## Direct Product (Tensor)

For a first-Rank Tensor (i.e., a Vector),

 (1)

which is a second-Rank Tensor. The Contraction of a direct product of first-Rank Tensors is the Scalar
 (2)

For a second-Rank Tensor,
 (3)

 (4)

For a general Tensor, 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.

References

Arfken, G. Contraction, Direct Product.'' §3.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 124-126, 1985.