Separable symmetric tensors and separable anti-symmetric tensors

Abstract

In this paper, we first introduce the invertibility of even-order tensors and the separable tensors, including separable symmetry tensors and separable anti-symmetry tensors, defined respectively as the sum and the algebraic sum of rank-1 tensors generated by the tensor product of some vectors, say, $v_1,v_2,\dots ,v_m$. We show that the $m!$ sumrands, each in form $v_{\sigma (1)}\times v_{\sigma (2)}\times \dots \times v_{\sigma (m)}$, are linearly independent if $v_1,v_2,\dots ,v_m$ are linearly independent, where $\sigma$ is any permutation on $\{1,2,\dots ,m\}$. We offer a class of tensors to achieve the upper bound for $\rank(A) \leq 6$ for all $A\in R^{3\times 3\times 3}$. We also show that each $3\times 3\times 3$ anti-symmetric tensor is separable.

18 pages, 0 figures