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# Communications in Mathematical Sciences

## Volume 13 (2015)

### Number 1

### Hankel tensors: Associated Hankel matrices and Vandermonde decomposition

Pages: 113 – 125

DOI: https://dx.doi.org/10.4310/CMS.2015.v13.n1.a6

#### Author

#### Abstract

Hankel tensors arise from applications such as signal processing. In this paper, we make an initial study on Hankel tensors. For each Hankel tensor, we associate a Hankel matrix and a higher order two-dimensional symmetric tensor, which we call the associated plane tensor. If the associated Hankel matrix is positive semi-definite, we call such a Hankel tensor a strong Hankel tensor. We show that an $m$ order $n$-dimensional tensor is a Hankel tensor if and only if it has a Vandermonde decomposition. We call a Hankel tensor a complete Hankel tensor if it has a Vandermonde decomposition with positive coefficients. We prove that if a Hankel tensor is copositive or an even order Hankel tensor is positive semi-definite, then the associated plane tensor is copositive or positive semi-definite, respectively. We show that even order strong and complete Hankel tensors are positive semi-definite, the Hadamard product of two strong Hankel tensors is a strong Hankel tensor, and the Hadamard product of two complete Hankel tensors is a complete Hankel tensor. We show that all the H-eigenvalues of a complete Hankel tensors (maybe of odd order) are nonnegative. We give some upper bounds and lower bounds for the smallest and the largest Z-eigenvalues of a Hankel tensor, respectively. Further questions on Hankel tensors are raised.

#### Keywords

Hankel tensors, Hankel matrices, plane tensors, positive semi-definiteness, co-positiveness, generating functions, Vandermonde decomposition, eigenvalues of tensors

#### 2010 Mathematics Subject Classification

15A18, 15A69

Published 16 July 2014