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Title: Hankel tensors : associated Hankel matrices and Vandermonde decomposition
Authors: Qi, L 
Issue Date: 2014
Source: Communications in mathematical sciences, 2014, v. 13, no. 1, p. 113-125
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: Co-positiveness
Eigenvalues of tensors
Generating functions
Hankel matrices
Hankel tensors
Plane tensors
Positive semi-definiteness
Vandermonde decomposition
Publisher: International Press of Boston, Inc.
Journal: Communications in Mathematical Sciences 
ISSN: 1539-6746
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