has two meanings:
If a tensor is dyadic then its contraction is a scalar obtained by dotting each pair of base vectors in each dyad. E.g. Let
be a dyadic tensor, then its contraction is ,
a scalar of rank 0.
E.g. Let be a dyadic tensor.
This tensor does not contract; if its base vectors are dotted the result is the contravariant metric tensor, , whose rank is 2.
More generally, if V is a vector space over the field k and V* is its dual vector space, then the contraction is the linear transformation given by =a(b).
References. Mathematical Physics by Donald H. Menzel. Dover Publications, New York.