Real Eigenvalues of nonsymmetric tensors

TL;DR

This paper introduces semidefinite relaxation methods to compute real eigenvalues of nonsymmetric tensors, providing a systematic approach for tensors with finitely many eigenvalues, supported by practical examples.

Contribution

It proposes Lasserre type semidefinite relaxations for calculating all real Z- and H-eigenvalues of nonsymmetric tensors, including those with finitely many eigenvalues.

Findings

All real Z-eigenvalues can be computed for tensors with finitely many.

All real H-eigenvalues can be computed for any nonsymmetric tensor.

The methods are demonstrated through various examples.

Abstract

This paper discusses the computation of real Z-eigenvalues and H-eigenvalues of nonsymmetric tensors. A general nonsymmetric tensor has finitely many Z-eigenvalues, while there may be infinitely many ones for special tensors. In the contrast, every nonsymmetric tensor has finitely many H-eigenvalues. We propose Lasserre type semidefinite relaxation methods for computing such eigenvalues. For every nonsymmetric tensor that has finitely many real Z-eigenvalues, we can compute all of them; each of them can be computed by solving a finite sequence of semidefinite relaxations. For every nonsymmetric tensor, we can compute all its real H-eigenvalues; each of them can be computed by solving a finite sequence of semidefinite relaxations. Various examples are demonstrated.