Pseudo-spectra theory of tensors and tensor polynomial eigenvalue problems

This paper is devoted to the extension of the ϵ-pseudo-spectra theory from matrices to tensors. Based on the definition of an eigenpair of real symmetric tensors and results on the ϵ-pseudo-spectrum of square matrices, we first introduce the ϵ-pseudo-spectrum of a complex tensor and investigate its fundamental properties, such as its computational interpretations and the link with the stability of its related homogeneous dynamical system. We then extend the ϵ-pseudo-spectrum to the setting of tensor polynomial eigenvalue problems. We further derive basic structure of the ϵ-pseudo-spectrum for tensor polynomial eigenvalue problems including the symmetry, boundedness and number of connected components under suitable mild assumptions. Finally, we discuss the implication of the ϵ-pseudo-spectrum for computing the backward errors and the distance from a regular tensor polynomial to the nearest irregular tensor polynomial.