I am trying to understand rank of a $d \times d \times d$ tensor. The way that I understand the $d \times d$ case is that a rank $r$, $d \times d$ tensor is a tensor that can be written as the sum of $r$ rank 1, $d \times d$ tensors, and each rank 1 tensor can be written as the outer product of two vectors. I understand that a rank $r$, $d \times d \times d$ tensor is one that can be written as the sum of $r$, $d \times d \times d$ rank 1 tensors, but I don't understand how to show that $d \times d \times d$ tensor is rank $1$. For example, how would one show that the following $2 \times 2 \times 2$ tensor is rank 1?
\begin{equation*} \left( \begin{array}{cc|cc} a_1 b_1 c_1 & a_1 b_1 c_2 & a_2 b_1 c_1 & a_2 b_1 c_2 \\ a_1 b_2 c_1 & a_1 b_2 c_2 & a_2 b_2 c_1 & a_2 b_2 c_2 \end{array} \right) \end{equation*}
Is there a way to write this as an outer product?
Thank you in advance.
We could write your tensor as $$ (a_1,a_2) \otimes (b_1,b_2) \otimes (c_1, c_2) $$ Or, depending on your notation, perhaps $$ (b_1,b_2) \otimes (c_1, c_2) \otimes (a_1,a_2) $$ One way to check that this tensor is rank one is to note that one matrix is a multiple of the other, and that each matrix is rank one.