论文标题
可分离的对称张量和可分离的抗对称张量
Separable symmetric tensors and separable anti-symmetric tensors
论文作者
论文摘要
在本文中，我们首先介绍了均衡张量和可分离张量的可逆性，包括可分离的对称张量和可分离的抗对称张量，分别定义为总和和秩-1张量的代数和代数，由某些矢量的张量产物产生，例如，$ v_ $ v_ {$ v_ {v_ $ ld}，v_ $ ld}，mot}，mot}，mot}，m。我们表明$ m！$ sumrands，每个形式$ v_ {σ（1）} \ times v_ {σ（2）} \ times \ times \ times \ times \ times v_ {σ（m）} $是线性独立的，如果$ v_ {1}，v_ {1}，v_ {2}，v_ {2}，$ worly是在$ \ set {1,2，\ ldots，m} $上。我们提供一类张量，以在r^{3 \ times 3 \ times 3} $中获得$ \ rank（a）\ leq 6 $的上限（a）\ leq 6 $。我们还表明，每个$ 3 \ times 3 \ times 3 $反对称张量都是可分离的。
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.
