Tensor Direct Product

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 V tensor W=W tensor V. 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


which is a second-tensor rank tensor. The tensor contraction of a direct product of first-tensor rank tensors is the scalar


For second-tensor rank tensors,


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 a and b can be implemented in the Wolfram Language as

  TensorDirectProduct[a_List, b_List] :=
    Outer[Times, a, b]

See also

Direct Product, Kronecker Product, Vector Space Tensor Product

Portions of this entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd and Weisstein, Eric W. "Tensor Direct Product." From MathWorld--A Wolfram Web Resource.

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