Editing Tensor contraction (section)

{{Short description|Operation in mathematics and physics}}
{{for|the module-theoretic construction of tensor fields and their contractions|tensor product of modules#Example from differential geometry: tensor field}}

In [[multilinear algebra]], a '''tensor contraction''' is an operation on a [[tensor]] that arises from the [[dual system|canonical pairing]] of a [[vector space]] and its [[dual vector space|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 [[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 (linear algebra)|trace]].