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Title: Further results on cauchy tensors and hankel tensors
Authors: Chen, H
Li, G
Qi, L 
Keywords: Generalized Cauchy tensor
Hankel tensor
Positive semi-definiteness
SOS tensor
Issue Date: 2016
Publisher: Elsevier
Source: Applied mathematics and computation, 2016, v. 275, p. 50-62 How to cite?
Journal: Applied mathematics and computation 
Abstract: In this article, we present various new results on Cauchy tensors and Hankel tensors. We first introduce the concept of generalized Cauchy tensors which extends Cauchy tensors in the current literature, and provide several conditions characterizing positive semi-definiteness of generalized Cauchy tensors with nonzero entries. Furthermore, we prove that all even order generalized Cauchy tensors with positive entries are completely positive tensors, which means every such that generalized Cauchy tensor can be decomposed as the sum of nonnegative rank-1 tensors. We also establish that all the H-eigenvalues of nonnegative Cauchy tensors are nonnegative. Secondly, we present new mathematical properties of Hankel tensors. We prove that an even order Hankel tensor is Vandermonde positive semi-definite if and only if its associated plane tensor is positive semi-definite. We also show that, if the Vandermonde rank of a Hankel tensor A is less than the dimension of the underlying space, then positive semi-definiteness of A is equivalent to the fact that A is a complete Hankel tensor, and so, is further equivalent to the SOS property of A. Thirdly, we introduce a new class of structured tensors called Cauchy-Hankel tensors, which is a special case of Cauchy tensors and Hankel tensors simultaneously. Sufficient and necessary conditions are established for an even order Cauchy-Hankel tensor to be positive definite.
ISSN: 0096-3003
EISSN: 1873-5649
DOI: 10.1016/j.amc.2015.11.051
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