Positivities of Vandermonde tensors

Abstract

We continue our work on Vandermonde tensors. An mth order n-dimensional real tensor is called a Vandermonde tensor associated with a vector if for every element ,Vandermonde tensors are the generalization of Vandermonde matrices. A real tensor is called a totally positive (non-negative) tensor if each of its hypercubic subtensors has a positive (non-negative) (hyper)determinant. We first show that the product of a Vandermonde tensor with its associated Vandermonde matrix is a Hankel tensor. Then, we investigate the positivities of Vandermonde tensors, including their positive definiteness and total positivity. We also show that a Vandermonde tensor generated by a positive increasing ordered vector is totally positive.