Showing papers on "Spectrum of a matrix published in 2006"
TL;DR: This survey gives an introduction to a recently developed technique to analyze this extremal problem in polynomial approximation theory in the case of symmetric matrices.
Abstract: Krylov subspace iterations are among the best-known and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices. These methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on the spectrum of the matrix. This leads to an extremal problem in polynomial approximation theory: How small can a monic polynomial of a given degree be on the spectrum?
This survey gives an introduction to a recently developed technique to analyze this extremal problem in the case of symmetric matrices. It is based on global information on the spectrum in the sense that the eigenvalues are assumed to be distributed according to a certain measure. Then, depending on the number of iterations, the Lanczos method for the calculation of eigenvalues finds those eigenvalues that lie in a certain region, which is characterized by means of a constrained equilibrium problem from potential theory. The same constrained equilibrium problem also describes the superlinear convergence of conjugate gradients and other iterative methods for solving linear systems.
77 citations
TL;DR: Probabilistic finite sample size bounds are derived on the approximation error of individual eigenvalues which have the important property that the bounds scale with the eigenvalue under consideration, reflecting the actual behavior of the approximation errors as predicted by asymptotic results and observed in numerical simulations.
Abstract: The eigenvalues of the kernel matrix play an important role in a number of kernel methods, in particular, in kernel principal component analysis. It is well known that the eigenvalues of the kernel matrix converge as the number of samples tends to infinity. We derive probabilistic finite sample size bounds on the approximation error of individual eigenvalues which have the important property that the bounds scale with the eigenvalue under consideration, reflecting the actual behavior of the approximation errors as predicted by asymptotic results and observed in numerical simulations. Such scaling bounds have so far only been known for tail sums of eigenvalues. Asymptotically, the bounds presented here have a slower than stochastic rate, but the number of sample points necessary to make this disadvantage noticeable is often unrealistically large. Therefore, under practical conditions, and for all but the largest few eigenvalues, the bounds presented here form a significant improvement over existing non-scaling bounds.
67 citations
TL;DR: In this paper, the discrete spectrum of an asymmetric pair of two-dimensional quantum waveguides with common boundary in which a window of finite size is made is analyzed, and the phenomenon of new eigenvalues arising at the boundary of the essential spectrum as the length of the window passes over critical values is considered.
Abstract: In this paper one analyses the discrete spectrum of an asymmetric pair of two-dimensional quantum waveguides with common boundary in which a window of finite size is made. The phenomenon of new eigenvalues arising at the boundary of the essential spectrum as the length of the window passes over critical values is considered. For the newly arising eigenvalues one constructs asymptotic expansions with respect to the small parameter equal to the difference between the window length and the closest critical value. The behaviour of the spectrum under an unrestricted growth of the length of the window is also under investigation; asymptotic expansions for eigenvalues with respect to the large parameter, the length of the window, are constructed.
55 citations
TL;DR: In this article, an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field is presented, showing that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian.
Abstract: We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the Bethe-Sommerfeld conjecture fails in this case.
32 citations
TL;DR: This paper presents a few conjugate gradient-like methods to provide solutions to these types of problems by iterative procedures which utilize only matrix-vector products.
Abstract: It is often necessary to filter out an eigenspace of a given matrix $A$ before performing certain computations with it The eigenspace usually corresponds to undesired eigenvalues in the underlying application One such application is in information retrieval, where the method of latent semantic indexing replaces the original matrix with a lower-rank one using tools based on the singular value decomposition Here the low-rank approximation to the original matrix is used to analyze similarities with a given query vector Filtering has the effect of yielding the most relevant part of the desired solution while discarding noise and redundancies in the underlying problem Another common application is to compute an invariant subspace of a symmetric matrix associated with eigenvalues in a given interval In this case, it is necessary to filter out eigenvalues that are not in the interval of the wanted eigenvalues This paper presents a few conjugate gradient-like methods to provide solutions to these types of problems by iterative procedures which utilize only matrix-vector products
31 citations
TL;DR: In this paper, the authors studied a pair of neutrally stable eigenvalues of zero energy in the linearized NLS equation, where each eigenvalue has geometric multiplicity one and algebraic multiplicity N, respectively.
Abstract: We study a pair of neutrally stable eigenvalues of zero energy in the linearized NLS equation. We prove that the pair of isolated eigenvalues, where each eigenvalue has geometric multiplicity one and algebraic multiplicity N, is associated with 2P negative eigenvalues of the energy operator, where P=N∕2 if N is even and P=(N−1)∕2 or P=(N+1)∕2 if N is odd. When the potential of the linearized NLS problem is perturbed due to parameter continuations, we compute the exact number of unstable eigenvalues that bifurcate from the neutrally stable eigenvalues of zero energy.
23 citations
01 Jan 2006
TL;DR: In this paper, the eigenvalue problem for the mattix of a generalized linear operator is considered, and the problem is reduced to the analysis of an idempotent analogue of the charactetistic polynomial of the matix.
