 Tensor rank

For a tensor $T\in V_1\otimes \cdots \otimes V_d$ over a field $k$, its tensor rank is the smallest number of pure tensors needed in an exact expressionThe zero tensor has rank zero. The field and the specified tensor factors are part of the definition. For two factors, the rank agrees with the matrix rank after choosing bases; for three or more factors, one matrix does not capture the full decomposition problem.^[ref]

For instance, $e_1\otimes f_1+e_2\otimes f_2$ has rank two. It cannot be a pure tensor, because its coefficient matrix has two independent columns. Every pure tensor corresponds to an outer-product matrix of rank at most one. In a bilinear algorithm, rank counts the scalar products of linear input expressions needed to form all outputs. Border rank asks the different question of how many terms suffice after allowing a limiting process.