For a firstRank Tensor (i.e., a Vector),

(1) 
which is a secondRank Tensor. The Contraction of a
direct product of firstRank Tensors is the Scalar

(2) 
For a secondRank 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. 124126, 1985.
© 19969 Eric W. Weisstein
19990524