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# 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

#### 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