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 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 language | English |
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Pages (from-to) | 941-952 |
Number of pages | 12 |
Journal | Communications in Mathematical Sciences |
Volume | 14 |
Issue number | 4 |
DOIs | |
Publication status | Published - 1 Jan 2016 |
Keywords
- Anti-circulant tensors
- Circulant index
- Generalized anti-circulant tensor
- Generating vectors
- Positive semi-definiteness
ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics