The necessary and sufficient conditions of copositive tensors

TL;DR

This paper establishes necessary and sufficient conditions for copositivity of symmetric tensors, linking it to eigenvalues of principal sub-tensors, and provides a practical method for testing copositivity through lower-dimensional tensors.

Contribution

It introduces a new characterization of copositive tensors via eigenvalues of principal sub-tensors, enabling easier testing of copositivity.

Findings

Copositivity is equivalent to all principal sub-tensors having no negative $H^{++}$-eigenvalues.

Provides a method to test copositivity using lower-dimensional tensors.

Defines strict copositivity in terms of eigenvalues on the entire space.

Abstract

In this paper, it is proved that (strict) copositivity of a symmetric tensor $\mathcal{A}$ is equivalent to the fact that every principal sub-tensor of $\mathcal{A}$ has no a (non-positive) negative $H^{++}$-eigenvalue. The necessary and sufficient conditions are also given in terms of the $Z^{++}$-eigenvalue of the principal sub-tensor of the given tensor. This presents a method of testing (strict) copositivity of a symmetric tensor by means of the lower dimensional tensors. Also the equivalent definition of strictly copositive tensors is given on entire space $\mathbb{R}^n$.