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 rank(A) ≤ 6 for all tensors of size 3×3×3. We also show that each 3×3×3 anti-symmetric tensor is separable.

Key words: S-product, Invertible tensor, Separable symmetric tensor, Separable antisymmetric tensor