contraction (mathematics)

In mathematics, contraction has several 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 notation). 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.

More generally, if V is a vector space over the field k and V* is its dual vector space, then the contraction is the linear transformation

<\cdot,\cdot>:V^*\times V\rightarrow k

given by =a(b).

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

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