Tensor product

The tensor product is a very versatile concept of mathematics: in linear algebra and differential geometry it is the description of multi- linear maps in commutative algebra and algebraic geometry, it corresponds both to the limitation of geometric structures on subsets, on the other hand, the Cartesian product of geometric objects.

In physics, called elements of the tensor product

(for a vector space with dual space, often ) as tensors, contravariant and covariant level of the stage. Short one speaks of tensors of type.

This article describes the mathematical and coordinate- free aspects of the tensor product. For individual tensors and coordinate representations see tensor.

• 2.1 The basic construction
• 2.2 Special Cases
• 2.3 Categorical properties
• 2.4 Examples

• 3.1 Elementary tensors
• 3.2 General form

Tensor product of vector spaces

Definition

Are and two vector spaces over a common Skalarkörper, then the tensor product

A vector space, which can be constructed as follows: If a base of and a base of, then a vector space, called Tensorproduktraum is, where there is a base in a unique way with the ordered pairs of the Cartesian product

The bases of the output spaces can be identified. The dimension is therefore equal to the product of the dimensions of and. The element of this base corresponding to the ordered pair is as noted. The symbol has thereby no deeper meaning to this place. An arbitrary element of the tensor then takes the form

Where and are finite subsets of the index sets and and for each and.

It is now possible with the help of this base product of vectors, and define, which is listed with the same shortcut icon. Naturally is the product of two basis vectors and the straight base vector, which was designated. The product of any vectors can now be obtained by bilinear continuation,

And

With finally the product is

Assigned.

Finite-dimensional case

For finite dimensional vector spaces V and W, the tensor can be constructed directly as a space of matrices. The lines are numbered with the base index of V, the column with the base index of W. The tensor product of two vectors is that matrix, the entry at the location (i, j ) is the i - th coordinate of v times the j-th coordinate of w is. The columns multiple of v, the line is a multiple of w. ( In the language of matrices, this design is also called dyadic product. )

Properties

For the tensor product of vectors following calculation rules apply to everyone, and:

In other words, the mapping; is bilinear. These rules look like distributive and associative laws; and hence the name tensor.

A commutative law does not, in general, because for include the vectors

Only on the same vector space if the spaces V and W are identical; and even then must apply no equality.

Tensor product of linear maps

There are

Two linear mappings between vector spaces. Your tensor is the linear map

Defined by

Are finite -dimensional, then you can choice of bases for, for, for and for linear mappings and their imaging matrices

Describe. The linear image is then related to the bases and to and through the Kronecker product

The dies and described. For example, with standard base and by the matrices and, optionally

Then is given by the Kronecker product

Universal definition

So far, the question has been circumvented, which nature is because with designated vector space in the general case. The specified invoices to this vector space can be condensed and expressed in unambiguous mathematical point of view in the form of a universal definition.

As a tensor product of vector spaces V and W, which is vector space in which the tensor products of vectors in V and W " live ", each vector space X ( over the common Skalarenkörper of V and W ) refers to the there is a bilinear Figure out there that satisfies the following universal property:

Is there such a vector space, it is ( up to isomorphism ) unique. It is and noted. The universal property can therefore be written as, often you decided not to award different names because the domain is implicit in the argument.

To actually specify vector spaces that meet this definition, there are two common ways. Once specified in the finite case, the space of bilinear forms on the dual spaces, as in the following, and on the other by constructing a simple to be specified, but the large room, from which a quotient space of a suitable subspace receives the properties of the tensor product. The latter design is discussed further below in the context of modules over rings.

Tensor and bilinear forms

Bilinear forms correspond to linear maps.

It is a bilinear form. Then one can show that

A well-defined linear map.

Conversely, if

A linear mapping, then the Figure

Bilinear.

In the case of finite dimensional vector spaces, one can define the tensor product of and so as the dual space of the vector space of all bilinear mappings.

One reason why one does not work, instead of the tensor product with the space of bilinear forms is the following: multi- linear forms, say for example pictures

, which are linear for three - vector spaces in each component correspond to linear maps

But there is no similar express simple way, spaces of multilinear forms through spaces of bilinear forms; denotes

The rooms

The use of

Can be canonically identified. This identification corresponds to the fact that one out of a multi- linear form

On the one hand by holding the argument of a bilinear form

On the other hand by holding the argument of a bilinear form

Can get.

Extension of scalars

If V is a vector space over K and L is an extension field of K, one can the tensor product

Form by also conceives L as a K- vector space; This is symbolized by. VL is a vector space over L, if one

Sets. The dimension of VL as an L- vector space is equal to the dimension of V as a K- vector space: { ei} is a K- basis of V, so is the amount

An L- basis of VL.

Tensor product of representations

There are

Representations of a group on vector spaces over the same body, then defined

A representation

On the tensor product.

Tensor product over a ring

Let R be a ring ( having 1, but not necessarily commutative ). Let M be a right R- module and N a left R- module. The tensor product over R is defined by an abelian group

Together with a bilinear map

Satisfies the following universal property:

Using the universal property of a up to isomorphism uniquely determined tensor is defined.

The basic construction

However, this is not yet proven the existence. This proves one in which one the abelian group constructed as follows:

Considering all pairs of the generated free - module and the corresponding sub-module formed by all the elements

Is generated.

Is then defined by the quotient of by, in symbols:

Special cases

• M is a - SR bimodul with another ring S, then

• If R is commutative, then

• If A is an R- algebra, then

• If R is a commutative ring, and A and B are associative algebras R, then

Categorical properties

Different variants of the tensor have rechtsadjungierte functors:

• If R is a ring, M a right R- module, N a left R- module and P is an abelian group, then:

• R is a ring, R a S -algebra M is a R- link module, and N is an S- link module, then:

• If R is a commutative ring with identity and M, N, P three R-modules, then:

In particular, the tensor product is a right exact functor.

The tensor product is the pushout in the category of commutative rings with identity; in particular, for a commutative ring with unity, the tensor product via the coproduct ( for a finite number of objects ) in the category of algebras.

Examples

• If R is a ring, I a two-sided ideal and M is a left R- module, then

• If R is a commutative ring with identity, then is

• Localizations of modules are tensor products with the localized rings, that is, for example,

Structure of the elements

Elementary tensors

A fundamental tensor or in pure tensor is a tensor element of the mold,.

General shape

Each element of the tensor product is a finite sum of elementary tensors. In general, it can not be written as an elementary tensor of each tensor.

For example, the tensor no elementary tensor in the tensor product, the standard basis vectors.

If R is a commutative ring and M is an element of a generated R - module, then any tensor of the tensor product is an elementary tensor for any R- module N.

Further terms

In algebra:

• Flatness
• Brauer group

In the differential geometry:

• Tensor field
• Differential form
• Vector bundles

In the functional analysis

• Projective tensor product ( Banach spaces, locally convex spaces )
• Injective tensor product ( Banach spaces, locally convex spaces )
• Hilbert space tensor
• Tensor product of von Neumann algebras
• Spatial tensor product (C *-algebras )
• Maximum tensor product (C *-algebras )