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Showing papers on "Spectrum of a matrix published in 2007"


Journal ArticleDOI
TL;DR: In this paper, a detailed derivation of the Pfaffian integration theorem for real eigenvalues in the spectrum of an n×n real asymmetric matrix drawn at random from Ginibre's Orthogonal Ensemble is presented.
Abstract: In the recent publication (E. Kanzieper and G. Akemann in Phys. Rev. Lett. 95:230201, 2005), an exact solution was reported for the probability p n,k to find exactly k real eigenvalues in the spectrum of an n×n real asymmetric matrix drawn at random from Ginibre’s Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof of the Pfaffian integration theorem, the key ingredient of our analysis of the statistics of real eigenvalues in the GinOE. We also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a GinOE matrix restricted to have exactly k real eigenvalues. In the particular case of k=0, all correlation functions of complex eigenvalues are determined.

84 citations


Journal Article
TL;DR: In this paper, a model operator associated with the energy operator of a system of three non conserved number of particles is considered, and the essential spectrum of the operator is described by the spectrum of a family of the generalized Friedrichs model.
Abstract: A model operator associated with the energy operator of a system of three non conserved number of particles is considered. The essential spectrum of the operator is described by the spectrum of a family of the generalized Friedrichs model. It is shown that there are infinitely many eigenvalues lying below the bottom of the essential spectrum, if a generalized Friedrichs model has a zero energy resonance.

34 citations


Journal ArticleDOI
TL;DR: In this paper, a model operator H associated to a system of three particles on the threedimensional lattice ℤ3 that interact via nonlocal pair potentials is studied.
Abstract: A model operator H associated to a system of three particles on the threedimensional lattice ℤ3 that interact via nonlocal pair potentials is studied. The following results are established. (i) The operator H has infinitely many eigenvalues lying below the bottom of the essential spectrum and accumulating at this point if both the Friedrichs model operators $$h_{\mu _\alpha } $$ (0), α = 1, 2, have threshold resonances. (ii) The operator H has finitely many eigenvalues lying outside the essential spectrum if at least one of the operators $$h_{\mu _\alpha } $$ (0), α = 1, 2, has a threshold eigenvalue.

27 citations


Journal ArticleDOI
TL;DR: The sampling method is extended to compute eigenvalues of singular non-self-adjoint Sturm-Liouville problems in the presence of a continuous spectrum and a new sampling formula is developed for its reconstruction.
Abstract: In this work we extend the sampling method to compute eigenvalues of singular non-self-adjoint Sturm-Liouville problems in the presence of a continuous spectrum. We first show that the characteristic function, whose zeros are the eigenvalues, belongs to a Hardy space, and then develop a new sampling formula for its reconstruction. We estimate the truncation error, obtain computable error bounds, and test the method with a few numerical experiments.

19 citations


Journal ArticleDOI
TL;DR: A key feature of the method that leads to a fast algorithm is to combine generating functions with the Laplace transform to compute explicitly the entries of the matrix without numerical integration.

13 citations


Journal ArticleDOI
TL;DR: In this paper, the spectrum of eigenvalues of the correlation matrix is studied under a model where most of the true eigen values are zero and the parameters are non-stationary.
Abstract: The exact meaning of the noise spectrum of eigenvalues of the correlation matrix is discussed. In order to better understand the possible phenomena behind the observed noise, the spectrum of eigenvalues of the correlation matrix is studied under a model where most of the true eigenvalues are zero and the parameters are non-stationary. The results are compared with real observation of Brazilian assets, suggesting that, although the non-stationarity seems to be an important aspect of the problem, partially explaining some of the eigenvalues as well as part of the kurtosis of the assets, it cannot, by itself, provide all the corrections needed to make the proposed model fit the data perfectly.

