Abstractly, the tensor direct product is the same as the vector space tensor product. However, it reflects an approach toward calculation using
coordinates, and indices in particular. The notion of tensor product is more algebraic,
intrinsic, and abstract. For instance, up to isomorphism,
the tensor product is commutative because 
. Note this does not mean that the tensor
product is symmetric.
For two first-tensor rank tensors (i.e., vectors), the tensor direct product is defined as
 |
(1)
|
which is a second-tensor rank tensor. The tensor contraction of a direct product of first-tensor rank tensors is the scalar
 |
(2)
|
For second-tensor rank tensors,
 |
(3)
|
 |
(4)
|
In general, the direct product of two tensors is a tensor of rank equal to the sum of the two initial ranks. The direct product is associative, but not commutative.
The tensor direct product of two tensors 
and 
can be implemented in the Wolfram
Language as
TensorDirectProduct[a_List, b_List] :=
Outer[Times, a, b]
