Perturbation analysis on T-eigenvalues of third-order tensors

TL;DR

This paper conducts a comprehensive perturbation analysis of T-eigenvalues in third-order tensors, extending classical matrix theorems, introducing pseudospectra concepts, and providing numerical visualizations to understand tensor sensitivity.

Contribution

It extends classical eigenvalue perturbation theorems to third-order tensors, introduces tensor pseudospectra, and offers new analytical tools for tensor eigenvalue sensitivity analysis.

Findings

All T-eigenvalues lie within union of Gershgorin discs for certain tensors.

Extended Bauer-Fike theorem applicable to F-diagonalizable tensors.

Defined tensor pseudospectra and demonstrated their properties with numerical examples.

Abstract

Perturbation analysis has emerged as a significant concern across multiple disciplines, with notable advancements being achieved, particularly in the realm of matrices. This study centers on specific aspects pertaining to tensor T-eigenvalues within the context of the tensor-tensor multiplication. Initially, an analytical perturbation analysis is introduced to explore the sensitivity of T-eigenvalues. In the case of third-order tensors featuring square frontal slices, we extend the classical Gershgorin disc theorem and show that all T-eigenvalues are located inside a union of Gershgorin discs. Additionally, we extend the Bauer-Fike theorem to encompass F-diagonalizable tensors and present two modified versions applicable to more general scenarios. The tensor case of the Kahan theorem, which accounts for general perturbations on Hermite tensors, is also investigated. Furthermore, we propose the concept of pseudospectra for third-order tensors based on tensor-tensor multiplication. We develop four definitions that are equivalent under the spectral norm to characterize tensor $\varepsilon$-pseudospectra. Additionally, we present several pseudospectral properties. To provide visualizations, several numerical examples are also provided to illustrate the $\varepsilon$-pseudospectra of specific tensors at different levels.