How close are Galerkin eigenvectors to the best approximation available out of the trial subspace? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace—and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonal-Ga...

There are methods to compute error bounds for a multiple eigenvalue together with an inclusion of a basis of the corresponding invariant subspace. Those methods have no restriction with respect to the structure of Jordan blocks, but they do not provide an inclusion of a single eigenvector. In this note we first show under general assumptions that a narrow inclusion of a single eigenvector is no...

In this paper, we thoroughly investigate correlations of eigenvector centrality to five centrality measures, including degree centrality, betweenness centrality, clustering coefficient centrality, closeness centrality, and farness centrality, of various types of network (random network, smallworld network, and real-world network). For each network, we compute those six centrality measures, from...

Crisp comparison matrices produce crisp weight estimates. It is logical for an interval or fuzzy comparison matrix to give an interval or fuzzy weight estimate. In this paper, an eigenvector method (EM) is proposed to generate interval or fuzzy weight estimate from an interval or fuzzy comparison matrix, which differs from Csutora and Buckley’s LambdaMax method in several aspects. First, the pr...

In many engineering applications, the physical quantities that have to be computed are obtained by solving a related eigenvalue problem. The matrix under consideration and thus its eigenvalues usually depend on some parameters. A natural question then is how sensitive the physical quantity is with respect to (some of) these parameters, i.e., how it behaves for small changes in the parameters. T...

The topic of this paper is a convergence analysis of preconditioned inverse iteration (PINVIT). A sharp estimate for the eigenvalue approximations is derived; the eigenvector approximations are controlled by an upper bound for the residual vector. The analysis is mainly based on extremal properties of various quantities which define the geometry of PINVIT.

A multifractal analysis is performed on the universality classes of random matrices and the transition ones. Our results indicate that the eigenvector probabi;ity distribution is a linear sum of two χ 2-distribution throughout the transition between the universality ensembles of random matrix theory and Poisson.