Tensor Contraction

Tensor contraction is an operation on a tensor performed by setting unlike indices equal and summing according to the Einstein summation convention. Contraction reduces the tensor rank by 2. For example, for a tensor of tensor rank two,

The contraction operation is invariant under coordinate transformations since

and must therefore be a scalar.

When is interpreted as a matrix, the contraction is the same as the matrix trace.

Sometimes, two tensors are contracted using an upper index of one tensor and a lower of the other tensor. In this context, contraction occurs after forming their tensor product.