Tensor rank

Summary

The rank of a tensor is the smallest number of simple tensors $v_1 \otimes \cdots \otimes v_d$ whose sum is the tensor. For tensors of order $2$, that is matrices, it is the usual rank of a matrix. For order $3$ and higher it is much harder to understand: it can depend on the field, and computing it is NP-hard. The rank of the matrix multiplication tensor governs the matrix multiplication exponent.