Further results on Cauchy tensors and Hankel tensors

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• 作者：Haibin Chen ; ¹ ; ^a ; ^{chenhaibin508@163.com" class="auth_mail" title="E-mail the corresponding author} ; Guoyin Li² ; ^b ; ^{g.li@unsw.edu.au" class="auth_mail" title="E-mail the corresponding author} ; Liqun Qi³ ; ^a ; ^{maqilq@polyu.edu.hk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Generalized Cauchy tensor ; SOS tensor ; Hankel tensor ; Positive semi-definiteness ; H-eigenvalue
• 刊名：Applied Mathematics and Computation
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：275
• 期：Complete
• 页码：50-62
• 全文大小：352 K

文摘

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$\mathcal{A}$ is less than the dimension of the underlying space, then positive semi-definiteness of A$\mathcal{A}$ is equivalent to the fact that A$\mathcal{A}$ is a complete Hankel tensor, and so, is further equivalent to the SOS property of A$\mathcal{A}$. 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.