Showing papers on "Spectrum of a matrix published in 2001"

THE ROTATION NUMBER APPROACH TO EIGENVALUES OF THE ONE-DIMENSIONAL p-LAPLACIAN WITH PERIODIC POTENTIALS

Abstract: The paper studies the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic potential. After a rotation number function () has been introduced, it is proved that for any non-negative integer n, the endpoints of the interval 1 (n=2) in R yield the corresponding periodic or anti-periodic eigenvalues. However, as in the Dirichlet problem of the higher dimensional p-Laplacian, it remains open if these eigenvalues represent all periodic and anti-periodic eigenvalues. The result obtained ^

The Spectrum of Random Matrices

Abstract: The Frobenius eigenvector of a positive square matrix is obtained by iterating the multiplication of an arbitrary positive vector by the matrix. Brody (1997) noticed that, when the entries of the matrix are independently and identically distributed, the speed of convergence increases statistically with the dimension of the matrix. As the speed depends on the ratio between the subdominant and the dominant eigenvalues, Brody's conjecture amounts to stating that this ratio tends to zero when the dimension tends to infinity. The paper provides a simple proof of the result. Some mathematical and economic aspects of the problem are discussed.

Numerical Determination of a Canonical Form of a Symplectic Matrix

Abstract: We propose an algorithm that transforms a real symplectic matrix with a stable structure to a block diagonal form composed of three main blocks. The two extreme blocks of the same size are associated respectively with the eigenvalues outside and inside the unit circle. Moreover, these eigenvalues are symmetric with respect to the unit circle. The central block is in turn composed of several diagonal blocks whose eigenvalues are on the unit circle and satisfy a modification of the Krein-Gelfand-Lidskii criterion. The proposed algorithm also gives a qualitative criterion for structural stability.

The computation of negative eigenvalues of singular Sturm–Liouville problems

Abstract: In this work we shall develop a new interpolation method for the computation of eigenvalues of singular Sturm-Liouville problems. Basic properties of the Jost solutions are used to determine the growth of the boundary function and an appropriate interpolation basis. This leads to a good approximation of the negative eigenvalues.

Rational and radical fixed point functions for the eigenvalue problem and polynomials

Abstract: The derivation and implementation of many algorithms in signal/image processing and control involve some form of polynomial root-finding and/or matrix eigendecomposition. In this paper, higher order fixed point functions in rational and/or radical forms are developed. This set of iterations can be considered as extensions of known methods such as the Newton, Lagurre and Halley methods and can be applied to compute all zeros of a polynomial as well as all eigenvalues of a complex matrix. One of the main features of the proposed algorithms is that they could have any predetermined rate of convergence regardless of the multiplicity of the zeros or eigenvalues. Additionally, eigenvalues and eigenvectors are computed using fast matrix inverse free algorithms which are based on the QR factorization.

Approach to energy eigenvalues and eigenfunctions from nonperturbative regions of eigenfunctions

Abstract: We study the approach to energy eigenvalues and eigenfunctions of Hamiltonian matrices with band structure from diagonalization of their truncated matrices. Making use of a generalization of Brillouin-Wigner perturbation theory, it is shown that in order to obtain approximate energy eigenvalues and eigenfunctions the sizes of truncated matrices should be larger than the nonperturbative regions of the eigenfunctions by several band width of the Hamiltonian matrix, with the nonperturbative regions being able to be estimated before the eigenfunctions are known. This prediction is checked numerically by the Wigner-band random-matrix model, which shows that 99% of eigenfunctions can be obtained when the sizes of truncated matrices are larger than those of the nonperturbative regions of the eigenfunctions by three band widths of the Hamiltonian matrix, on average.

The inverse eigenvalue problem for a weighted helmholtz equation

Abstract: Given the m lowest eigenvalues, we seek to recover an approximation to the density function ρ in the weighted Helmholtz equation -Δ=λρu on a rectangle with Dirchlet boundary conditions. The density ρ is assumed to be symmetric with respect to the midlines of the rectangle. Projection of the boundary value problem and the unknown density function onto appropriate vector spaces leads to a matrix inverse problem. Solutions of the matrix inverse problem exist provided that the reciprocals of the prescribed eigenvalues are close to the reciprocals of the simple eigenvalues of the base problem with ρ = 1. The matrix inverse problem is solved by a fixed—point iterative method and a density function ρ* is constructed which has the same m lowest eigenvalues as the unknown ρ. The algorithm can be modified when multiple base eigenvalues arise, although the success of the modification depends on the symmetry properties of the base eigenfunctions.

