Generalized Tensor Eigenvalue Problems

Abstract

This paper is devoted to generalized tensor eigenvalue problems. We focus on the properties and perturbations of the spectra of regular tensor pairs. Employing different techniques, we extend several classical results from matrices or matrix pairs to tensor pairs, such as the Gershgorin circle theorem, the Collatz--Wielandt formula, the Bauer--Fike theorem, the Rayleigh--Ritz theorem, backward error analysis, the componentwise distance of a nonsingular tensor to singularity, etc. Some of these results preserve their original forms, while others change when being extended.