Tensor convolutions and Hankel tensors

Abstract

Let A be an mth order n-dimensional tensor, where m, nare some positive integers and N:= m(n−1).Then A is called a Hankel tensor associated with a vector v\in {ℝ}^{N+\mathrm{1}} if A_{\sigma }=v_k for each k= 0, 1, …,Nwhenever σ= (i₁, …,i_m) satisfies i₁ +…+i_m = m+k.We introduce the elementary Hankel tensors which are some special Hankel tensors, and present all the eigenvalues of the elementary Hankel tensors for k= 0, 1, 2. We also show that a convolution can be expressed as the product of some third-order elementary Hankel tensors, and a Hankel tensor can be decomposed as a convolution of two Vandermonde matrices following the definition of the convolution of tensors. Finally, we use the properties of the convolution to characterize Hankel tensors and (0,1) Hankel tensors.

Key words： Tensor; convolution; Hankel tensor; elementary Hankel tensor; symmetric tensor

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