Showing papers on "Spectrum of a matrix published in 2008"
TL;DR: In this paper, the authors established the limiting distributions of the extreme sample eigenvalues associated to spike eigen values when the population and the sample sizes become large, and provided a central limit theorem on random sesquilinear forms.
Abstract: In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). This model is proposed by Johnstone to cope with empirical findings on various data sets. The question is to quantify the effect of the perturbation caused by the spike eigenvalues. A recent work by Baik and Silverstein establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. This paper establishes the limiting distributions of these extreme sample eigenvalues. As another important result of the paper, we provide a central limit theorem on random sesquilinear forms.
218 citations
TL;DR: This paper derives a new test matrix for S-parameter models which is only half the size of the Hamiltonian matrix, leading to savings in the eigenvalue computation time by a factor of nearly eight.
Abstract: Rational models must be passive in order to ensure stable time domain simulations. The assessment of passivity properties is usually done via a Hamiltonian matrix that is associated with the state-space model, allowing precise characterization of passivity violations from its imaginary eigenvalues. The calculation of eigenvalues can be time consuming for large models as the matrix size is equal to twice the number of model states. In this paper, we derive for S-parameter models a new test matrix which is only half the size of the Hamiltonian matrix. This leads to savings in the eigenvalue computation time by a factor of nearly eight. The new test matrix takes into account that the model is symmetrical, in pole-residue form. Its application is demonstrated by three examples: a microwave filter, a package, and a synthetic model.
76 citations
TL;DR: In this article, the spectrum of a periodic self-adjoint differential operator on the real axis perturbed by a small localized non-selfadjoint operator was studied and it was shown that the continuous spectrum does not depend on the perturbation.
Abstract: The paper deals with the spectrum of a periodic self-adjoint differential operator on the real axis perturbed by a small localized non-self-adjoint operator. We show that the continuous spectrum does not depend on the perturbation, the residual spectrum is empty, and the point spectrum has no finite accumulation points. We study the problem of the existence of eigenvalues embedded in the continuous spectrum, obtain necessary and sufficient conditions for the existence of eigenvalues, construct asymptotic expansions of the eigenvalues and corresponding eigenfunctions and consider some examples.
41 citations
08 Dec 2008
TL;DR: This paper presents some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of Wishart matrices, using the tensor operator T (.), which was first introduced in .
Abstract: The distribution of the eigenvalues of Wishart matrices and Gaussian quadratic forms is of great interest in communication theory, especially in relation to multiple-input multiple-output (MIMO) systems. In this paper we present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of Wishart matrices, using the tensor operator T (.), which was first introduced in . We obtain both the joint probability distribution function (p.d.f.) of the eigenvalues and the expectation of arbitrary functions of the eigenvalues, including the moments, for the case of both ordered and unordered eigenvalues. These expressions are extremely compact and easy to handle. Application to MIMO systems are discussed.
39 citations
TL;DR: In this article, the authors define two entropic measures of nonclassical correlation of a multipartite quantum system, one is defined as the minimum uncertainty about a joint system after collecting outcomes of particular local measurements and the other is defined by taking the maximum over all local systems about the minimum distance between a genuine set and a mimic set of eigenvalues of a reduced density matrix of a local system.
Abstract: There is a commonly recognized paradigm in which a multipartite quantum system described by a density matrix having no product eigenbasis is considered to possess nonclassical correlation. Supporting this paradigm, we define two entropic measures of nonclassical correlation of a multipartite quantum system. One is defined as the minimum uncertainty about a joint system after we collect outcomes of particular local measurements. The other is defined by taking the maximum over all local systems about the minimum distance between a genuine set and a mimic set of eigenvalues of a reduced density matrix of a local system. The latter measure is based on an artificial game to create mimic eigenvalues of a reduced density matrix of a local system from eigenvalues of a density matrix of a global system. Numerical computation of these measures for several examples is performed.
30 citations
01 Jan 2008
TL;DR: In this article, the maximum eigenvalues of the reciprocal distance matrix and the reverse Wiener matrix of a connected graph were investigated and the Nordhaus-Gaddum-type results for them were obtained.
