Showing papers on "Spectrum of a matrix published in 2014"

Determining eigenvalues of a density matrix with minimal information in a single experimental setting

Abstract: Eigenvalues of a density matrix characterize well the quantum state's properties, such as coherence and entanglement. We propose a simple method to determine all the eigenvalues of an unknown density matrix of a finite-dimensional system in a single experimental setting. Without fully reconstructing a quantum state, eigenvalues are determined with the minimal number of parameters obtained by a measurement of a single observable. Moreover, its implementation is illustrated in linear optical and superconducting systems.

Generalized eigenvalue problems with specified eigenvalues

Abstract: We consider the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specied eigenvalues. We derive a singular value optimization characterization for this problem and illustrate its usefulness for two applications. First, the characterization yields a singular value formula for determining the nearest pencil whose eigenvalues lie in a specied region in the complex plane. For instance, this enables the numerical computation of the nearest stable descriptor system in control theory. Second, the characterization partially solves the problem posed in [Boutry et al. 2005] regarding the distance from a general rectangular pencil to the nearest pencil with a complete set of eigenvalues. The involved singular value optimization problems are solved by means of BFGS and Lipschitz-based global optimization algorithms.

Local Asymptotic Normality of the spectrum of high-dimensional spiked F-ratios

Abstract: We consider two types of spiked multivariate F distributions: a scaled distribution with the scale matrix equal to a rank-one perturbation of the identity, and a distribution with trivial scale, but rank-one non-centrality. The norm of the rank-one matrix (spike) parameterizes the joint distribution of the eigenvalues of the corresponding F matrix. We show that, for a spike located above a phase transition threshold, the asymptotic behavior of the log ratio of the joint density of the eigenvalues of the F matrix to their joint density under a local deviation from this value depends only on the largest eigenvalue λ1. Furthermore, λ1 is asymptotically normal, and the statistical experiment of observing all the eigenvalues of the F matrix converges in the Le Cam sense to a Gaussian shift experiment that depends on the asymptotic mean and variance of λ1. In particular, the best statistical inference about a sufficiently large spike in the local asymptotic regime is based on the largest eigenvalue only. As a by-product of our analysis, we establish joint asymptotic normality of a few of the largest eigenvalues of the multi-spiked F matrix when the corresponding spikes are above the phase transition threshold.

Outlier eigenvalues for deformed i.i.d. random matrices

Abstract: We consider a square random matrix of size N of the form A + Y where A is deterministic and Y has iid entries with variance 1/N. Under mild assumptions, as N grows, the empirical distribution of the eigenvalues of A+Y converges weakly to a limit probability measure \beta on the complex plane. This work is devoted to the study of the outlier eigenvalues, i.e. eigenvalues in the complement of the support of \beta. Even in the simplest cases, a variety of interesting phenomena can occur. As in earlier works, we give a sufficient condition to guarantee that outliers are stable and provide examples where their fluctuations vary with the particular distribution of the entries of Y or the Jordan decomposition of A. We also exhibit concrete examples where the outlier eigenvalues converge in distribution to the zeros of a Gaussian analytic function.

Full eigenvalues of the Markov matrix for scale-free polymer networks.

Abstract: Much important information about the structural and dynamical properties of complex systems can be extracted from the eigenvalues and eigenvectors of a Markov matrix associated with random walks performed on these systems, and spectral methods have become an indispensable tool in the complex system analysis. In this paper, we study the Markov matrix of a class of scale-free polymer networks. We present an exact analytical expression for all the eigenvalues and determine explicitly their multiplicities. We then use the obtained eigenvalues to derive an explicit formula for the random target access time for random walks on the studied networks. Furthermore, based on the link between the eigenvalues of the Markov matrix and the number of spanning trees, we confirm the validity of the obtained eigenvalues and their corresponding degeneracies.

Google matrix of the citation network of Physical Review.

Abstract: We study the statistical properties of spectrum and eigenstates of the Google matrix of the citation network of Physical Review for the period 1893--2009. The main fraction of complex eigenvalues with largest modulus is determined numerically by different methods based on high-precision computations with up to $p=16\phantom{\rule{0.16em}{0ex}}384$ binary digits that allow us to resolve hard numerical problems for small eigenvalues. The nearly nilpotent matrix structure allows us to obtain a semianalytical computation of eigenvalues. We find that the spectrum is characterized by the fractal Weyl law with a fractal dimension ${d}_{f}\ensuremath{\approx}1$. It is found that the majority of eigenvectors are located in a localized phase. The statistical distribution of articles in the PageRank-CheiRank plane is established providing a better understanding of information flows on the network. The concept of ImpactRank is proposed to determine an influence domain of a given article. We also discuss the properties of random matrix models of Perron-Frobenius operators.

On the simplicity of the eigenvalues of non-self-adjoint mathieu-hill operators

Abstract: We find conditions on the potential of the non-self-adjoint Mathieu-Hill operator such that the all eigenvalues of the periodic, antiperiodic, Dirichlet and Neumann boundary value problems are simple.

