Positive semi-definiteness of generalized anti-circulant tensors

Guoyin Li, Liqun Qi, Qun Wang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

5 Citations (Scopus)


Anti-circulant tensors have applications in exponential data fitting. They are special Hankel tensors. In this paper, we extend the definition of anti-circulant tensors to generalized anticirculant tensors by introducing a circulant index r such that the entries of the generating vector of a Hankel tensor are circulant with module r. In the special case when r=n, where n is the dimension of the Hankel tensor, the generalized anti-circulant tensor reduces to the anti-circulant tensor. Hence, generalized anti-circulant tensors are still special Hankel tensors. For the cases that GCD(m,r)=1, GCD(m,r)=2, and some other cases, including the matrix case that m=2, we give necessary and sufficient conditions for positive semi-definiteness of even-order generalized anti-circulant tensors and show that, in these cases, they are sum-of-squares tensors. This shows that, in these cases, there are no PNS (positive semi-definite tensors which are not sum-of-squares) Hankel tensors.
Original languageEnglish
Pages (from-to)941-952
Number of pages12
JournalCommunications in Mathematical Sciences
Issue number4
Publication statusPublished - 1 Jan 2016


  • Anti-circulant tensors
  • Circulant index
  • Generalized anti-circulant tensor
  • Generating vectors
  • Positive semi-definiteness

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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