Separable Symmetric Tensors and Separable Anti-symmetric Tensors

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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.