Tensor contraction

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In multilinear algebra, a tensor contraction is a sum of products of scalar components of one or more tensors caused by applying the summation convention to a pair of dummy indices which are bound to each other in an expression. The contraction of a single mixed tensor occurs when a pair of literal indices (one a subscript, the other a superscript) of the 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.

Tensor contraction can be seen as a generalization of matrix multiplication.

Contents 1 Contraction of a tensor with itself 1.1 Contraction of a dyadic tensor 1.2 Tensor divergence 2 Contraction of a pair of tensors 2.1 Matrix multiplication 3 Contraction between tensors seen as a self-contraction of a composite tensor 4 See also 5 References

[edit] Contraction of a tensor with itself

Given a mixed tensor of type (m, n) with m≥1 and n≥1, then letting a pair of indices, one contravariant and one covariant, be labeled with the same letter will imply a summation over those two indices. The result of the summation will be a new tensor of type (m−1, n−1) which will inherit the indices of the pre-contracted tensor except for the pair of indices which were bound to each other and over which the contraction took place. Example:

T^αβ_γβ = T^α0_γ0 + T^α1_γ1 + T^α2_γ2 + T^α3_γ3 = U^α_γ

[edit] Contraction of a dyadic tensor

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

be a dyadic tensor. Then its contraction is

,

a scalar (rank 0).

For example: Let

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

,

whose rank is 2.

[edit] Tensor divergence

From here onwards, assume that tensors are four-dimensional.

Let

be the covariant derivative of vector in Cartesian coordinates.

Then changing index β to α causes the pair of indices to become bound to each other, so that the derivative contracts with itself to obtain the following sum:

V^α_,α = V⁰_,0 + V¹_,1 + V²_,2 + V³_,3

which is a four-dimensional divergence. Then

V^α_,α = 0

is a continuity equation for .

[edit] Contraction of a pair of tensors

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

given by

.

In abstract index notation, such contraction is denoted as

and is shorthand for the summation

a_γb^γ = a₀b⁰ + a₁b¹ + a₂b² + a₃b³

which yields a scalar.

[edit] Matrix multiplication

Matrices can be represented as tensors of type (1,1) with the first index being contravariant and the second index being covariant. Let Λ^α_β be the components of one matrix and let Μ^β_γ be the components of a second matrix. Then their multiplication is given by the following contraction:

Λ^α_βΜ^β_γ = Ν^α_γ.

[edit] Contraction between tensors seen as a self-contraction of a composite tensor

In abstract index notation, a prerequisite for a pair of tensors to contract with each other is for them to be placed side by side (juxtaposed) as factors of the same term, but doing so implicitly yields components of a composite tensor which is the tensor product of the two factors. For example, given vector and one-form , juxtaposition of their components,

v^αu_β,

yields a composite tensor whose components are

.

Then binding the pair of indices to each other yields a self-contraction of tensor W which yields a scalar (a tensor of rank zero):

W^α_α = v⁰u₀ + v¹u₁ + v²u₂ + v³u₃ = k.

This example between a pair of first rank tensors can be generalized to contractions between tensors of arbitrary rank: such contractions can be seen as the result of first juxtaposing tensors whose indices are not yet bound to each other, to produce a composite tensor which is their tensor product. Then bind a pair of indices to each other, producing self-contraction of the composite tensor, which is equivalent to the contraction between distinct tensors.

[edit] See also

• Tensor product
• Interior product

[edit] References

• Donald H. Menzel. Mathematical Physics. Dover Publications, New York.