Method for Kernel Correlation-Based Spectral Data Processing

Abstract: Data points of input data are processed by first determining a Laplacian matrix for the data. A spectrum of the Laplacian matrix includes an attractive spectrum of positive eigenvalues, a repulsive spectrum of negative eigenvalues, and a neutral spectrum of zero eigenvalues. An operation for the processing is determined using the Laplacian matrix, using information about the attractive spectrum, the repulsive spectrum, and the neutral spectrum, wherein the information includes the spectra and properties derived from the Spectra. Then, the operation is performed to produce processed data.

A note on graphs whose largest eigenvalues of the modularity matrix equals zero

Abstract: Informally, a community within a graph is a subgraph whose vertices are more connected to one another than to the vertices outside the community. One of the most popular community detection methods is the Newman’s spectral modularity maximization algorithm, which divides a graph into two communities based on the signs of the principal eigenvector of its modularity matrix in the case that the modularity matrix has positive largest eigenvalue. Newman defined a graph to be indivisible if its modularity matrix has no positive eigenvalues. It is shown here that a graph is indivisible if and only if it is a complete multipartite graph.

On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case

Abstract: A lattice model of radiative decay (so-called spin-boson model) of a two level atom and at most two photons is considered. The location of the essential spectrum is described. For any coupling constant the finiteness of the number of eigenvalues below the bottom of its essential spectrum is proved. The results are obtained by considering a more general model $H$ for which the lower bound of its essential spectrum is estimated. Conditions which guarantee the finiteness of the number of eigenvalues of $H,$ below the bottom of its essential spectrum are found. It is shown that the discrete spectrum might be infinite if the parameter functions are chosen in a special form.

Spectrum and Fine Spectrum Generalized Difference Operator Over The Sequence Space ℓ1

Abstract: In this paper, we examined the fine spectrum of upper triangul ar double-band matrices over the sequence spaces `1. Also, we determined the point spectrum, the residual spectrum and the continuous spectrum of the operator A(e;e) on `1. Further, we derived the approximate point spectrum, defect spectrum and compression spectrum of the matrix operator A(e;e) over the space `1.

Shape sensitivity analysis of the eigenvalues of the Reissner-Mindlin system

Abstract: We consider the eigenvalue problem for the Reissner-Mindlin system arising in the study of the free vibration modes of an elastic clamped plate. We provide quantitative estimates for the variation of the eigenvalues upon variation of the shape of the plate. We also prove analyticity results and establish Hadamard-type formulas. Finally, we address the problem of minimization of the eigenvalues in the case of isovolumetric domain perturbations. In the spirit of the Rayleigh conjecture for the biharmonic operator, we prove that balls are critical points with volume constraint for all simple eigenvalues and the elementary symmetric functions of multiple eigenvalues.

Inverse covariance principal component analysis for power system stability studies

Abstract: The dominant poles (eigenvalues) of system matrices are used extensively in determining the power system stability analysis. The challenge is to find an accurate and efficient way of computing these dominant poles, especially for large power systems. Here we present a novel way for finding the system stability based on inverse covariance principal component analysis (ICPCA) to compute the eigenvalues of large system matrices. The efficacy of the proposed method is shown by numerical calculations over realistic power system data and we also prove the possibility of using ICPCA to determine the eigenvalues closest to any damping ratio and repeated eigenvalues. Our proposed method can also be applied for stability analysis of other engineering applications.

Infiniteness of the number of eigenvalues embedded in the essential spectrum of a 2 × 2 operator matrix

Abstract: In the present paper a 2 × 2 block operator matrix H is considered as a bounded self-adjoint operator in the direct sum of two Hilbert spaces. The structure of the essential spectrum of H is studied. Under some natural conditions the infiniteness of the number of eigenvalues is proved, located inside, in the gap or below the bottom of the essential spectrum of H.

On efficiently computing the eigenvalues of limited-memory quasi-Newton matrices

Abstract: In this paper, we consider the problem of efficiently computing the eigenvalues of limited-memory quasi-Newton matrices that exhibit a compact formulation. In addition, we produce a compact formula for quasi-Newton matrices generated by any member of the Broyden convex class of updates. Our proposed method makes use of efficient updates to the QR factorization that substantially reduces the cost of computing the eigenvalues after the quasi-Newton matrix is updated. Numerical experiments suggest that the proposed method is able to compute eigenvalues to high accuracy. Applications for this work include modified quasi-Newton methods and trust-region methods for large-scale optimization, the efficient computation of condition numbers and singular values, and sensitivity analysis.

Convergence of eigenvalues to the support of the limiting measure in critical $\beta$ matrix models

Abstract: We consider the convergence of the eigenvalues to the support of the equilibrium measure in the $\beta$ ensemble models under a critical condition. We show a phase transition phenomenon, namely that, with probability one, all eigenvalues will fall in the support of the limiting spectral measure when $\beta>1$, whereas this fails when $\beta<1$.

Estimates of eigenvalues of a boundary value problem with a parameter

Abstract: We study an eigenvalue problem for theLaplace operator with boundary condition containing a parameter. We estimate the rate of convergence of the eigenvalues to the eigenvalues of the Dirichlet problem for large positive values of the parameter

Computation of Eigenvalues

Abstract: This chapter deals with eigenvalues and eigenvectors of matrices. The material here is a sequel to Chapter 2 dealing with the solution of linear equations. Eigenvalues are very important since many engineering problems naturally lead to eigenvalue problems. When the size of a matrix is large special numerical methods are necessary for obtaining eigenvalues and eigenvectors.

