Tensor convolutions and Hankel tensors |
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Authors: | Changqing Xu Yiran Xu |
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Institution: | 1.School of Mathematics and Physics,Suzhou University of Science and Technology,Suzhou,China;2.Department of Geophysics,Institute of Disaster Prevention,Beijing,China |
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Abstract: | Let A be an mth order n-dimensional tensor, where m, n are some positive integers and N:= m(n?1). Then A is called a Hankel tensor associated with a vector v ∈ ?N+1 if Aσ = v k for each k = 0, 1,...,N whenever σ = (i1,..., im) satisfies i1 +· · ·+im = m+k. We introduce the elementary Hankel tensors which are some special Hankel tensors, and present all the eigenvalues of the elementary Hankel tensors for k = 0, 1, 2. We also show that a convolution can be expressed as the product of some third-order elementary Hankel tensors, and a Hankel tensor can be decomposed as a convolution of two Vandermonde matrices following the definition of the convolution of tensors. Finally, we use the properties of the convolution to characterize Hankel tensors and (0,1) Hankel tensors. |
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