Abstract
In this paper, we first initialize the S-product of tensors to unify the outer product, contractive product, and the inner product of tensors. Then, we introduce the separable symmetry tensors and separable anti-symmetry tensors, which are defined, respectively, as the sum and the algebraic sum of rank-one tensors generated by the tensor product of some vectors. We offer a class of tensors to achieve the upper bound for \(\texttt {rank}({\mathcal {A}}) \leqslant 6\) for all tensors of size \(3\times 3\times 3\). We also show that each \(3\times 3\times 3\) anti-symmetric tensor is separable.
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Notes
A general permutation tensor can be defined without the restriction of a constant dimension in the first m modes. Here we simplify it to fit our purpose.
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Acknowledgements
The authors would like to thank Professor Fuzhen Zhang of Nova Southeastern University for his remarks for the proof of Theorem 8. Thanks are also given to the anonymous referees for their valuable suggestions and remarks which lead to the improvement of the manuscript.
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Xu, C., Xu, K. Separable Symmetric Tensors and Separable Anti-symmetric Tensors. Commun. Appl. Math. Comput. 5, 1509–1523 (2023). https://doi.org/10.1007/s42967-022-00217-x
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DOI: https://doi.org/10.1007/s42967-022-00217-x