Abstract: We report some properties of the maximum eigenvalues of the reciprocal distance matrix and the reverse Wiener matrix of a connected graph, in particular, various lower and upper bounds, and the Nordhaus-Gaddum-type results for them. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem 108: 858 - 864, 2008
26 citations
Journal Article•
TL;DR: In this article, the eigenvalues of Schrodinger operator with a weight on compact Riernainnian manifolds with boundary (possibly empty) and prove a general inequality for them.
Abstract: In this paper we consider eigenvalues of Schrodinger operator with a weight on compact Riernainnian manifolds with boundary (possibly empty) and prove a general inequality for them. By using this inequality, we study eigenvalues of Schrodinger operator with a weight on compact domains in a unit sphere, a complex projective space and a minimal submanifold in a Euclidean space. We also study the same problem on closed minimal submanifolds in a sphere, compact homogeneous space and closed complex hypersurfaces in a complex projective space. We give explict bound for the (k + 1)-th eigenvalue of the Schrodinger operator on such objects in terms of its first k eigenvalues. Our results generalize many previous estimates on eigenvalues of the Laplacian.
22 citations
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TL;DR: In this article, it was shown that the properly rescaled eigenvalues of a continuum Anderson Hamiltonian are distributed as a Poisson point process with intensity measure given by the density of states.
Abstract: We show absence of energy levels repulsion for the eigenvalues of random Schrodinger operators in the continuum. We prove that, in the localization region at the bottom of the spectrum, the properly rescaled eigenvalues of a continuum Anderson Hamiltonian are distributed as a Poisson point process with intensity measure given by the density of states. We also obtain simplicity of the eigenvalues. We derive a Minami estimate for continuum Anderson Hamiltonians. We also give a simple and transparent proof of Minami's estimate for the (discrete) Anderson model.
17 citations
TL;DR: In this paper, the authors give a sufficient condition for a one-dimensional Schrodinger operator with point δ-interactions, which contain m points of interaction with negative intensities, to have at least m negative eigenvalues.
Abstract: We give a sufficient condition for a one-dimensional Schrodinger operator with point δ-interactions, which contain m points of interaction with negative intensities, to have at least m negative eigenvalues.
16 citations
TL;DR: In this article, the consistency and asymptotic efficiency of an estimator of the drift matrix, $F$, of Ornstein-Uhlenbeck processes that are not necessarily stable are investigated.
Abstract: In this paper, we investigate the consistency and asymptotic efficiency of an estimator of the drift matrix, $F$, of Ornstein-Uhlenbeck processes that are not necessarily stable. We consider all the cases. (1) The eigenvalues of $F$ are in the right half space (i.e., eigenvalues with positive real parts). In this case the process grows exponentially fast. (2) The eigenvalues of $F$ are on the left half space (i.e., the eigenvalues with negative or zero real parts). The process where all eigenvalues of $F$ have negative real parts is called a stable process and has a unique invariant (i.e., stationary) distribution. In this case the process does not grow. When the eigenvalues of $F$ have zero real parts (i.e., the case of zero eigenvalues and purely imaginary eigenvalues) the process grows polynomially fast. Considering (1) and (2) separately, we first show that an estimator, $\hat{F}$, of $F$ is consistent. We then combine them to present results for the general Ornstein-Uhlenbeck processes. We adopt similar procedure to show the asymptotic efficiency of the estimator.
10 citations
Patent•
17 Jul 2008TL;DR: In this article, the largest eigenvalue is determined among eigenvalues corresponding to a correlation matrix indicating correlations between a plurality of channels receiving incoming radar waves from an object that reflects a radar wave as a reference eigen value λ 1.
Abstract: A largest eigenvalue is determined among eigenvalues corresponding to a correlation matrix indicating correlations between a plurality of channels receiving incoming radar waves from an object that reflects a radar wave as a reference eigenvalue λ1. A ratio Rλi (=10 log 10(λi/λ1)) is calculated of each eigenvalue λ2 to λN to the reference eigenvalue λ1. Eigenvalues among the reference eigenvalue λ1 and the eigenvalues λ2 to λN of which the eigenvalue ratio Rλi is greater than a noise threshold TH are identified as eigenvalues in signal space. Eigenvalues of which the eigenvalue ratio Rλi is equal to the noise threshold TH or less are identified as eigenvalues in noise space. The number of eigenvalues identified as the eigenvalues in signal space is counted as the number of arrival signals.
TL;DR: In this article, it was shown that the eigenvalues of a highly non-self-adjoint operator arise in fluid mechanics and correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent.