12 citations



Book ChapterDOI
04 Sep 2007
TL;DR: In this paper a method is presented to fair the limit surface of a subdivision algorithm around an extraordinary point with the dominant, subdominant and subsub-dominant eigenvalues.
Abstract: In this paper a method is presented to fair the limit surface of a subdivision algorithm around an extraordinary point. The eigenvalues and eigenvectors of the subdivision matrix determine the continuity and shape of the limit surface. The dominant, subdominant and subsub-dominant eigenvalues should satisfy linear and quadratic equality- and inequality-constraints to guarantee continuous normal and bounded curvature globally. The remaining eigenvalues need only satisfy linear inequality-constraints. In general, except for the dominant eigenvalue, all eigenvalues can be used to optimize the shape of the limit surface with our method.

9 citations


Journal ArticleDOI
TL;DR: This paper bounds the smallest nonzero eigenvalue, which serves as an indicator of how difficult it is to correctlycompute the desired null space of the approximate alignment matrix.
Abstract: The alignment algorithm of Zhang and Zha is an effective method recently proposed for nonlinear manifold learning (or dimensionality reduction). By first computing local coordinates of a data set, it constructs an alignment matrix from which a global coordinate is obtained from its null space. In practice, the local coordinates can only be constructed approximately and so is the alignment matrix. This together with roundoff errors requires that we compute the the eigenspace associated with a few smallest eigenvalues of an approximate alignment matrix. For this purpose, it is important to know the first nonzero eigenvalue of the alignment matrix or a lower bound in order to computationally separate the null space. This paper bounds the smallest nonzero eigenvalue, which serves as an indicator of how difficult it is to correctly compute the desired null space of the approximate alignment matrix.

9 citations


Patent
Teruyoshi Washizawa1
04 Jun 2007
TL;DR: In this article, a variance-covariance matrix of a matrix having a combination of multivariate data and objective variables is obtained, and multiple eigenvalues and their corresponding eigenvectors are calculated by eigenvalue decomposition of the variance covariance matrix.
Abstract: A variance-covariance matrix of a matrix having a combination of multivariate data and objective variables is obtained, and multiple eigenvalues and their corresponding eigenvectors are calculated by eigenvalue decomposition of the variance-covariance matrix. Accumulated contributions are calculated from the multiple eigenvalues in descending order of absolute value of the eigenvalues. Regression coefficients are calculated from eigenvalues and eigenvectors that correspond to accumulated contributions that exceed a predetermined value.

7 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the asymptotic behavior of the Steklov eigenvalues of the pLaplacian and derived lower and upper bounds of a Weyl-type expansion of the function.
Abstract: In this paper we study the asymptotic behavior of the Steklov eigenvalues of the pLaplacian. We show the existence of lower and upper bounds of a Weyl-type expansion of the function N( ) which counts the number of eigenvalues less than or equal to , and we derive from them asymptotic bounds for the eigenvalues.

Proceedings ArticleDOI
01 Dec 2007
TL;DR: It is shown that the feedback U = [N, A + AT] + p[AT, A], N diagonal, rho > 0 allows to solve the diagonalization problem under the assumption that the to be diagonalized matrix has real spectrum.
Abstract: The present paper deals with the problem of diagonalizing matrices using a control system of the form A = [U, A], where [U, A] = UA - AU and A, U are real matrices. It is shown that the feedback U = [N, A + AT] + p[AT, A], N diagonal, rho > 0 allows to solve the diagonalization problem under the assumption that the to be diagonalized matrix has real spectrum. Moreover, in the case of a complex spectrum, the feedback allows to check if a matrix is stable or to compute all eigenvalues of a matrix or roots of a polynomial.

Journal ArticleDOI
TL;DR: The classical Brauer-Ostrowski Theorem gives a localization of the spectrum of a matrix by a union of Cassini ovals as mentioned in this paper, which is a result for operator matrices.
Abstract: The classical Brauer-Ostrowski Theorem gives a localization of the spectrum of a matrix by a union of Cassini ovals. In this paper we prove a corresponding result for operator matrices.

Journal ArticleDOI
TL;DR: In this article, the authors considered a bound that relates the distance between X and Y to the eigenvalues of the normalized Laplacian matrix for G,t he volumes ofX and Y, and the volumes of their complements.
Abstract: Let G be a connected graph, and let X and Y be subsets of its vertex set. A previously published bound is considered that relates the distance between X and Y to the eigenvalues of the normalized Laplacian matrix for G ,t he volumes ofX and Y , and the volumes of their complements. A counterexample is given to the bound, and then a corrected version of the bound is provided.

