Spectrum of a matrix

About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.

On the higher eigenvalues for the ∞-eigenvalue problem

Abstract: We study the higher eigenvalues and eigenfunctions for the so-called ∞-eigenvalue problem. The problem arises as an asymptotic limit of the nonlinear eigenvalue problems for the p-Laplace operators and is very closely related to the geometry of the underlying domain. We are able to prove several properties that are known in the linear case p = 2 of the Laplacian, but are unknown for other values of p. In particular, we establish the validity of the Payne-Polya-Weinberger conjecture regarding the ratio of the first two eigenvalues and the Payne nodal conjecture, which deals with the zero set of a second eigenfunction. The limit problem also exhibits phenomena that are not encountered for any 1 < p < ∞.

Finite Element Methods for Maxwell's Transmission Eigenvalues

Abstract: The transmission eigenvalue problem plays a critical role in the theory of qualitative methods for inhomogeneous media in inverse scattering theory. Efficient computational tools for transmission eigenvalues are needed to motivate improvements to theory, and, more importantly, are parts of inverse algorithms for estimating material properties. In this paper, we propose two finite element methods to compute a few lowest Maxwell's transmission eigenvalues which are of interest in applications. Since the discrete matrix eigenvalue problem is large, sparse, and, in particular, non-Hermitian due to the fact that the problem is neither elliptic nor self-adjoint, we devise an adaptive method which combines the Arnoldi iteration and estimation of transmission eigenvalues. Exact transmission eigenvalues for balls are derived and used as a benchmark. Numerical examples are provided to show the viability of the proposed methods and to test the accuracy of recently derived inequalities for transmission eigenvalues.

A note on correlations between eigenvalues of a random matrix

Abstract: Dyson's method is adopted here for the so called Gaussian ensembles. Incidently this confirms the long cherished belief that the statistical properties of a small number of eigenvalues is the same for the two kinds of ensembles, the circular and the Gaussian ones.

Integrable Structure of Ginibre’s Ensemble of Real Random Matrices and a Pfaffian Integration Theorem

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.

Eigenvalue translation based preconditioners for the GMRES(k) method

Abstract: The paper considers a possible approach to the construction of high-quality preconditionings for solving large sparse unsymmetric offdiagonally dominant, possibly indefinite linear systems. We are interested in the construction of an efficient iterative method which does not require from the user a prescription of several problem-dependent parameters to ensure the convergence, which can be used in the case when only a procedure for multiplying the coefficient matrix by a vector is available and which allows for an efficient parallel/vector implementation with only one additional assumption that the most of eigenvalues of the coefficient matrix are condensed in a vicinity of the point 1 of the complex plane. The suggested preconditioning strategy is based on consecutive translations of groups of spread eigenvalues into a vicinity of the point 1. Approximations to eigenvalues to be translated are computed by the Arnoldi procedure at several GMRES(k) iterations. We formulate the optimization problem to find optimal translations, present its suboptimal solution and prove the numerical stability of consecutive translations. The results of numerical experiments with the model CFD problem show the efficiency of the suggested preconditioning strategy.