An S-type eigenvalue localization set for tensors

TL;DR

This paper introduces a new eigenvalue localization set for tensors that improves upon previous bounds, providing practical criteria for tensor positive definiteness and semi-definiteness.

Contribution

It proposes a tighter S-type eigenvalue localization set for tensors by partitioning index sets, enhancing existing eigenvalue bounds and applications.

Findings

The new set is tighter than previous eigenvalue bounds.

Provides checkable conditions for tensor positive definiteness.

Offers criteria for tensor positive semi-definiteness.

Abstract

An S-type eigenvalue localization set for a tensor is given by breaking N={1,2,...,n} into disjoint subsets S and its complement. It is shown that the new set is tighter than those provided by L. Qi (Journal of Symbolic Computation 40 (2005) 1302-1324) and Li et al. (Numer. Linear Algebra Appl. 21 (2014) 39-50). As applications of the results, a checkable sufficient condition for the positive definiteness of tensors and a checkable sufficient condition of the positive semi-definiteness of tensors are given.