In
multilinear algebra, a
tensor contraction is an operation on one or more
tensors 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 which 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.