Abstract: We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at $\pm \infty$. We use this result to determine the asymptotic distribution of the eigenvalues and to compute some of the eigenvalues numerically. We compare these to earlier calculations by other authors.
TL;DR: In this paper, the eigenvalues of the Laplace operator in the space of functions of mean zero and having a constant (unprescribed) boundary value were investigated and the existence of an optimal domain in the class of convex sets was discussed.
Abstract: Abstract We investigate the eigenvalues of the Laplace operator in the space of functions of mean zero and having a constant (unprescribed) boundary value. The first eigenvalue of such problem lies between the first two eigenvalues of the Laplacian with homogeneous Dirichlet boundary conditions and satisfies an isoperimetric inequality: in the class of open bounded sets of prescribed measure, it becomes minimal for the union of two disjoint balls having the same radius. Existence of an optimal domain in the class of convex sets is also discussed.
TL;DR: Ndjinga et al. as discussed by the authors presented two algorithms for the computation of the matrix sign and absolute value functions, which avoid a complete diagonalization of the matrices, but require some informations regarding the eigenvalues location.
01 Jan 2008
TL;DR: The relation between the main eigenvalues and associated eigenvectors of adjacency matrix and Seidel matrix of a graph is investigated in this paper, where a simple graph with vertex set V (G) and (0, 1)-adjacency matrix A is considered.
Abstract: Let G be a simple graph with vertex set V (G) and (0, 1)-adjacency matrix A. As usual, A*(G) = J - I - 2A denotes the Seidel matrix of the graph G. The eigenvalue λ of A is said to be a main eigenvalue of G if the eigenspace e(λ) is not orthogonal to the all-1 vector e. In this paper, relations between the main eigenvalues and associated eigenvectors of adjacency matrix and Seidel matrix of a graph are investigated. .
TL;DR: In this article, loop connections of two linear systems are studied and the possible eigenvalues of a matrix of a system obtained as a result of these connections are determined, and the main result is that loop connections can be used to determine the eigenvectors of a linear system.
Abstract: In this paper, loop connections of two linear systems are studied. As the main result, the possible eigenvalues of a matrixof a system obtained as a result of these connections are determined.
TL;DR: In this paper, the authors considered a vector Sturm-Liouville operator with a potential that is unbounded below, and derived asymptotic formulas for the spectrum, which involve the eigenvalues of the matrix potential as well as the rotation velocities of the Eigenvectors.
Abstract: On the half-line, we consider a vector Sturm-Liouville operator with a potential that is unbounded below. Asymptotic formulas for the spectrum are given. These formulas involve the eigenvalues of the matrix potential as well as the “rotational velocities” of the eigenvectors.
TL;DR: In this article, the eigenvalues of the p-adic curvature transformations on buildings were studied and the maximal eigenvectors of these transformations were determined, in particular, for buildings.
Abstract: We study the eigenvalues of the p-adic curvature transformationson buildings. In particular, we determine the maximal eigenvalues ofthese transformations.
01 Jan 2008
TL;DR: In this article, the problem of designing discrete-time control systems with closed-loop eigenvalues in a prescribed region of stability is addressed, where a state feedback matrix is obtained by elementary similarity operations and using the Gerschgorin theorem.
Abstract: This paper is concerned with the problem of designing discrete-time control systems with closed-loop eigenvalues in a prescribed region of stability. First, we obtain a state feedback matrix which assigns all the eigenvalues to zero and then by elementary similarity operations and using the Gerschgorin theorem we find a state feedback which assigns the eigenvalues inside a circle with center c and radius r. This new algorithm can also be used for the placement of closed-loop eigenvalues in a specified disc in z-plane and can be employed for large-scale discrete-time linear control systems. Some illustrative examples are presented to show the advantages of this new technique.
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TL;DR: The existence of infinitely many eigenvalues below the bottom of the essential spectrum of a model operator corresponding to the energy operator of a system with non-conserved number n � 3 of particles is proved in this paper.