Journal ArticleDOI
TL;DR: In this article, the authors consider the self-adjoint operator of a generalized Friedrichs model whose essential spectrum may contain lacunas and obtain a formula for the number of eigenvalues lying on an arbitrary interval outside the essential spectrum.
Abstract: We consider the self-adjoint operator of a generalized Friedrichs model whose essential spectrum may contain lacunas. We obtain a formula for the number of eigenvalues lying on an arbitrary interval outside the essential spectrum of this operator. We find a sufficient condition for the discrete spectrum to be finite. Applying the formula for the number of eigenvalues, we show that there exist an infinite number of eigenvalues on the lacuna for a particular Friedrichs model and obtain the asymptotics for the number of eigenvalues.

Proceedings ArticleDOI
TL;DR: In this paper, it was shown that a bounded linear operator T acting on a complex Banach space X has SVEP if there is no analytic function f : U → X defined on a nonempty open subset U of the complex plane such that f 6 ≥ 0 and (T − z)f(z) ≡ 0 (z ∈ U).
Abstract: The single-valued extension property (SVEP) plays an important role in the local spectral theory. A bounded linear operator T acting on a complex Banach space X is said to have SVEP if there is no analytic function f : U → X defined on a nonempty open subset U of the complex plane such that f 6≡ 0 and (T − z)f(z) ≡ 0 (z ∈ U). By definition, every operator without SVEP has a nonempty open set consisting of eigenvalues. So if the point spectrum of an operator has empty interior, then the operator has automatically SVEP. On the other hand, it is easy to construct an operator T with SVEP such that the interior of the point spectrum σp(T ) is nonempty. As a simple example, see [LN], p. 15, let D := {z ∈ C; |z| < 1} be the open unit disc, let X be the Banach space of all bounded complex-valued functions on D with the supremum norm and let T ∈ B(X) be the operator of multiplication by the independent variable z. It is easy to see that T has SVEP and σp(T ) = D. Moreover, T is decomposable. Note that the Banach space in the last example is non-separable. At first glance it seems that a separable Banach space is too ”small” for the existence of an operator T with SVEP and with nonempty interior of the point spectrum. This motivates the following question which was raised at the Workshop on Operator Theory in Warsaw, 2004.

Proceedings ArticleDOI
01 Nov 2007
TL;DR: In this article, the authors derived new simplified analytical cumulative density functions for the eigenvalues of complex noncentral Wishart matrix of size 2 times 2 such distributions are often encountered in multiple input multiple output (MIMO) Ricean channels.
Abstract: This paper derives new simplified analytical cumulative density functions for the eigenvalues of complex noncentral Wishart matrix of size 2times2 Such distributions are often encountered in multiple input multiple output (MIMO) Ricean channels The results derived herein are general for any arbitrary non-centrality matrix, and they account the cases having identical or non-identical eigenvalues of the underlying non-centrality matrix When compared to the generalized distribution recently found in literature which only treated the case of non-identical eigenvalue of the non-centrality matrix, the expressions presented in this paper exhibit a larger parameters range to which the numerical calculation remains computationally viable

Journal ArticleDOI
TL;DR: A procedure for detecting the extreme eigenvalues that relies on the relationship between the acceleration rate of polynomial acceleration iteration and the norm of the matrix via the spectral theorem, applicable to normal matrices is described.
Abstract: For a self-adjoint linear operator with a discrete spectrum or a Hermitian matrix, the ''extreme'' eigenvalues define the boundaries of clusters in the spectrum of real eigenvalues. The outer extreme ones are the largest and the smallest eigenvalues. If there are extended intervals in the spectrum in which no eigenvalues are present, the eigenvalues bounding these gaps are the inner extreme eigenvalues. We will describe a procedure for detecting the extreme eigenvalues that relies on the relationship between the acceleration rate of polynomial acceleration iteration and the norm of the matrix via the spectral theorem, applicable to normal matrices. The strategy makes use of the fast growth rate of Chebyshev polynomials to distinguish ranges in the spectrum of the matrix which are devoid of eigenvalues. The method is numerically stable with regard to the dimension of the matrix problem and is thus capable of handling matrices of large dimension. The overall computational cost is quadratic in the size of a dense matrix; linear in the size of a sparse matrix. We verify computationally that the algorithm is accurate and efficient, even on large matrices.

