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.
More things to try:
Arfken, G. "Contraction, Direct Product." §3.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 124-126, 1985. Jeffreys, H. and Jeffreys, B. S. "Transformation of Coordinates." §3.02 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 86-87, 1988.