Dirac eigenvalues and eigenvectors at finite temperature

Abstract: We investigate the eigenvalues and eigenvectors of the staggered Dirac operator in the vicinity of the chiral phase transition of quenched SU(3) lattice gauge theory. We consider both the global features of the spectrum and the local correlations. In the chirally symmetric phase, the local correlations in the bulk of the spectrum are still described by random matrix theory, and we investigate the dependence of the bulk Thouless energy on the simulation parameters. At and above the critical point, the properties of the low-lying Dirac eigenvalues depend on the Z 3 -phase of the Polyakov loop. In the real phase, they are no longer described by chiral random matrix theory. We also investigate the localization properties of the Dirac eigenvectors in the different Z 3 -phases.

On the spectrum of a matrix model for the D=11 supermembrane compactified on a torus with non-trivial winding

Abstract: The spectrum of the Hamiltonian of the double compactified D=11 supermembrane with non-trivial central charge or equivalently the non-commutative symplectic super Maxwell theory is analyzed. In distinction to what occurs for the D=11 supermembrane in Minkowski target space where the bosonic potential presents string-like spikes which render the spectrum of the supersymmetric model continuous, we prove that the potential of the bosonic compactified membrane with non-trivial central charge is strictly positive definite and becomes infinity in all directions when the norm of the configuration space goes to infinity. This ensures that the resolvent of the bosonic Hamiltonian is compact. We find an upper bound for the asymptotic distribution of the eigenvalues.

Eigenvalues in Non-Linear Potential Theory

Abstract: In this work, we study the existence of eigenvalues for a type of non-linear equations and we give some conditions to get positive eigenfunctions. Particularly, we show that under some appropriate conditions, the set of eigenvalues is an interval.

Eigenvalues of non-Hermitian Fibonacci Hamiltonians

Abstract: We study, using numerical methods, the effects of a non-Hermitian field h on the eigenvalues of a tight-binding Fibonacci system. For vanishing non-Hermitian field, all eigenvalues are real and correspond to critical eigenfunctions. The eigenvalues become complex and eigenfunctions tend to be delocalized for non-zero values of the parameter h. The transition from critical to extended states is monitored through the inverse participation ratio as a function of h. A simple two-band model is introduced to explain the behavior of the eigenvalues on the complex plane.

An Further Generalizations of Ky Fan's Theorem of Matrix Eigenvalues

Abstract: In this paper, Ky Fan's theorem of matrix eigenvalues is generalized, its several generalizations are gotten and a kind of method of estimation of matrix eigenvalues is given.

Eigenstates of Large Profiled Matrices

Abstract: Publisher Summary The coefficient matrices of large systems of equations frequently have a profile structure. For dynamic and stability analyses, a considerable number of eigenstates of such matrices must be determined, for instance the eigenstates with the smallest eigenvalues. The algorithm of the eigenstate solver should be capable of handling multiple eigenvalues, which are common in symmetric structural systems. Conventional algorithms, such as algorithms based on the method of Lanczos, do not yield the eigenvalues in sequence and require special handling of multiple eigenvalues. This chapter presents a new algorithm for the special eigenvalue problem that preserves convex matrix profiles during iteration, yields the eigenvalues in ascending order, and does not require special precautions for multiple eigenvalues. The algorithm is applied to dam vibration problems. Multiple eigenvalues do not require special treatment. Eigenvalues of equal magnitude, but opposite sign, require special consideration.

A universal matrix perturbation technique for complex modes

Abstract: A universal matrix perturbation technique for complex modes is presented. This technique is applicable to all the three cases of complex eigenvalues: distinct, repeated and closely spaced eigenvalues. The lower order perturbation formulas are obtained by performing two complex eigensubspace condensations, and the higher order perturbation formulas are derived by a successive approximation process. Three illustrative examples are given to verify the proposed method and satisfactory results are observed.