The contraction of a Tensor is obtained by setting unlike indices equal and summing according to the
Einstein Summation convention. Contraction reduces the Rank of a Tensor by 2.
For a second Rank Tensor,

Therefore, the contraction is invariant, and must be a Scalar. In fact, this Scalar is known as the Trace of a Matrix in Matrix theory.

**References**

Arfken, G. ``Contraction, Direct Product.'' §3.2 in
*Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 124-126, 1985.

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1999-05-26