We consider a large class of residuum based a posteriori eigenvalue/eigenvector estimates and present an abstract framework for proving their asymptotic exactness. Equivalence of the estimator and the error is also established. To demonstrate the strength of our abstract approach we present a detailed study of hierarchical error estimators for Laplace eigenvalue problems in planar polygonal reg...

In 7] and 8], the power and Lanczos algorithms with random start for estimating the largest eigenvalue of an n n large symmetric positive deenite matrix were analyzed. In this paper we continue this study by estimating an eigenvector corresponding to the largest eigenvalue. We analyze polynomial algorithms using Krylov information for two error criteria: the randomized error and the randomized ...

The notes below will serve as the basis for the module lectures to be given on the topics of Jordan form, nonnegative matrices, and Markov chains. As you'll see, most of the proofs have been omitted in the notes. The intention is to sketch the proofs during the lectures, but also to have the students work through some of the details (using references such as those listed at the end of this docu...

A matrix A is said to be strictly totally positive (STP) if all its minors are strictly positive. STP matrices were independently introduced by Schoenberg in 1930 (see [13, 14]) and by Krein and Gantmacher in the 1930s. The main results concerning eigenvalues and eigenvectors of STP matrices were proved by Gantmacher and Krein in their 1937 paper [6]. (An announcement appeared in 1935 in [5]. C...

Notation: λ1(A) = largest eigenvalue of A. Motivation: The largest eigenvalue tells us the spectrum of a matrix and thus can be somewhat useful. More crucially for the adjacency matrix of a graph, we know exactly that the all one’s vector is the largest eigenvector, and thus by working orthogonal to this vector we can approximate the second largest eigenvalue of the graph, which tells us how we...

In this paper, we present an edge detection method based on wavelet transform and Hessian matrix of image at each pixel. Many methods which based on wavelet transform, use wavelet transform to approximate the gradient of image and detect edges by searching the modulus maximum of gradient vectors. In our scheme, we use wavelet transform to approximate Hessian matrix of image at each pixel, too. ...

A signless Laplacian eigenvalue of a graph G is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, some necessary and sufficient conditions for a graph with one main signless Laplacian eigenvalue or two main signless Laplacian eigenvalues are given. And the trees and unicyclic graphs with exactly two main signless L...

We first provide existence and uniqueness conditions for the solvability of an algebraic eigenvalue problem with eigenvector nonlinearity. We then present a local and global convergence analysis for a self-consistent field (SCF) iteration for solving the problem. The well-known sin Θ theorem in the perturbation theory of Hermitian matrices plays a central role. The near-optimality of the local ...

Refined estimates for finite element or, more generally, Galerkin approximations of the eigenvalues and eigenvectors of selfadjoint eigenvalue problems are presented. More specifically, refined results on the asymptotic behavior of the eigenvalue and eigenvector errors are proved. Both simple and multiple eigenvalues are treated.