Communications in Mathematical Sciences

Volume 14 (2016)

Number 4

Positive semi-definiteness of generalized anti-circulant tensors

Pages: 941 – 952

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n4.a3

Authors

Guoyin Li (Department of Applied Mathematics, University of New South Wales, Sydney, Australia)

Liqun Qi (Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong)

Qun Wang (Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong)

Abstract

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 anti-circulant 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 $\mathrm{GCD}(m,r)=1$, $\mathrm{GCD}(m,r)=2$, 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.

Keywords

anti-circulant tensors, generalized anti-circulant tensor, generating vectors, circulant index, positive semi-definiteness

2010 Mathematics Subject Classification

15A18, 15A69

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