Inmultilinear algebra, atensor contraction is an operation on atensor that arises from thecanonical pairing of avector space and itsdual. In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying thesummation convention to a pair of dummy indices that are bound to each other in an expression. The contraction of a singlemixed 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. InEinstein notation this summation is built into the notation. The result is anothertensor with order reduced by 2.

Tensor contraction can be seen as ageneralization of thetrace.

LetV be a vector space over afieldk. The core of the contraction operation, and the simplest case, is the canonical pairing ofV with itsdual vector spaceV^∗. The pairing is thelinear map from thetensor product of these two spaces to the fieldk:

${\displaystyle C:V\otimes V^{\ast }\to k}$

corresponding to thebilinear form

${\displaystyle ⟨v,f⟩=f(v)}$

wheref is inV^∗ andv is inV. The mapC defines the contraction operation on a tensor of type(1, 1), which is an element of${\displaystyle V\otimes V^{\ast }}$. Note that the result is ascalar (an element ofk). Infinite dimensions, using thenatural isomorphism between${\displaystyle V\otimes V^{\ast }}$ and the space of linear maps fromV toV,^[1] one obtains a basis-free definition of thetrace.

In general, atensor of type(m,n) (withm ≥ 1 andn ≥ 1) is an element of the vector space

${\displaystyle V\otimes \cdots \otimes V\otimes V^{\ast }\otimes \cdots \otimes V^{\ast }}$

(where there arem factorsV andn factorsV^∗).^[2][3] Applying the canonical pairing to thekthV factor and thelthV^∗ factor, and using the identity on all other factors, defines the (k,l) contraction operation, which is a linear map that yields a tensor of type(m − 1,n − 1).^[2] By analogy with the(1, 1) case, the general contraction operation is sometimes called the trace.

Intensor index notation, the basic contraction of a vector and a dual vector is denoted by

${\displaystyle \tilde{f}(\overrightarrow{v})=f_{\gamma }{v}^{\gamma },}$

which is shorthand for the explicit coordinate summation^[4]

${\displaystyle f_{\gamma }{v}^{\gamma }=f_1{v}^1+f_2{v}^2+\cdots +f_n{v}^n}$

(wherevⁱ are the components ofv in a particular basis andf_i are the components off in the corresponding dual basis).

Since a general mixeddyadic tensor is a linear combination of decomposable tensors of the form${\displaystyle f\otimes v}$, the explicit formula for the dyadic case follows: let

${\displaystyle \mathbf{T}=T_j^i{\mathbf{e}}_i\otimes {\mathbf{e}}^j}$

be a mixed dyadic tensor. Then its contraction is

${\displaystyle T_j^i{\mathbf{e}}_i\cdot {\mathbf{e}}^j=T_j^i{\delta }_i{}^j=T_j^j=T_1^1+\cdots +T_n^n}$.

A general contraction is denoted by labeling onecovariant index and onecontravariant index with the same letter, summation over that index being implied by thesummation convention. The resulting contracted tensor inherits the remaining indices of the original tensor. For example, contracting a tensorT of type (2,2) on the second and third indices to create a new tensorU of type (1,1) is written as

${\displaystyle T^{ab}{}_{bc}=\sum _b{T}^{ab}{}_{bc}=T^{a1}{}_{1c}+T^{a2}{}_{2c}+\cdots +T^{an}{}_{nc}=U^a{}_c.}$

By contrast, let

${\displaystyle \mathbf{T}={\mathbf{e}}^i\otimes {\mathbf{e}}^j}$

be an unmixed dyadic tensor. This tensor does not contract; if its base vectors are dotted,^{[clarification needed]} the result is the contravariantmetric tensor,

${\displaystyle g^{ij}={\mathbf{e}}^i\cdot {\mathbf{e}}^j}$,

whose rank is 2.

As in the previous example, contraction on a pair of indices that are either both contravariant or both covariant is not possible in general. However, in the presence of aninner product (also known as ametric)g, such contractions are possible. One uses the metric to raise or lower one of the indices, as needed, and then one uses the usual operation of contraction. The combined operation is known asmetric contraction.^[5]

Contraction is often applied totensor fields over spaces (e.g.Euclidean space,manifolds, orschemes^{[citation needed]}). Since contraction is a purely algebraic operation, it can be applied pointwise to a tensor field, e.g. ifT is a (1,1) tensor field on Euclidean space, then in any coordinates, its contraction (a scalar field)U at a pointx is given by

${\displaystyle U(x)=\sum _i{T}_i^i(x)}$

Since the role ofx is not complicated here, it is often suppressed, and the notation for tensor fields becomes identical to that for purely algebraic tensors.

