Tensor contraction

Let V be a vector space over a field k. The core of the contraction operation, and the simplest case, is the natural pairing of V with its dual vector space V^∗. The pairing is the linear transformation from the tensor product of these two spaces to the field k:

${\displaystyle C:V\otimes V^{*}\rightarrow k}$

corresponding to the bilinear form

${\displaystyle \langle v,f\rangle =f(v)}$

where f is in V^∗ and v is in V. The map C defines the contraction operation on a tensor of type (1, 1), which is an element of ${\displaystyle V\otimes V^{*}}$. Note that the result is a scalar (an element of k). Using the natural isomorphism between ${\displaystyle V\otimes V^{*}}$ and the space of linear transformations from V to V,[1] one obtains a basis-free definition of the trace.

In general, a tensor of type (m, n) (with m ≥ 1 and n ≥ 1) is an element of the vector space

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

(where there are m factors V and n factors V^∗).[2][3] Applying the natural pairing to the kth V factor and the lth V^∗ factor, and using the identity on all other factors, defines the (k, l) contraction operation, which is a linear map which 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.

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

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

which is shorthand for the explicit coordinate summation[4]

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

(where vⁱ are the components of v in a particular basis and f_i are the components of f in the corresponding dual basis).

Since a general mixed dyadic 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 one covariant index and one contravariant index with the same letter, summation over that index being implied by the summation convention. The resulting contracted tensor inherits the remaining indices of the original tensor. For example, contracting a tensor T of type (2,2) on the second and third indices to create a new tensor U 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, the result is the contravariant metric 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 an inner product (also known as a metric) 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 as metric contraction.[5]

Contraction is often applied to tensor fields over spaces (e.g. Euclidean space, manifolds, or schemes). Since contraction is a purely algebraic operation, it can be applied pointwise to a tensor field, e.g. if T is a (1,1) tensor field on Euclidean space, then in any coordinates, its contraction (a scalar field) U at a point x is given by

${\displaystyle U(x)=\sum _{i}T_{i}^{i}(x)}$

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

Over a Riemannian manifold, a metric (field of inner products) is available, and both metric and non-metric contractions are crucial to the theory. For example, the Ricci tensor is a non-metric contraction of the Riemann curvature tensor, and the scalar 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.

One can generalize the core contraction operation (vector with dual vector) in a slightly different way, by considering a pair of tensors T and U. The tensor 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 where T is a vector and U 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 ${\displaystyle \Lambda ^{\alpha }{}_{\beta }}$ 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:

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

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

Let R be a commutative ring and let M be a finite free module over R. Then contraction operates on the full (mixed) tensor algebra of M in exactly the same way as it does in the case of vector spaces over a field. (The key fact is that the natural pairing is still perfect in this case.)

More generally, let O_X be a sheaf of commutative rings over a topological space X, e.g. O_X could be the structure sheaf of a complex manifold, analytic space, or scheme. Let M be a locally free sheaf of modules over O_X of finite rank. Then the dual of M 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.