Journal ArticleDOI
TL;DR: This paper proposes a numerical method to verify for nearly multiple eigenvalues of a Hermitian matrix not being strictly multiple Eigenvalues, and suggests several methods for enclosing multiple and nearly multipleEigenvalues.

Journal ArticleDOI
TL;DR: In this article, the authors present a technique to efficiently calculate a projection without knowledge of the spectrum, which requires only few matrix-matrix products and inversions, which for some classes of matrices, like the $${\mathcal{H}}-matrices, can be computed in almost linear complexity.
Abstract: To efficiently calculate only part of the spectrum of a matrix, one can use a projection onto a suitable subspace. In this work, we present a technique to efficiently calculate such a projection without knowledge of the spectrum. The technique requires only few matrix–matrix products and inversions, which for some classes of matrices, like the $${\mathcal{H}}$$-matrices, can be computed in almost linear complexity.

Dissertation
01 Oct 2007
TL;DR: In this article, the authors consider the quadratic eigenvalue problem with singular leading and trailing coefficients and present two types of algorithms: linearization and Householder reflectors.
Abstract: In this thesis we consider algorithms for solving the quadratic eigenvalue problem, (lambda^2*A_2 + lambda*A_1 + A_0)x=0 when the leading or trailing coefficient matrices are singular. In a finite element discretization this corresponds to the mass or stiffness matrices being singular and reflects modes of vibration (or eigenvalues) at zero or ``infinity''. We are interested in deflation procedures that enable us to utilize knowledge of the presence of these (or any) eigenvalues to reduce the overall cost in computing the remaining eigenvalues and eigenvectors of interest. We first give an introduction to the quadratic eigenvalue problem and explain how it can be solved by a process called linearization. We present two types of algorithms, firstly a modification of an algorithm published by Kublanovskaya, Mikhailov, and Khazanov in the 1970s that has recently been translated into English. Using these ideas we present algorithms that are able to reduce the size of the problem by ``deflating'' infinite and zero eigenvalues that arise when the mass or stiffness matrix (or both) are singular. Secondly we look at methods that deflate zero and infinite eigenvalues by the use of Householder reflectors; this requires a basis for the null space of the mass or stiffness matrix (or both), so we also summarize various decompositions that can be used to give this information. We consider different applications that yield a quadratic eigenvalue problem with singular leading and trailing coefficients and after testing the implementations of the algorithms on some of these problems we comment on their stability.

Journal ArticleDOI
TL;DR: The potential of the two-dimensional discrete Schrödinger equation can be reconstructed from a part of its spectrum and prescribed symmetry conditions of the basis eigenfunctions, which provides the natural initial data for the corresponding missing eigenvalues.
Abstract: The potential of the two-dimensional discrete Schrodinger equation can be reconstructed from a part of its spectrum and prescribed symmetry conditions of the basis eigenfunctions. The discrete potential along with the missing eigenvalues is found by solving a polynomial system of equations, which is derived and solved using the REDUCE computer algebra system. To ensure the convergence of the iterative process implemented in the Numeric package in REDUCE, proper initial data must be specified. The prescribed eigenvalues are perturbed original eigenvalues corresponding to the zero discrete potential. The original eigenvalues provide the natural initial data for the corresponding missing eigenvalues. In the case of a square, there are many multiple eigenvalues among the original eigenvalues. The direct application of the variant of the Newton method implemented in the Numeric package in REDUCE is impossible in the case of multiple initial data. A modification of the method proposed earlier for calculating the discrete potential of the two-dimensional discrete Schrodinger equation in a square is illustrated by an example.