to find other sensitive eigenvalues. Simulation results illustrate that the generalized algorithm is able to not only maintain the excellent properties of TSA-SPA in terms of convergence and robustness, but also consider various parameter changes effectively in large-scale power systems.
8 citations
01 Sep 2016
TL;DR: In this article, only the unstable eigenvalues and eigen values which are close to the imaginary axis of the complex eigenvalue plane are assigned due to their predominant influence on the response of the system.
Abstract: Generally, a mechanical system always has symmetric system matrices. Nevertheless, when some non-conservative forces are included, such as friction and aerodynamic force, the symmetry of the stiffness matrix or damping matrix or both violated. Moreover, such an asymmetric system is prone to dynamic instability. Distinct from the eigenvalue assignment for symmetric systems to reassign their natural frequencies, the main purpose of eigenvalue assignment for asymmetric systems is to shift the unstable eigenvalues to the stable region. In this research, only the unstable eigenvalues and eigenvalues which are close to the imaginary axis of the complex eigenvalue plane are assigned due to their predominant influence on the response of the system. The remaining eigenvalues remain unchanged. The state-feedback control gains are obtained by solving the constrained linear least-squares problems in which the linear system matrices are deduced based on the receptance method and the constraint is derived from the unobservability condition. The numerical simulation results demonstrate that the proposed method is capable of partially assigning those targeted eigenvalues of the system for stabilisation.
TL;DR: In this article, the authors studied the Dirac operator on a 3-sphere equipped with Riemannian metric and derived explicit perturbation formulae for the two eigenvalues closest to zero, taking account of the second variations.
Abstract: We study the (massless) Dirac operator on a 3-sphere equipped with Riemannian metric. For the standard metric the spectrum is known. In particular, the eigenvalues closest to zero are the two double eigenvalues +3/2 and -3/2. Our aim is to analyse the behaviour of eigenvalues when the metric is perturbed in an arbitrary smooth fashion from the standard one. We derive explicit perturbation formulae for the two eigenvalues closest to zero, taking account of the second variations. Note that these eigenvalues remain double eigenvalues under perturbations of the metric: they cannot split because of a particular symmetry of the Dirac operator in dimension three (it commutes with the antilinear operator of charge conjugation). Our perturbation formulae show that in the first approximation our two eigenvalues maintain symmetry about zero and are completely determined by the increment of Riemannian volume. Spectral asymmetry is observed only in the second approximation of the perturbation process. As an example we consider a special family of metrics, the so-called generalized Berger spheres, for which the eigenvalues can be evaluated explicitly.
01 Sep 2016
TL;DR: The network resonance method is investigated, and it is shown that the method can estimate eigenvalues of the scaled Laplacian matrix of the entire network through observations of oscillation dynamics even if observable nodes are restricted to a part of network.
Abstract: Eigenvalues of the Laplacian matrix play an important role in characterizing structural and dynamical properties of networks In the procedure for calculating eigenvalues of the Laplacian matrix, we need to get the Laplacian matrix that represents structures of the network Since the actual structure of networks and the strength of links are difficult to know, it is difficult to determine elements of the Laplacian matrix To solve this problem, our previous study proposed a concept of the network resonance method, which is for estimating eigenvalues of the scaled Laplacian matrix using resonance of oscillation dynamics on networks This method does not need a priori information about the network structure In this research, we investigate feasibility of the network resonance method, and show that the method can estimate eigenvalues of the scaled Laplacian matrix of the entire network through observations of oscillation dynamics even if observable nodes are restricted to a part of network
TL;DR: In this article, bounds on the absolute values of the left and right eigenvalues of quaternionic matrix polynomials are devised and illustrated for the matrix p norm, where p = 1, 2, \infty, F.
Abstract: Localization theorems are discussed for the left and right eigenvalues of block quaternionic matrices. Basic definitions of the left and right eigenvalues of quaternionic matrices are extended to quaternionic matrix polynomials. Furthermore, bounds on the absolute values of the left and right eigenvalues of quaternionic matrix polynomials are devised and illustrated for the matrix p norm, where \({p = 1, 2, \infty, F}\). The above generalizes the bounds on the absolute values of the eigenvalues of complex matrix polynomials, which give sharper bounds to the bounds developed in [LAA, 358, pp. 5–22 2003] for the case of 1, 2, and \({\infty}\) matrix norms.
