The Z-eigenpairs of orthogonally diagonalizable symmetric tensors

TL;DR

This paper studies Z-eigenpairs of orthogonally diagonalizable symmetric tensors, providing explicit formulas, counting eigenpairs based on tensor properties, and analyzing their local optimality conditions.

Contribution

It introduces a unified expression for eigenpairs, determines their total number from tensor order and rank, and examines their local optimality conditions.

Findings

Eigenpairs can be uniformly expressed using basic eigenpairs.

The total number of eigenpairs is uniquely determined by tensor order and rank.

Local optimality of eigenpairs is analyzed via second-order conditions.

Abstract

In this paper, we focus on a special class of symmetric tensors, which can be orthogonally diagonalizable, and investigate their Z-eigenpairs problem. We show that the eigenpairs can be uniformly expressed using several basic eigenpairs, and the number of all the eigenpairs is uniquely determined by the order and rank of the symmetric tensor. In addition, we exploit the local optimality of each eigenpair by checking the second-order necessary condition.