可分对称张量与可分anti-symmetric张量
Separable symmetric tensors and separable anti-symmetric tensors
摘要：
本文首先介绍了偶数阶张量的可逆性和可分张量，包括可分对称张量和可分反对称张量，分别定义为若干向量的张量积生成的秩为1的张量的和和代数和，如$v_{1}, v_{2}, \ldots, v_{m}$ .我们证明了如果$v_{1},v_{2}, \ldots, v_{m}$是线性无关的，则每个形式为$v_{σ(1)}\times v_{σ(2)}\times\ldots\times v_{σ(m)}$的$m!$和阶是线性无关的，其中$σ$是$\set{1,2,\ldots,m}$上的任意置换.我们提供了一类张量，以实现所有$A\in R^{3\times 3\times 3}$的$\rank(A) \leq 6$的上限。我们还证明了每个$3\times 3\times 3$反对称张量是可分的。
英文摘要：
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 $v_{σ(1)}\times v_{σ(2)}\times\ldots\times v_{σ(m)}$, are linearly independent if $v_{1},v_{2}, \ldots, v_{m}$ are linearly independent, where $σ$ 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.
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