We give general spectral and eigenvalue perturbation bounds for a selfadjoint operator perturbed in the sense of the pseudo-Friedrichs extension. We also give several generalisations of the aforementioned extension. The spectral bounds for finite eigenvalues are obtained by using analyticity and monotonicity properties (rather than variational principles) and they are general enough to include ...

The class of connected 3-colored digraphs containing exactly one nonsingular cycle is considered in this article. The main objective is to study the smallest Laplacian eigenvalue and the corresponding eigenvectors of such graphs. It is shown that the smallest Laplacian eigenvalue of such a graph can be realized as the algebraic connectivity (second smallest Laplacian eigenvalue) of a suitable u...

This work considers eigenvalue problems that are nonlinear in the eigenvalue parameter. Given such a nonlinear eigenvalue problem T , we are concerned with finding the minimal backward error such that T has a set of prescribed eigenvalues with prescribed algebraic multiplicities. While the usual resolvent norm addresses this question for a single eigenvalue of multiplicity one, the general sett...

Presented are two related numerical methods, one for the inverse eigenvalue problem for nonnegative or stochastic matrices and another for the inverse eigenvalue problem for symmetric nonnegative matrices. The methods are iterative in nature and utilize alternating projection ideas. For the symmetric problem, the main computational component of each iteration is an eigenvalue-eigenvector decomp...

In this paper we construct a class of random matrix ensembles labelled by a real parameter α ∈ (0, 1), whose eigenvalue density near zero behaves like |x|α. The eigenvalue spacing near zero scales like 1/N1/(1+α) and thus these ensembles are representatives of a continous series of new universality classes. We study these ensembles both in the bulk and on the scale of eigenvalue spacing. In the...

The solution of eigenvalue problems for partial differential operators by using boundary integral equation methods usually involves some Newton potentials which may be resolved by using a multiple reciprocity approach. Here we propose an alternative approach which is in some sense equivalent to the above. Instead of a linear eigenvalue problem for the partial differential operator we consider a...

For standard eigenvalue problems, closed-form expressions for the condition numbers of a multiple eigenvalue are known. In particular, they are uniformly 1 in the Hermitian case and generally take different values in the non-Hermitian case. We consider the generalized eigenvalue problem and identify the condition numbers. Our main result is that a multiple eigenvalue generally has multiple cond...

An efficient semi-analytic method is developed for computing the band structures of two-dimensional photonic crystals which are triangular lattices of circular cylinders. The problem is formulated as an eigenvalue problem for a given frequency using the Dirichlet-to-Neumann (DtN) map of a hexagon unit cell. This is a linear eigenvalue problem even if the material is dispersive, where the eigenv...

The problem of nding the smallest eigenvalue and the corresponding eigenspace of a symmetric matrix is stated as a semide nite optimization problem. A straightforward application of nowadays more or less standard routines for the solution of semide nite problems yields a new algorithm for the smallest eigenvalue problem; the approach not only yields the smallest eigenvalue, but also a symmetric...

We present a new computational approach for a class of large-scale nonlinear eigenvalue problems (NEPs) that are nonlinear in the eigenvalue. The contribution of this paper is two-fold. We derive a new iterative algorithm for NEPs, the tensor infinite Arnoldi method (TIAR), which is applicable to a general class of NEPs, and we show how to specialize the algorithm to a specific NEP: the wavegui...