Abstract: This paper is a continuation of our recent work on the localization of the eigenvalues of matrices. We give new bounds for the real and imaginary parts of the eigenvalues of matrices. Applications to the localization of the zeros of polynomials are also given.
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TL;DR: The number of distinct singular values of a matrix after perturbation is investigated and some refined bounds that only rely on the information of the matrix in question and the low-rank update are provided are provided.
Abstract: The eigenproblem of low-rank updated matrices are of crucial importance in many applications. Recently, an upper bound on the number of distinct eigenvalues of a perturbed matrix was established. The result can be applied to estimate the number of Krylov iterations required for solving a perturbed linear system. In this paper, we revisit this problem and establish some refined bounds. Some {\it a prior} upper bounds that only rely on the information of the matrix in question and the low-rank update are provided. Examples show the superiority of our theoretical results over the existing ones. The number of distinct singular values of a matrix after perturbation is also investigated.
18 Apr 2016
TL;DR: In this paper, a Sturm-liouville equation on finite number disjoint intervals together with transmission conditions at the points of interaction is considered, and the eigenvalues parameter appears linearly in the boundary conditions.
Abstract: In this work we consider a Sturm-liouville equation on finite number disjoint intervals together with transmission conditions at the points of interaction. Moreover, the eigenvalues parameter appears linearly in the boundary conditions. We investigate some properties of eigenvalues and eigenfunctions of the considered problem and derive asymptotic formulas for the eigenvalues.
TL;DR: In this article, the spectral properties of large, time-lagged correlation matrices were analyzed using the tools of random matrix theory, and the spectral features of the large lagged correlation matrix were analyzed as a function of the depth of the time-lag.
Abstract: We analyze the spectral properties of large, time-lagged correlation matrices using the tools of random matrix theory. We compare predictions of the one-dimensional spectra, based on approaches already proposed in the literature. Employing the methods of free random variables and diagrammatic techniques, we solve a general random matrix problem, namely the spectrum of a matrix $\frac{1}{T}XAX^{\dagger}$, where $X$ is an $N\times T$ Gaussian random matrix and $A$ is \textit{any} $T\times T$, not necessarily symmetric (Hermitian) matrix. As a particular application, we present the spectral features of the large lagged correlation matrices as a function of the depth of the time-lag. We also analyze the properties of left and right eigenvector correlations for the time-lagged matrices. We positively verify our results by the numerical simulations.
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TL;DR: In this article, an improved upper bound for the number of distinct eigenvalues of a matrix after perturbation is presented, and some results based on the improved estimate are presented.
Abstract: An upper bound for the number of distinct eigenvalues of a perturbed matrix has been recently established by P. E. Farrell [1, Theorem 1.3]. The estimate is the central result in Farrell's work and can be applied to estimate the number of Krylov iterations required for solving a perturbed linear system. In this paper, we present an improved upper bound for the number of distinct eigenvalues of a matrix after perturbation. Furthermore, some results based on the improved estimate are presented.
TL;DR: In this article, the authors consider a matrix A with eigenvalues λ and define a homogeneous system of linear equation (A-λI) v = 0, where V is the characteristic polynomial whose roots are equal to λ.
Abstract: Eigenvalues are special sets of scalars associated with a given matrix. In other words for a given matrix A, if there exist a non-zero vector V such that, AV= λV for some scalar λ, then λ is called the eigenvalue of matrix A with corresponding eigenvector V. The set of all nxm matrices over a field F is denoted by Mnm (F). If m = n, then the matrices are square, and which is denoted by Mn (F). We omit the field F = C and in this case we simply write Mnm or Mn as appropriate. Each square matrix AϵMnm has a value in R associated with it and it is called its determinant which is use full for solving a system of linear equation and it is denoted by det (A). Consider a square matrix AϵMn with eigenvalues λ, and then by definition the eigenvectors of A satisfy the equation, AV = λV, where v={v1, v2, v3…………vn}. That is, AV=λV is equivalent to the homogeneous system of linear equation (A-λI) v=0. This homogeneous system can be written compactly as (A-λI) V = 0 and from Cramer’s rule, we know that a linear system of equation has a non-trivial solution if and only if its determinant is zero, so the solution λ is given by det (A-λI) =0. This is called the characteristic equation of matrix A and the left hand side of the characteristic equation is the characteristic polynomial whose roots are equals to λ.
