Symmetric Nonnegative Tensors and Copositive Tensors

TL;DR

This paper establishes new spectral properties of symmetric nonnegative tensors, introduces the concept of copositive tensors extending matrices, and provides conditions for copositivity with implications for tensor analysis.

Contribution

It proves spectral bounds for symmetric nonnegative tensors, introduces copositive tensors, and characterizes their properties and conditions for copositivity.

Findings

Largest H-eigenvalue bounds via row sums

Eigenvalue positivity linked to eigenvector positivity

Conditions for copositivity based on tensor row sums

Abstract

We first prove two new spectral properties for symmetric nonnegative tensors. We prove a maximum property for the largest H-eigenvalue of a symmetric nonnegative tensor, and establish some bounds for this eigenvalue via row sums of that tensor. We show that if an eigenvalue of a symmetric nonnegative tensor has a positive H-eigenvector, then this eigenvalue is the largest H-eigenvalue of that tensor. We also give a necessary and sufficient condition for this. We then introduce copositive tensors. This concept extends the concept of copositive matrices. Symmetric nonnegative tensors and positive semi-definite tensors are examples of copositive tensors. The diagonal elements of a copositive tensor must be nonnegative. We show that if each sum of a diagonal element and all the negative off-diagonal elements in the same row of a real symmetric tensor is nonnegative, then that tensor is a copositive tensor. Some further properties of copositive tensors are discussed.