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 rank (or order) reduced by 2.
Tensor contraction can be seen as a generalization of the trace.
Read more about Tensor Contraction: Abstract Formulation, Contraction in Index Notation, Metric Contraction, Application To Tensor Fields, Contraction of A Pair of Tensors, More General Algebraic Contexts