TL;DR: The error of calculating the higher eigenvalues does not grow, but it also vanishes as eigen values tend to infinity, and the proposed method gives good estimates of eigenfunctions corresponding to high eigen Values.
TL;DR: In this article, the fine spectrum of the matrix operator (Δ2uvw)t on l 1 was determined, which is defined generalized upper triangular triple band matrix on l1.
Abstract: In this work, we determine the fine spectrum of the matrix operator (Δ2uvw)t
which is defined generalized upper triangular triple band matrix on l1. Also,
we give the approximate point spectrum, defect spectrum and compression
spectrum of the matrix operator (Δ2uvw)t on l1.
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TL;DR: The setting of inhomogeneous random graphs is used to present an extension of the Kato-Temple inequality for bounding eigenvalues and it is proved under mild assumptions that the top eigen values of the adjacency matrix concentrate around the corresponding top eigens of the edge probability matrix with high probability, even when the latter have multiplicity.
Abstract: In this paper we use the setting of inhomogeneous random graphs to present an extension of the Kato-Temple inequality for bounding eigenvalues. We prove under mild assumptions that the top eigenvalues (i.e. largest in magnitude and ordered) of the adjacency matrix concentrate around the corresponding top eigenvalues of the edge probability matrix with high probability, even when the latter have multiplicity. The proof techniques we employ do not strictly depend upon the underlying random graph setting and are readily extended to yield high-probability bounds for perturbations of singular values of rectangular matrices in a quite general random matrix setting.
15 Jan 2016
TL;DR: In this article, a non iterative method for computing the eigenvalues of non-self-adjoint operators is developed on the basis of the theory of regularized traces.
Abstract: One can find the eigenvalues of non-selfadjoint operators only by numerical methods. The use of these methods is associated with large computational difficulties. Therefore, the development of a new method for calculating of eigenvalues of non-self-adjoint operators has great theoretical and practical interest. Non iterative method for finding of eigenvalues of perturbed self-adjoint operators is developed on the basis of the theory of regularized traces. This method is called the method of regularized traces. The linear formulas for computing of the eigenvalues of the discrete operators, which are semi-bounded from below, were found. Using them, one can compute the eigenvalues of perturbed self-adjoint operator with any their number. Note that for this computation it does not matter whether the eigenvalues with less number are known or not. Numerical calculations of eigenvalues for the spectral problems, which are generated by the equations of mathematical physics, show that for large numbers of eigenvalues the proposed formulas give more exact result than the Galerkin method. In addition, the obtained formulas allow to compute the eigenvalues of perturbed self-adjoint operator with very large number, such that the use of the Galerkin method becomes difficult. The algorithm of application of the method of regularized traces for finding of eigenvalues of the Couette spectral problem of hydrodynamic stability theory is constructed. This problem studies the stability of the flow of a tough liquid between two rotating axisymmetric cylinders to small perturbations of the basic flow. A feature of this problem is the fact that the differential operator is a matrix one. Numerical experiments have shown the high computational efficiency of the proposed algorithm of computing of the eigenvalues of the studied spectral problem. The algorithm of application of the method of regularized traces for spectral problems, which are generated by the matrix discrete operators limited from below, is constructed in the paper.
TL;DR: This paper presents a highly efficient algorithm, named EvArnoldi, for solving the large-scale eigenvalues problem, and in its basic formulation, is mathematically equivalent to ARPACK.
