Tensor Direct Product

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 vector space tensor product is commutative because . This does not imply that an individual tensor , even when , is a symmetric tensor.

For a covector and a vector , both of tensor rank 1, the tensor direct product has the transformation law

(1)

which is a tensor of rank 2. The tensor contraction of this opposite-variance direct product is the scalar

(2)

For tensors of rank 2,

(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]