Tensor Contraction

The contraction of a tensor is obtained by setting unlike indices equal and summing according to the Einstein summation convention. Contraction reduces the tensor rank by 2. For example, for a second-rank tensor,

The contraction operation is invariant under coordinate changes since

and must therefore be a scalar.

When is interpreted as a matrix, the contraction is the same as the 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 tensor multiplication.