Abstract: Eigenvalues and eigenvectors are an essential theme in numerical linear algebra. Their study is mainly motivated by their high importance in a wide range of applications. Knowledge of eigenvalues is essential in quantum molecular science. Solutions of the Schrodinger equation for the electrons composing the molecule are the basis of electronic structure theory. Electronic eigenvalues compose the potential energy surfaces for nuclear motion. The eigenvectors allow calculation of diople transition matrix elements, the core of spectroscopy. The vibrational dynamics molecule also requires knowledge of the eigenvalues of the vibrational Hamiltonian. Typically in these problems, the dimension of Hilbert space is huge. Practically, only a small subset of eigenvalues is required. In this paper, we present a highly efficient algorithm, named EvArnoldi, for solving the large-scale eigenvalues problem. The algorithm, in its basic formulation, is mathematically equivalent to ARPACK (Sorensen, D. C. Implicitly Restart...
01 Jan 2016
TL;DR: In this article, the singular values of a Hermitian matrix C = A + B in terms of the combined list of eigenvalues of A and B are characterized by Horn-type linear inequalities.
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. Introduction. Let X be the upper right p by n- p submatrix of an n by n matrix Z, with 2p • • • > 7n and s\ > - • > sp be the singular values of Z and X, respectively. We prove that the possible pairs of sequences (sk) and (7^) are characterized by the inequalities
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TL;DR: In this paper, a rigorous derivation of asymptotic formula for the largest eigenvalues using the convergence estimation of the eigen values of a sequence of self-adjoint compact operators of perturbations resulting from the presence of small inhomogeneities is provided.
Abstract: In this paper, we provide a rigorous derivation of asymptotic formula for the largest eigenvalues using the convergence estimation of the eigenvalues of a sequence of self-adjoint compact operators of perturbations resulting from the presence of small inhomogeneities.
01 Jan 2016
TL;DR: It is claimed that £1.2bn has been invested in the construction of this building in the last five years solely on the basis of its potential as a result of leaks from the USGS.
Abstract: Матрицы, имеющие кратные собственные числа, рассматривались ранее в основном с теоретической точки зрения. Однако в последнее время такие вырожденные матрицы представляют и практический интерес, поскольку они возникают в задачах квантовой механики, ядерной физики, оптики, динамики механических систем. В данной работе рассматривается квадратная матрица, элементы которой суть линейные функции параметра. Предлагается метод, позволяющий за конечное число алгебраических операций над элементами матрицы построить полином, корни которого — значения параметра, соответствующие кратным собственным числам матрицы. Имеется возможность обобщения предложенного метода для матриц, элементы которых являются полиномами от параметра степени выше первой. Приводится численный пример, иллюстрирующий работу этого метода. Библиогр. 16 назв. Ключевые слова: кронекеровское произведение, метод Леверье, суммы Ньютона.
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TL;DR: In this paper, the authors quantify the distribution of real eigenvalues for products of finite size real Gaussian matrices by giving an explicit Pfaffian formula for the probability that there are exactly $k$ real eigvalues as a determinant with entries involving particular Meijer G-functions.
Abstract: Random matrices formed from i.i.d. standard real Gaussian entries have the feature that the expected number of real eigenvalues is non-zero. This property persists for products of such matrices, independently chosen, and moreover it is known that as the number of matrices in the product tends to infinity, the probability that all eigenvalues are real tends to unity. We quantify the distribution of the number of real eigenvalues for products of finite size real Gaussian matrices by giving an explicit Pfaffian formula for the probability that there are exactly $k$ real eigenvalues as a determinant with entries involving particular Meijer G-functions. We also compute the explicit form of the Pfaffian correlation kernel for the correlation between real eigenvalues, and the correlation between complex eigenvalues. The simplest example of these - the eigenvalue density of the real eigenvalues - gives by integration the expected number of real eigenvalues. Our ability to perform these calculations relies on the construction of certain skew-orthogonal polynomials in the complex plane, the computation of which is carried out using their relationship to particular random matrix averages.
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TL;DR: In this paper, the Steklov eigenvalues of the Laplace operator were considered as limiting Neumann eigen values in a problem of mass concentration at the boundary of a ball.
Abstract: We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of mass concentration at the boundary of a ball. We discuss the asymptotic behavior of the Neumann eigenvalues and find explicit formulas for their derivatives at the limiting problem. We deduce that the Neumann eigenvalues have a monotone behavior in the limit and that Steklov eigenvalues locally minimize the Neumann eigenvalues.