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, contr(T_j^i) congruent T_i^i. The contraction operation is invariant under coordinate changes since T_i^(, i) = (dx_i^, )/(dx_k) (dx_l)/(dx_i^, ) T_l^k = (dx_l)/(dx_k) T_l^k = δ_k^l T_l^k = T_k^k, and must therefore be a scalar. When T_j^i is interpreted as a matrix, the contraction is the same as the trace.
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