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 T_j^i 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.

See also

Index Gymnastics, Tensor, Tensor Calculus

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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.

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Tensor Contraction

Cite this as:

Weisstein, Eric W. "Tensor Contraction." From MathWorld--A Wolfram Web Resource.

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