Separable symmetric tensors and separable anti-symmetric tensors

TL;DR

This paper explores the properties of separable symmetric and anti-symmetric tensors, introduces their invertibility, and provides bounds on tensor rank, demonstrating that all 3x3x3 anti-symmetric tensors are separable.

Contribution

It defines separable symmetry and anti-symmetry tensors, proves linear independence of tensor sums, and establishes rank bounds for 3x3x3 tensors, including the separability of all anti-symmetric tensors.

Findings

Linearly independent tensor sums for linearly independent vectors.

Upper bound of tensor rank at 6 for 3x3x3 tensors.

All 3x3x3 anti-symmetric tensors are separable.

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, $v_{1}, v_{2}, \ldots, v_{m}$. We show that the $m!$ sumrands, each in form $v_{\sigma(1)}\times v_{\sigma(2)}\times\ldots\times v_{\sigma(m)}$, are linearly independent if $v_{1},v_{2}, \ldots, v_{m}$ are linearly independent, where $\sigma$ is any permutation on $\set{1,2,\ldots,m}$. 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.