Tensor convolutions and Hankel tensors

doi:10.1007/s11464-017-0666-8

Front. Math. China

2017, Vol. 12

Issue (6) : 1357-1373 https://doi.org/10.1007/s11464-017-0666-8

RESEARCH ARTICLE

Tensor convolutions and Hankel tensors

Changqing XU¹(

), Yiran XU²

¹. School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China
². Department of Geophysics, Institute of Disaster Prevention, Beijing 101601, China

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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 $v ∈ ? N + 1$ if $A σ = v k$ 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.

Keywords Tensor convolution Hankel tensor elementary Hankel tensor symmetric tensor

Corresponding Author(s): Changqing XU

Issue Date: 27 November 2017

Cite this article:

Changqing XU,Yiran XU. Tensor convolutions and Hankel tensors[J]. Front. Math. China, 2017, 12(6): 1357-1373.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-017-0666-8
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I6/1357

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