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. In mathematics, multilinear algebra extends the methods of linear algebra. ... In tensor analysis, a mixed tensor is a tensor which is neither covariant nor contravariant. ... For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ... In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalars, vectors, and linear operators. ... Note: This is a fairly abstract mathematical approach to tensors. ...

Tensor contraction can be seen as a generalization of matrix multiplication. This article gives an overview of the various ways to multiply matrices. ...

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: In tensor analysis, a mixed tensor is a tensor which is neither covariant nor contravariant. ...

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

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 A dyadic tensor in multilinear algebra is a second rank tensor written in a special notation, formed by juxtaposing pairs of vectors, i. ... Scalar is a concept that has meaning in mathematics, physics, and computing. ...

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.

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 In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...

V^α_,α = 0

is a continuity equation for . Note that all the examples given below express the same idea (i. ...

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 A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

given by

.

In abstract index notation, such contraction is denoted as Abstract index notation - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...

and is shorthand for the summation

a_γb^γ = a₀b⁰ + a₁b¹ + a₂b² + a₃b³

which yields a scalar.

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 Contravariant is a mathematical term with a precise definition in tensor analysis. ... In category theory, see covariant functor. ...

Λ^α_βΜ^β_γ = Ν^α_γ.

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, In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... VECTOR is the name of a Human Capital Management tool from the UK company Vector Management Systems. ... A one-form is a linear function which maps each vector in a vector space to a real number, such that the mapping is invariant with respect to coordinate transformations of the vector space. ...

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.

See also

• Tensor product

In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

References

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