Contraction (Tensor)

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,

$${\rm contr}(B'^i_j) \equiv B'^i_i$$

$$B'^i_i = {\partial x_i'\over \partial x_k} {\partial x_l\ove... ...partial x_l\over\partial x_k} B^k_l = \delta_k^lB^k_l = B^k_k.$$

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.