Tensor Contraction - Abstract Formulation

Abstract Formulation

Let V be a vector space over a field k. The core of the contraction operation, and the simplest case, is the natural pairing of V with its dual vector space V*. The pairing is the linear transformation from the tensor product of these two spaces to the field k:

corresponding to the bilinear form

where f is in V* and v is in V. The map C defines the contraction operation on a tensor of type (1,1), which is an element of . Note that the result is a scalar (an element of k). Using the natural isomorphism between and the space of linear transformations from V to V, one obtains a basis-free definition of the trace.

In general, a tensor of type (m, n) (with m ≥ 1 and n ≥ 1) is an element of the vector space

(where there are m V factors and n V* factors). Applying the natural pairing to the kth V factor and the lth V* factor, and using the identity on all other factors, defines the (k, l) contraction operation, which is a linear map which yields a tensor of type (m − 1, n − 1). By analogy with the (1,1) case, the general contraction operation is sometimes called the trace.