On some properties of nonnegative weakly irreducible tensors

2011/11/03 by Yuning Yang, Yang, Yuning, Qingzhi Yang +1 · 4 citations
Mathematics · Medicine · Computer Science · #Tensor decomposition and applications #Advanced Neuroimaging Techniques and Applications #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.1111.0713

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.

Citations

• Primitive tensors and convergence of an iterative process for the eigenvalues of a primitive tensor
• Perron–Frobenius theorem for nonnegative multilinear forms and extensions

Cited by

• Tensor Spectral Stability for Uniform Hypergraphs with Bounded Matching Number
• The trace of uniform hypergraphs with application to Estrada index
• The matching polynomials and spectral radii of uniform supertrees
• A modified Newton iteration for finding nonnegative Z-eigenpairs of a nonnegative tensor

Related

• Further Results for Perron-Frobenius Theorem for Nonnegative Tensors II
• Sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor and its applications
• Characterizations of the spectral radius of nonnegative weakly irreducible tensors via digraph
• A note on the geometric simplicity of the spectral radius of nonnegative irreducible tensors
• A Necessary and Sufficient Condition for Existence of a Positive Perron Vector