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Contraction (mathematics)

In mathematics, contraction has two meanings:

• Contraction of a tensor. It occurs when a pair of literal indices (one a subscript, the other a superscript) of a mixed tensor are set equal to each other so that a summation over that index takes place (due to the Einstein summation convention). The result is another tensor whose rank is reduced by 2.

If a tensor is dyadic then its contraction is a scalar obtained by dotting each pair of base vectors in each dyad. E.g. Let $\mathbf{T} = T^i{}_j \mathbf{e_i e^j}$ be a dyadic tensor, then its contraction is $T^i {}_j \mathbf{e_i} \cdot \mathbf{e^j} = T^i {}_j \delta_i^j T^j {}_j T^1 {}_1 + T^2 {}_2 + T^3 {}_3$,

a scalar of rank 0.

E.g. Let $\mathbf{T} = \mathbf{e^i e^j}$ be a dyadic tensor.
This tensor does not contract; if its base vectors are dotted the result is the contravariant metric tensor, $g^{ij}= \mathbf{e^i} \cdot \mathbf{e^j}$, whose rank is 2.

References. Mathematical Physics by Donald H. Menzel. Dover Publications, New York.