The Relation Between a Tensor and Its Associated Semi-Symmetric Form

Abstract

It is known that every tensor has an associated semi-symmetric tensor. The purpose of this paper is to investigate the shared properties of a tensor and its semi-symmetric form. In particular, a corresponding semi-symmetric tensor has smaller Frobenius norm under some conditions and can be used to get smaller bounds for eigenvalues and solutions of dynamical systems and tensor complementarity problems. In addition, every tensor has the same eigenvalues as its corresponding semi-symmetric form, also a corresponding semi-symmetric tensor inherits properties like being circulant, Toeplitz, $Z$-tensor, $M$-tensor, $H$-tensor and some others. Also, there are a two-way connection for properties like being positive definite, $P$-tensor, semi-positive, primitive and several others.