On some properties of nonnegative weakly irreducible tensors

TL;DR

This paper extends properties known for nonnegative irreducible tensors to the broader class of nonnegative weakly irreducible tensors, introducing new lemmas and counterexamples to clarify their differences.

Contribution

It introduces a key lemma using new tools, defines stochastic tensors, and proves diagonal similarity to stochastic tensors for weakly irreducible tensors, generalizing existing results.

Findings

Every weakly irreducible tensor with spectral radius one is diagonally similar to a unique stochastic tensor.

Some properties of irreducible tensors do not extend to weakly irreducible tensors, as shown by counterexamples.

New lemmas facilitate the generalization of spectral properties from irreducible to weakly irreducible tensors.

Abstract

In this paper, we mainly focus on how to generalize some conclusions from nonnegative irreducible tensors to nonnegative weakly irreducible tensors. To do so, a basic and important lemma is proven using new tools. First, we give the definition of stochastic tensors. Then we show that every nonnegative weakly irreducible tensor with spectral radius being one is diagonally similar to a unique weakly irreducible stochastic tensor. Based on it, we prove some important lemmas, which help us to generalize the results related. Some counterexamples are provided to show that some conclusions for nonnegative irreducible tensors do not hold for nonnegative weakly irreducible tensors.