There is in fact another definition of “tensor rank” which has everything to do with the rank of a matrix. For a tensor t in a tensor product of vector spaces V

There is in fact another definition of “tensor rank” which has everything to do with the rank of a matrix. For a tensor t in a tensor product of vector spaces VxW, define the rank of t to be the least number of summands possible in an expression t = v1xw1 + … + vnxwn. If t is zero, then it rank is zero. If the tensor product is Vx(dual V), ie of type (n,m)=(1,1), then a tensor t can be considered as a matrix, and its tensor rank is the same thing as its matrix rank. You’re basically looking for the smallest way of writing the matrix as a sum of outer products of row and column vectors. If you’re into quantum physics, then tensors of rank 0 or 1 are non-entangled, and tensors of rank 2 or more are entangled states.