Median modal control of interval continuous-time plants

Abstract: In this paper we consider a system with interval state matrix. Control law ensures desired eigenvalues and relative uncertainty of the closed-loop system state matrix. An example is given.

Eigenvalues for a nonlinear time-delay system

Abstract: The paper introduces the notion of an eigenvalue of a nonlinear time-delay system. It is shown that the respective object, playing the role of an eigenvalue, is a specific non-commutative polynomial, as it is invariant under a change of coordinates. The basic properties and possible applications of such a formalism are then discussed. Namely, the transformation of a nonlinear time-delay system into the diagonal form.

A new Gershgorin-type result for the localisation of the spectrum of matrices

Abstract: We present a Gershgorin's type result on the localisation of the spectrum of a matrix. Our method is elementary and relies upon the method of Schur complements, furthermore it outperforms the one based on the Cassini ovals of Ostrovski and Brauer. Furthermore, it yields estimates that hold without major differences in the cases of both scalar and operator matrices. Several refinements of known results are obtained.

A lower bound for the number of negative eigenvalues on a Euclidean space and on a complete Riemannian manifold

Abstract: The purpose of the present manuscript is to provide a lower bound for the number of negative eigenvalues associated to a generalized Schrodinger operator, this lower bound is given in terms of a finite number of cubes.

An Inequality Constrained SL/QP Method for Minimizing the Spectral Abscissa

Abstract: We consider a problem in eigenvalue optimization, in particular finding a local minimizer of the spectral abscissa - the value of a parameter that results in the smallest value of the largest real part of the spectrum of a matrix system. This is an important problem for the stabilization of control systems. Many systems require the spectra to lie in the left half plane in order for them to be stable. The optimization problem, however, is difficult to solve because the underlying objective function is nonconvex, nonsmooth, and non-Lipschitz. In addition, local minima tend to correspond to points of non-differentiability and locally non-Lipschitz behavior. We present a sequential linear and quadratic programming algorithm that solves a series of linear or quadratic subproblems formed by linearizing the surfaces corresponding to the largest eigenvalues. We present numerical results comparing the algorithms to the state of the art.

Multi-sensor data fusion based on the eigenvector of relation matrix

Abstract: A data fusion approach based on eigenvalues and eigenvectors of relation matrix of multi-sensor is proposed. Features of relation matrix are extracted and the information of relation matrix is fully utilized by eigenvalues and eigenvectors of relation matrix. This approach overcomes the problem of influenced by expert experience in threshold settings by the optimal sensor group selected through fusion weight. Finally, the result of digital simulation demonstrates that this approach can be simply calculated with high accuracy and excellent robustness.

On Eigenvalues of Random Complexes

Abstract: We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial-Meshulam model $X^k(n,p)$ of random $k$-dimensional simplicial complexes on $n$ vertices. We show that for $p=\Omega(\log n/n)$, the eigenvalues of these matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of $(k-2)$-dimensional faces. Garland's result concerns the Laplacian; we develop an analogous result for the adjacency matrix. The same arguments apply to other models of random complexes which allow for dependencies between the choices of $k$-dimensional simplices. In the second part of the paper, we apply this to the question of possible higher-dimensional analogues of the discrete Cheeger inequality, which in the classical case of graphs relates the eigenvalues of a graph and its edge expansion. It is very natural to ask whether this generalizes to higher dimensions and, in particular, whether the higher-dimensional Laplacian spectra capture the notion of coboundary expansion - a generalization of edge expansion that arose in recent work of Linial and Meshulam and of Gromov. We show that this most straightforward version of a higher-dimensional discrete Cheeger inequality fails, in quite a strong way: For every $k\geq 2$ and $n\in \mathbb{N}$, there is a $k$-dimensional complex $Y^k_n$ on $n$ vertices that has strong spectral expansion properties (all nontrivial eigenvalues of the normalised $k$-dimensional Laplacian lie in the interval $[1-O(1/\sqrt{n}),1+O(1/\sqrt{n})]$) but whose coboundary expansion is bounded from above by $O(\log n/n)$ and so tends to zero as $n\rightarrow \infty$; moreover, $Y^k_n$ can be taken to have vanishing integer homology in dimension less than $k$.

On the Hermitian matrix with distinct eigenvalues and its applications

Abstract: We rst investigate the Hermitian matrices with k distinct eigenvalues, and then give an algebraic characterization to a graph with k distinct eigenvalues with respect to the adjacency and (normalized) Laplacian matrix.

Asymptotic Behaviors of the Eigenvalues of Schrödinger Operator with Critical Potential

Abstract: We study the asymptotic behaviors of the discrete eigenvalue of Schrodinger operator with We obtain the leading terms of discrete eigenvalues of when the eigenvalues tend to 0. In particular, we obtain the asymptotic behaviors of eigenvalues when has singularity at .