Tensor contraction

From Academic Kids

In mathematics, the contraction of a tensor in multilinear algebra occurs when a pair of literal indices (one a subscript, the other a superscript) of a mixed tensor are set equal to each other and summed over. In the Einstein notation this summation is built into the notation. The result is another tensor with rank reduced by 2.

If a tensor is dyadic then its contraction is a scalar, which is obtained by dotting each pair of base vectors in each dyad. Let

<math>

\mathbf{T} = T^i{}_j \mathbf{e_i e^j} <math>

be a dyadic tensor. Then its contraction is

<math> 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 <math>,

a scalar (rank 0).

For example: Let

<math> \mathbf{T} = \mathbf{e^i e^j} <math>

be a dyadic tensor. This tensor does not contract; if its base vectors are dotted the result is the contravariant metric tensor,

<math> g^{ij}= \mathbf{e^i} \cdot

\mathbf{e^j} <math>,

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

<math><\cdot,\cdot>:V^*\otimes V\rightarrow k<math>

given by

<a,b> = a(b).

References

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