Abstract: A model operator H corresponding to the energy operator of a system with non-conserved number n � 3 of particles is considered The precise location and struct ure of the essential spectrum of H is described The existence of infinitely many eigenvalues below the bottom of the essential spectrum of H is proved if the generalized Friedrichs model has a virtual level at the bottom of the essential spectrum and for the number N(z) of eigenvalues below z < 0 an asymptotics established The finiteness of eigenvalues o f H below the bottom of the essential spectrum is proved if the generalized Friedrichs model has a zero eigenvalue at the bottom of its essential spectrum
TL;DR: A computational procedure for finding eigenvalues of a real matrix based on Alternate Quadrant Interlocking Factorization, a parallel direct method developed by Rao in 1994 for the solution of the general linear system Ax=b is presented.
Abstract: This paper presents a computational procedure for finding eigenvalues of a real matrix based on Alternate Quadrant Interlocking Factorization, a parallel direct method developed by Rao in 1994 for the solution of the general linear system Ax=b. The computational procedure is similar to LR algorithm as studied by Rutishauser in 1958 for finding eigenvalues of a general matrix. After a series of transformations the eigenvalues are obtained from simple 2×2 matrices derived from the main and cross diagonals of the limit matrix. A sufficient condition for the convergence of the computational procedure is proved. Numerical examples are given to demonstrate the method.
TL;DR: In this paper, the authors considered linear Schrodinger operator with double or resonant eigenvalues and derived the bound of the measure of the potentials leading to such double eigen values.
Journal Article•
TL;DR: In this article, the fundamental properties of eigenvalues for regular Sturm-Liouville problems are extended to special kind boundary value problems, which have discontinuities in the solution or its quasi-derivative at an interior point.
Abstract: Some fundamental properties of eigenvalues for regular Sturm-Liouville problems are extended to special kind boundary value problems,which have discontinuities in the solution or its quasi-derivative at an interior point.It is proved that such discontinuous problems has infinitely many eigenvalues and the eigenvalues are boundary from below.
TL;DR: In this article, the Sturm-Liouville operator on the unit interval is considered and two-sided a priori estimates of potential in terms of Dirichlet and Neumann eigenvalues and eigen values for two types of mixed boundary conditions are obtained.
Abstract: We consider the Sturm–Liouville operator on the unit interval. We obtain two-sided a priori estimates of potential in terms of Dirichlet and Neumann eigenvalues and eigenvalues for 2 types of mixed boundary conditions.
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TL;DR: In this article, Huang and So gave a complete characterization of the $2\times 2$ symplectic matrices having an infinite number of left eigenvalues, which was later improved by applying an algorithm for the resolution of equations due to De Leo et al.
Abstract: We obtain a complete characterization of the $2\times 2$ symplectic matrices having an infinite number of left eigenvalues. Previously, we give a new proof of a result from Huang and So about the number of eigenvalues of a quaternionic matrix. This is achieved by applying an algorithm for the resolution of equations due to De Leo et al.
01 Jan 2008
TL;DR: The perturbations of Nevanlinna type functions which preserve the set of zeros of this function or add to this set new points are discussed in this article, where the eigenvalues of the perturbed operator are obtained as a set of zero points of the function.
Abstract: The perturbations of Nevanlinna type functions which preserve the set of zeros of this function or add to this set new points are discussed. 1. Statement of the problem The point spectrum in the case of rank one perturbation of purely continuous spectrum may be very rich. In general, this spectrum contains the eigenvalues as well the spectral singularities. We will not give a review of the references (we only indicate (1) and (2)). Let us consider the case where non-perturbed continuous spectrum coincides with the half line (0,1). The eigenvalues of the perturbed operator is obtained as a set of zeros of the function
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TL;DR: In this paper, the authors proved conditions on potentials which imply that the sum of the negative eigenvalues of the Schroeodinger operator is finite, based on estimates of the Hilbert-Schmidt norm of semigroup differences.
Abstract: We prove conditions on potentials which imply that the sum of the negative eigenvalues of the Schroeodinger operator is finite. We use a method for bounding eigenvalues based on estimates of the Hilbert-Schmidt norm of semigroup differences and on complex analysis
01 Dec 2008
TL;DR: An algorithm to assign eigenvalues in a multi-input system using a block-Hessenberg matrix is presented and has a parallel structure which can be exploited to implement it in a multiprocessor computer.
Abstract: An algorithm to assign eigenvalues in a multi-input system using a block-Hessenberg matrix is presented in this paper. The present algorithm can assign multiple eigenvalues and also complex conjugate eigenvalues without involving complex arithmetic. The proposed algorithm is robust and has a parallel structure which can be exploited to implement it in a multiprocessor computer.