An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors

TL;DR

This paper introduces an unconstrained optimization method to find real eigenvalues of even order symmetric tensors, extending classical matrix eigenvalue principles to higher-order tensors with practical numerical applications.

Contribution

It develops a novel unconstrained optimization framework for tensor eigenvalues, generalizing variational principles from matrices to higher-order tensors.

Findings

Effective in computing Z-eigenvalues of tensors

Assists in determining tensor positive semidefiniteness

Numerical results demonstrate practical applicability

Abstract

Let $n$ be a positive integer and $m$ be a positive even integer. Let ${\mathcal A}$ be an $m^{th}$ order $n$-dimensional real weakly symmetric tensor and ${\mathcal B}$ be a real weakly symmetric positive definite tensor of the same size. $\lambda \in R$ is called a ${\mathcal B}_r$-eigenvalue of ${\mathcal A}$ if ${\mathcal A} x^{m-1} = \lambda {\mathcal B} x^{m-1}$ for some $x \in R^n \backslash \{0\}$. In this paper, we introduce two unconstrained optimization problems and obtain some variational characterizations for the minimum and maximum ${\mathcal B}_r$--eigenvalues of ${\mathcal A}$. Our results extend Auchmuty's unconstrained variational principles for eigenvalues of real symmetric matrices. This unconstrained optimization approach can be used to find a Z-, H-, or D-eigenvalue of an even order weakly symmetric tensor. We provide some numerical results to illustrate the effectiveness of this approach for finding a Z-eigenvalue and for determining the positive semidefiniteness of an even order symmetric tensor.