Over aRiemannian manifold, a metric (field of inner products) is available, and both metric and non-metric contractions are crucial to the theory. For example, theRicci tensor is a non-metric contraction of theRiemann curvature tensor, and thescalar curvature is the unique metric contraction of the Ricci tensor.

One can also view contraction of a tensor field in the context of modules over an appropriate ring of functions on the manifold^[5] or the context of sheaves of modules over the structure sheaf;^[6] see the discussion at the end of this article.

As an application of the contraction of a tensor field, letV be avector field on aRiemannian manifold (for example,Euclidean space). Let${\displaystyle V^{\alpha }{}_{\beta }}$ be thecovariant derivative ofV (in some choice of coordinates). In the case ofCartesian coordinates in Euclidean space, one can write

${\displaystyle V^{\alpha }{}_{\beta }=\frac{\mathrm{\partial }{V}^{\alpha }}{\mathrm{\partial }{x}^{\beta }}.}$

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

${\displaystyle V^{\alpha }{}_{\alpha }=V^0{}_0+\cdots +V^n{}_n,}$

which is thedivergence divV. Then

${\displaystyle \mathrm{div}V=V^{\alpha }{}_{\alpha }=0}$

is acontinuity equation forV.

In general, one can define various divergence operations on higher-ranktensor fields, as follows. IfT is a tensor field with at least one contravariant index, taking thecovariant differential and contracting the chosen contravariant index with the new covariant index corresponding to the differential results in a new tensor of rank one lower than that ofT.^[5]

One can generalize the core contraction operation (vector with dual vector) in a slightly different way, by considering a pair of tensorsT andU. Thetensor product${\displaystyle T\otimes U}$ is a new tensor, which, if it has at least one covariant and one contravariant index, can be contracted. The case whereT is a vector andU is a dual vector is exactly the core operation introduced first in this article.

In tensor index notation, to contract two tensors with each other, one places them side by side (juxtaposed) as factors of the same term. This implements the tensor product, yielding a composite tensor. Contracting two indices in this composite tensor implements the desired contraction of the two tensors.

For example, matrices can be represented as tensors of type (1,1) with the first index being contravariant and the second index being covariant. Let${\displaystyle {\mathrm{\Lambda }}^{\alpha }{}_{\beta }}$ be the components of one matrix and let${\displaystyle {\mathrm{M}}^{\beta }{}_{\gamma }}$ be the components of a second matrix. Then their multiplication is given by the following contraction, an example of the contraction of a pair of tensors:

${\displaystyle {\mathrm{\Lambda }}^{\alpha }{}_{\beta }{\mathrm{M}}^{\beta }{}_{\gamma }={\mathrm{N}}^{\alpha }{}_{\gamma }}$.

Also, theinterior product of a vector with adifferential form is a special case of the contraction of two tensors with each other.

LetR be acommutative ring and letM be a finite freemodule overR. Then contraction operates on the full (mixed) tensor algebra ofM in exactly the same way as it does in the case of vector spaces over a field. (The key fact is that the canonical pairing is still perfect in this case.)

More generally, letO_X be asheaf of commutative rings over atopological spaceX, e.g.O_X could be thestructure sheaf of acomplex manifold,analytic space, orscheme. LetM be alocally free sheaf of modules overO_X of finite rank. Then the dual ofM is still well-behaved^[6] and contraction operations make sense in this context.

• Tensor product
• Partial trace
• Interior product
• Raising and lowering indices
• Musical isomorphism
• Ricci calculus

• Bishop, Richard L.; Goldberg, Samuel I. (1980).Tensor Analysis on Manifolds. New York: Dover.ISBN 0-486-64039-6.
• Menzel, Donald H. (1961).Mathematical Physics. New York: Dover.ISBN 0-486-60056-4.{{cite book}}:ISBN / Date incompatibility (help)

• Tensors

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