Some bounds for the Z-eigenpair of nonnegative tensors

Xiaoyu Ma,Yisheng Song

doi:10.1186/s13660-019-2226-0

Abstract

Tensor eigenvalue problem is one of important research topics in tensor theory. In this manuscript, we consider the properties of Z-eigenpair of irreducible nonnegative tensors. By estimating the ratio of the smallest and largest components of a positive Z-eigenvector for a nonnegative tensor, we present some bounds for the eigenvector and Z-spectral radius of an irreducible and weakly symmetric nonnegative tensor. The proposed bounds complement and extend some existing results. Finally, several examples are given to show that such a bound is different from one given in the literature.

Highlights

Matrix theory is one of the most fundamental tools of mathematics exploration and scientific research [2, 12]
As a higher-order generalization of a matrix, tensors and their properties are widely used in a great variety of fields, such as gravitational theory and quantum mechanics in physics [32, 42], large-scale date analysis [18], hypergraph spectral theory [33, 43], social network data analytics [16, 48], automatical control [27], the best rank-one approximations in statistical data analysis [17, 49], complementarity problems [1, 7, 9, 10, 15, 24,25,26, 37, 38, 40, 41], etc
By estimating the ratio of the smallest and largest components of a Perron vector, we present some bounds for the eigenvector and Z-spectral radius of an irreducible and weakly symmetric nonnegative tensor

Summary

Introduction

Matrix theory is one of the most fundamental tools of mathematics exploration and scientific research [2, 12]. (2019) 2019:271 methods of the spectral radius (H-eigenvalue) of nonnegative tensors [3, 5, 8, 10, 19, 21, 31, 34,35,36, 39, 43,44,45,46]. By estimating the ratio of the smallest and largest components of a Perron vector, we present some bounds for the eigenvector and Z-spectral radius of an irreducible and weakly symmetric nonnegative tensor.

Results

Conclusion