An eigenvalue localization set for tensors with applications to determine the positive (semi-)definiteness of tensors

Abstract

A new eigenvalue localization set for tensors is given, and proved to be tighter than those in [Qi L. Eigenvalues of a real supersymmetric tensor. J. Symbolic Comput. 2005;40:1302–1324] and [Li CQ, Li YT, Kong X. New eigenvalue inclusion sets for tensors. Numer. Linear Algebra Appl. 2014;21:39–50]. Based on this set, we give two checkable sufficient conditions of the positive definiteness of tensors, and two checkable sufficient conditions of the positive semi-definiteness of tensors.