Tensor rank

Remarks

For $d = 2$ and $V_1 = F^m$, $V_2 = F^n$, a tensor is an $m \times n$ matrix, a simple tensor is a rank-one matrix $u v^T$, and the tensor rank is the rank of the matrix. Example: in $F^2 \otimes F^2 \otimes F^2$ with standard basis $e_1, e_2$, the tensor $T = e_1 \otimes e_1 \otimes e_1 + e_2 \otimes e_2 \otimes e_2$ has rank $2$. The written sum shows rank at most $2$. The tensor is not simple: flattening it to a $2 \times 4$ matrix gives rank $2$, while a simple tensor flattens to a matrix of rank $1$. Non-example of rank $2$: over $\C$, the tensor $e_1 \otimes e_1 \otimes e_2 + e_1 \otimes e_2 \otimes e_1 + e_2 \otimes e_1 \otimes e_1$ has rank $3$, although it is a limit of tensors of rank $2$. So the set of tensors of rank at most $r$ need not be closed, which leads to the notion of border rank. The rank of a real tensor can be larger than its rank over $\C$. Other names are canonical polyadic rank and CP rank.