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Further results on Cauchy tensors and Hankel tensors

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  • Chen, Haibin
  • Li, Guoyin
  • Qi, Liqun

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.

Suggested Citation

  • Chen, Haibin & Li, Guoyin & Qi, Liqun, 2016. "Further results on Cauchy tensors and Hankel tensors," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 50-62.
    • Handle: RePEc:eee:apmaco:v:275:y:2016:i:c:p:50-62
    DOI: 10.1016/j.amc.2015.11.051
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    References listed on IDEAS

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    1. Shenglong Hu & Guoyin Li & Liqun Qi & Yisheng Song, 2013. "Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 717-738, September.
    2. Yisheng Song & Liqun Qi, 2015. "Properties of Some Classes of Structured Tensors," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 854-873, June.
    3. Laurent, M., 2009. "Sums of squares, moment matrices and optimization over polynomials," Other publications TiSEM 9fef820b-69d2-43f2-a501-e, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Jinyan Fan & Jiawang Nie & Anwa Zhou, 2019. "Completely Positive Binary Tensors," Mathematics of Operations Research, INFORMS, vol. 44(3), pages 1087-1100, August.

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