A modified \emph{S}-type eigenvalue localization set of tensors with its applications

Abstract

Based on the $S$-type eigenvalue localization set developed by Li et al. (Linear Algebra Appl. 493 (2016) 469-483) for tensors,  a modified \emph{S}-type eigenvalue localization set for tensors is established in this paper. The proposed set containing all eigenvalues of tensors is much sharper compared with that employed by Li et al.  As its applications, some checkable sufficient conditions, which can be utilized for judging the positive (semi-)definiteness of tensors, are developed. In addition, we also provide other results, which include a new upper bound for the spectral radius of nonnegative tensors and a new lower bound for the minimum \emph{H}-eigenvalue of weakly irreducible strong \emph{M}-tensors. These bounds are superior to some previous results. These findings are supported by some numerical examples.