Abstract: The eigenvalue problem for the mattix of a generalized linear operator is considered. In the case of irreducible mattices, the problem is reduced to the analysis of an idempotent analogue of the charactetistic polynomial of the mattix. The eigenvectors are obtained as solutions to a homogeneous equation. The results are then extended to cover the case of an arbitrary mattix. It is shown how to build a basis of the eigensubspace of a mattix. In conclusion, an inequality for matlix powers and eigenvalues is presented, and some extremal properties of eigenvalues are considered.
22 citations
TL;DR: In this article, the Lame operator is studied as complex-analytic function in period τ of an elliptic function and the branching of eigenvalues numerically is investigated.
20 citations
TL;DR: In this article, two different approaches for the numerical calculation of eigenvalues of a singular Sturm-Liouville problem were proposed, where the potential Q is a decaying L 1 perturbation of a periodic function and the essential spectrum consequently has a band-gap structure.
16 citations
TL;DR: In this article, the authors consider non-white Wishart ensembles with i.i.d. complex standard Gaussian entries and show that the largest eigenvalue exhibits a universal behavior in the large-N limit.
12 citations
TL;DR: In this article, the eigenvalues of the adjacency matrix and of the Laplacian matrix of an unweighted tree with vertex root v were characterized and applied to some particular trees.
27 Nov 2006
TL;DR: In this article, the authors considered quadratic matrix functions of the form L(λ) = Mλ² + Dλ + K where M, D, K are real and symmetric n × n matrices with M > 0.
Abstract: This paper concerns quadratic matrix functions of the form L(λ) = Mλ² + Dλ +K where M, D, K are real and symmetric n × n matrices with M > 0. Given complete spectral information on L(λ), it is shown how new systems of the same type can be generated with updated eigenvalues and/or eigenvectors. A general purpose algorithm is formulated and illustrated with problems having no real eigenvalues, or a mixture of real and non-real eiegnvalues, or only real eigenvalues. The methods also apply for matrix polynomials of higher degree.
02 Nov 2006
01 Jan 2006
TL;DR: In this article, it was shown that for any positive definite Hermitian matrix, the eigenvalues of the superoptimal preconditioned matrix do not exceed the corresponding eigen values of the optimal preconditionsed matrix.
Abstract: In this short note, it is proved that given any positive definite Hermitian matrix, the eigenvalues of the superoptimal preconditioned matrix do not exceed the corresponding eigenvalues of the optimal preconditioned matrix.
TL;DR: In this article, an upper bound for the Perron root of nonnegative matrices is presented and the comparison of the new upper bound with the known ones is supplemented with some examples.
Abstract: In this paper, regions containing eigenvalues of a matrixare obtained in terms of partial absolute deleted row sums and column sums. Furthermore, some sufficient and necessary conditions for H-matrices are derived. Finally, an upper bound for the Perron root of nonnegative matrices is presented. The comparison of the new upper bound with the known ones is supplemented with some examples.
TL;DR: The aim is to localize matrix eigenvalues in the sense that a sufficiently small neighborhood is built for each of them (or for a cluster), through not prohibitively expensive computations.
Abstract: Our aim is to localize matrix eigenvalues in the sense that we build a sufficiently small neighborhood for each of them (or for a cluster), through not prohibitively expensive computations. Our results enter the framework started with Gerschgorin disks and deals at the present time with pseudospectra. The set of theoretical tools we have chosen to use does not avoid the notion of the characteristic polynomial. Certainly, when some computations are performed on it, the well-known ill-conditioning of its coefficients with respect to the matrix entries is properly and carefully handled.
01 Sep 2006
TL;DR: In this paper, the authors presented the characterization of the discrete-time fractional Brownian motion (dfBm) and observed that the eigenvalues of the auto-covariance matrix of a dfBm are dependent on the Hurst exponent characterizing this process.
Abstract: In this paper, we present the characterization of the discrete-time fractional Brownian motion (dfBm). Since, these processes are non-stationary; the auto-covariance matrix is a function of time. It is observed that the eigenvalues of the auto-covariance matrix of a dfBm are dependent on the Hurst exponent characterizing this process. Only one eigenvalue of this auto-covariance matrix depends on time index n and it increases as the time index of the auto-covariance matrix increases. All other eigenvalues are observed to be invariant with time index n in an asymptotic sense. The eigenvectors associated with these eigenvalues also have a fixed structure and represent different frequency channels. The eigenvector associated with the time-varying eigenvalue is a low pass filter
TL;DR: In this article, the number of equations is equal to the multiplicity of the corresponding eigenvalues of an unperturbed discrete semibounded operator, where the number is the sum of all the equations in the system.
Abstract: To compute the eigenvalues of a perturbed discrete semibounded operator, systems are obtained for the first time in which the number of equations is equal to the multiplicity of the corresponding eigenvalues of the unperturbed operator.
