A Necessary and Sufficient Condition for Existence of a Positive Perron Vector

TL;DR

This paper establishes a precise condition called strong nonnegativity that determines when a nonnegative tensor has a positive Perron vector, extending classical matrix results to tensors.

Contribution

It introduces the concept of strongly nonnegative tensors and proves this condition is necessary and sufficient for the existence of a positive Perron vector in nonnegative tensors.

Findings

Characterization of strongly nonnegative tensors

Equivalence between strong nonnegativity and existence of positive Perron vector

Numerical methods for finding positive Perron vectors

Abstract

In 1907, Oskar Perron showed that a positive square matrix has a unique largest positive eigenvalue with a positive eigenvector. This result was extended to irreducible nonnegative matrices by Geog Frobenius in 1912, and to irreducible nonnegative tensors and weakly irreducible nonnegative tensors recently. This result is a fundamental result in matrix theory and has found wide applications in probability theory, internet search engines, spectral graph and hypergraph theory, etc. In this paper, we give a necessary and sufficient condition for the existence of such a positive eigenvector, i.e., a positive Perron vector, for a nonnegative tensor. We show that every nonnegative tensor has a canonical nonnegative partition form, from which we introduce strongly nonnegative tensors. A tensor is called strongly nonnegative, if the spectral radius of each genuine weakly irreducible block is equal to the spectral radius of the tensor, which is strictly larger than the spectral radius of any other block. We prove that a nonnegative tensor has a positive Perron vector if and only if it is strongly nonnegative. The proof is nontrivial. Numerical results for finding a positive Perron vector are reported.