Perturbation analysis on T-eigenvalues of third-order tensors
Changxin Mo, Weiyang Ding, Yimin Wei
TL;DR
The paper addresses the perturbation analysis of $T$-eigenvalues for third-order tensors under the tensor-tensor product, bridging matrix perturbation theory and tensor analysis through the $bcirc$ framework. It generalizes classical results—Gershgorin discs, Bauer-Fike, and Kahan theorems—to tensors, providing both diagonalizable and non-diagonalizable cases and introducing $\varepsilon$-pseudospectra with four equivalent definitions under the spectral norm. It offers theoretical developments plus numerical visualizations of tensor $\varepsilon$-pseudospectra to localize $T$-eigenvalues and identify more $T$-positive definite tensors, highlighting implications for $T$-semidefinite programming and multilinear dynamical systems. The work thus broadens stability analysis and optimization techniques in tensor-tensor settings and sets the stage for practical applications in tensor-based optimization and control.
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.