[Paper Review] Further Results for Perron-Frobenius Theorem for Nonnegative Tensors II

Yuning Yang, Qingzhi Yang|arXiv (Cornell University)|Apr 2, 2011

Tensor decomposition and applications15 references4 citations

TL;DR

This paper extends the Perron-Frobenius theorem to nonnegative weakly irreducible tensors, generalizing properties from irreducible to weakly irreducible settings. It establishes new spectral properties and eigenvalue behavior, including the existence of a positive eigenpair and uniqueness of the positive eigenvector under weak irreducibility.

ABSTRACT

In this paper, we generalize some conclusions from the nonnegative irreducible tensor to the nonnegative weakly irreducible tensor and give more properties of eigenvalue problems.

Motivation & Objective

• To generalize spectral theorems from nonnegative irreducible tensors to the broader class of nonnegative weakly irreducible tensors.
• To investigate the existence and uniqueness of positive eigenpairs under weak irreducibility.
• To explore additional spectral properties of eigenvalues and eigenvectors in the weakly irreducible setting.
• To provide a theoretical foundation for tensor eigenvalue problems beyond the irreducible case.

Proposed method

• Adapting techniques from matrix theory to higher-order tensors using weak irreducibility as a structural constraint.
• Employing the generalized Perron-Frobenius theorem framework for nonnegative tensors.
• Analyzing the spectral radius and its associated eigenvector under weak irreducibility conditions.
• Using topological and algebraic arguments to prove the existence of a positive eigenpair.
• Establishing uniqueness of the positive eigenvector up to scalar multiples.
• Extending known results on nonnegative irreducible tensors to the weakly irreducible case via continuity and perturbation arguments.

Experimental results

Research questions

• RQ1 How do spectral properties of nonnegative tensors change when irreducibility is relaxed to weak irreducibility?
• RQ2 Does a positive eigenpair still exist for nonnegative weakly irreducible tensors?
• RQ3 Is the positive eigenvector unique up to scaling in the weakly irreducible case?
• RQ4 What additional spectral characteristics emerge under weak irreducibility compared to irreducibility?
• RQ5 Can the Perron-Frobenius theorem be extended to include weakly irreducible tensors with similar conclusions?

Key findings

• A positive eigenpair exists for any nonnegative weakly irreducible tensor.
• The spectral radius is a positive eigenvalue with a corresponding positive eigenvector.
• The positive eigenvector associated with the spectral radius is unique up to scalar multiplication.
• The spectral radius is strictly greater than the modulus of any other eigenvalue under weak irreducibility.
• The results generalize classical Perron-Frobenius theorems from matrices and irreducible tensors to the broader class of weakly irreducible tensors.
• The framework supports further analysis of tensor eigenvalue problems in applications such as hypergraphs and multidimensional systems.