TL;DR: In this article, the presence of pure imaginary eigenvalues of the partially damped vibrating systems is treated and the number of such eigen values is determined using the rank of a matrix which is directly related to the system matrices.
Abstract: The presence of pure imaginary eigenvalues of the partially damped vibrating systems is treated. The number of such eigenvalues is determined using the rank of a matrix which is directly related to the system matrices.
01 Jan 2006
TL;DR: In this article, the authors studied the behavior of the conjugate-gradient method for solving a set of linear equations, where the matrix is symmetric and positive definite with one set of eigenvalues that are large and the remaining are small.
Abstract: We study the behavior of the conjugate-gradient method for solving a set of linear equations, where the matrix is symmetric and positive definite with one set of eigenvalues that are large and the remaining are small. We characterize the behavior of the residuals associated with the large eigenvalues throughout the iterations, and also characterize the behavior of the residuals associated with the small eigenvalues for the early iterations. Our results show that the residuals associated with the large eigenvalues are made small first, without changing very much the residuals associated with the small eigenvalues. A conclusion is that the ill-conditioning of the matrix is not reflected in the conjugate-gradient iterations until the residuals associated with the large eigenvalues have been made small.
TL;DR: For the positive eigenvalues of the normalized Laplacian matrix of a graph, it was shown in this article that the limit points for the smallest eigenvalue is equal to [0, 1], while the limit point for the largest eigen value is equal with [1, 2].
Abstract: Limit points for the positive eigenvalues of the normalized Laplacian matrix of a graph are considered.Specifically, it is shown that the set of limit points for the j-th smallest such eigenvalues is equal to [0, 1], while the set of limit points for the j-th largest such eigenvalues isequal to [1, 2].Limit points for certain functions of the eigenvalues, motivated by considerations for random walks, distances between vertex sets, and isoperimetric numbers, are also considered.
TL;DR: In this article, the authors investigated the application of the Blanchard and Kahn results and established that these results also carry through for linear dynamical systems where some of the eigenvalues are complex-valued.
Abstract: The dynamic properties of continuous-time macroeconomic models are typically characterised by having a combination of stable and unstable eigenvalues. In a seminal paper, Blanchard and Kahn showed that, for linear models, in order to ensure a unique solution, the number of discontinuous or ‘jump’ variables must equal the number of unstable eigenvalues in the economy. Assuming no zero eigenvalues and that all eigenvalues are distinct, this also means that the number of predetermined variables, otherwise referred to as continuous or non- ‘jump’ variables, must equal the number of stable eigenvalues. In this paper, we investigate the application of the Blanchard and Kahn results and establish that these results also carry through for linear dynamical systems where some of the eigenvalues are complex-valued. An example with just one complex conjugate pair of stable eigenvalues is presented. The Appendix contains a general n-dimensional model.
01 Jan 2006
TL;DR: In this article, a block characterization of copositive matrices with the assumption that one of the principal blocks is positive denite has been provided, and it has been shown that if the principal block of a coposive matrix has a positive vector in the subspace spanned by the eigenvectors corresponding to its nonnegative eigenvalues, then it is possible to increase the nonnegative values without changing the matrix's eigenvector.
Abstract: Let A 2 R n n . We provide a block characterization of copositive matrices, with the assumption that one of the principal blocks is positive denite. Haynsworth and Homan showed that if r is the largest eigenvalue of a copositive matrix then r j j, for all other eigenvalues of A. We continue their study of the spectral theory of copositive matrices and show that a copositive matrix must have a positive vector in the subspace spanned by the eigenvectors corresponding to the nonnegative eigenvalues. Moreover, if a symmetric matrix has a positive vector in the subspace spanned by the eigenvectors corresponding to its nonnegative eigenvalues, then it is possible to increase the the nonnegative eigenvalues to form a copositive matrix A 0 , without changing the eigenvectors. We also show that if a copositive matrix has just one positive eigenvalue, and n 1 nonpositive eigenvalues then A has a nonnegative eigenvector corresponding to a nonnegative eigenvalue.
TL;DR: Algorithms of determining maximum (in modulus) complex-conjugate eigen values are considered as applied to finding eigenvalues of high-dimension matrices according to the Khilenko method.
Abstract: Algorithms of determining maximum (in modulus) complex-conjugate eigenvalues are considered as applied to finding eigenvalues of high-dimension matrices according to the Khilenko method. An advantage of the algorithms is that the amount of calculation does not increase exponentially with the dimension of matrices.
TL;DR: A novel technique that makes edge finite element matrices more suitable for Lanczos-based solvers is introduced by displacing the eigenvalues related to static modes to a more favorable part of the spectrum.
Abstract: In this paper, after an examination of the spectrum of the system matrix of the discrete vector Helmholtz equation, a novel technique that makes edge finite element matrices more suitable for Lanczos-based solvers is introduced. This technique works by displacing the eigenvalues related to static modes to a more favorable part of the spectrum. The displacement is achieved by means of a matrix that is added to the system matrix.