Tensor contraction

In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices that are bound to each other in an expression. The contraction of a single mixed tensor occurs when a pair of literal indices (one a subscript, the other a superscript) of the tensor are set equal to each other and summed over. In the Einstein notation this summation is built into the notation. The result is another tensor with order reduced by 2. In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices that are bound to each other in an expression. The contraction of a single mixed tensor occurs when a pair of literal indices (one a subscript, the other a superscript) of the tensor are set equal to each other and summed over. In the Einstein notation this summation is built into the notation. The result is another tensor with order reduced by 2. Tensor contraction can be seen as a generalization of the trace. 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 V ∗ ⊗ V {displaystyle V^{*}otimes V} . Note that the result is a scalar (an element of k). Using the natural isomorphism between V ∗ ⊗ V {displaystyle V^{*}otimes V} 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 factors V and n factors V∗). 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. In tensor index notation, the basic contraction of a vector and a dual vector is denoted by which is shorthand for the explicit coordinate summation