Tensor product

In mathematics, the tensor product $V\otimes W$ of two vector spaces $V$ and $W$ (over the same field) is a vector space to which is associated a bilinear map $V\times W\to V\otimes W$ that maps a pair $(v,w)$, where $v\in V,w\in W$, to an element of $V\otimes W$ denoted Template:Tmath.

An element of the form $v\otimes w$ is called the tensor product of $v$ and $w$. An element of $V\otimes W$ is a tensor, and the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span $V\otimes W$ in the sense that every element of $V\otimes W$ is a sum of elementary tensors. If bases are given for $V$ and $W$, a basis of $V\otimes W$ is formed by all tensor products of a basis element of $V$ and a basis element of $W$.

The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from $V\times W$ into another vector space $Z$ factors uniquely through a linear map $V\otimes W\to Z$ (see Template:Slink), i.e. the bilinear map is associated to a unique linear map from the tensor product $V\otimes W$ to $Z$.

Tensor products are used in many application areas, including physics and engineering. For example, in general relativity, the gravitational field is described through the metric tensor, which is a tensor field with one tensor at each point of the space-time manifold, and each belonging to the tensor product of the cotangent space at the point with itself.

Definitions and constructions

The tensor product of two vector spaces is a vector space that is defined up to an isomorphism. There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.

The tensor product can also be defined through a universal property; see Template:Slink, below. As for every universal property, all objects that satisfy the property are isomorphic through a unique isomorphism that is compatible with the universal property. When this definition is used, the other definitions may be viewed as constructions of objects satisfying the universal property and as proofs that there are objects satisfying the universal property, that is that tensor products exist.

From bases

Let Template:Mvar and Template:Mvar be two vector spaces over a field Template:Mvar, with respective bases $B_V$ and Template:Tmath.

The tensor product $V\otimes W$ of Template:Mvar and Template:Mvar is a vector space that has as a basis the set of all $v\otimes w$ with $v\in B_V$ and Template:Tmath, regarded purely as symbols with no further meaning. This definition can be formalized in the following way (this formalization is rarely used in practice, as the preceding informal definition is generally sufficient): $V\otimes W$ is the set of functions from the Cartesian product $B_V\times B_W$ to Template:Mvar that have a finite number of nonzero values. The pointwise operations make $V\otimes W$ a vector space. The function that maps $(v,w)$ to Template:Math and the other elements of $B_V\times B_W$ to Template:Math is denoted Template:Tmath.

The set $\{v\otimes w\mid v\in B_V,w\in B_W\}$, called the tensor product of the bases $B_V$ and Template:Tmath, is straightforwardly a basis of Template:Tmath.

Equivalently, we can define $V\otimes W$ to be the set of bilinear forms on $V\times W$ that are nonzero at only a finite number of elements of Template:Tmath. To see this, given $(x,y)\in V\times W$ and a bilinear form Template:Tmath, we can decompose $x$ and $y$ in the bases $B_V$ and $B_W$ as: $$x=\sum _{v\in B_V}{x}_{v}v\text{and}y=\sum _{w\in B_W}{y}_{w}w,$$ where only a finite number of $x_v$'s and $y_w$'s are nonzero, and find by the bilinearity of $B$ that: $$B(x,y)=\sum _{v\in B_V}\sum _{w\in B_W}{x}_v{y}_{w}B(v,w)$$

Hence, we see that the value of $B$ for any $(x,y)\in V\times W$ is uniquely and totally determined by the values that it takes on Template:Tmath. This lets us extend the maps $v\otimes w$ defined on $B_V\times B_W$ as before into bilinear maps $v\otimes w:V\times W\to F$ , by letting: $$(v\otimes w)(x,y):=\sum _{v^{\prime }\in B_V}\sum _{w^{\prime }\in B_W}{x}_{v^{\prime }}{y}_{w^{\prime }}(v\otimes w)(v^{\prime },w^{\prime })=x_v{y}_w.$$

Then we can express any bilinear form $B$ as a (potentially infinite) formal linear combination of the $v\otimes w$ maps according to: $$B=\sum _{v\in B_V}\sum _{w\in B_W}B(v,w)(v\otimes w)$$ making these maps similar to a Schauder basis for the vector space $\text{Hom}(V,W;F)$ of all bilinear forms on Template:Tmath. To instead have it be a proper Hamel basis, it only remains to add the requirement that $B$ is nonzero at an only a finite number of elements of Template:Tmath, and consider the subspace of such maps instead.

In either construction, the tensor product of two vectors is defined from their decomposition on the bases. More precisely, taking the basis decompositions of $x\in V$ and $y\in W$ as before: $$\begin{array}{cc}x\otimes y& =\left(\sum _{v\in B_V}{x}_{v}v\right)\otimes \left(\sum _{w\in B_W}{y}_{w}w\right)\\ & =\sum _{v\in B_V}\sum _{w\in B_W}{x}_v{y}_{w}v\otimes w.\end{array}$$

This definition is quite clearly derived from the coefficients of $B(v,w)$ in the expansion by bilinearity of $B(x,y)$ using the bases $B_V$ and Template:Tmath, as done above. It is then straightforward to verify that with this definition, the map $\otimes :(x,y)\mapsto x\otimes y$ is a bilinear map from $V\times W$ to $V\otimes W$ satisfying the universal property that any construction of the tensor product satisfies (see below).

If arranged into a rectangular array, the coordinate vector of $x\otimes y$ is the outer product of the coordinate vectors of $x$ and Template:Tmath. Therefore, the tensor product is a generalization of the outer product, that is, an abstraction of it beyond coordinate vectors.

A limitation of this definition of the tensor product is that, if one changes bases, a different tensor product is defined. However, the decomposition on one basis of the elements of the other basis defines a canonical isomorphism between the two tensor products of vector spaces, which allows identifying them. Also, contrarily to the two following alternative definitions, this definition cannot be extended into a definition of the tensor product of modules over a ring.

As a quotient space

A construction of the tensor product that is basis independent can be obtained in the following way.

Let Template:Mvar and Template:Mvar be two vector spaces over a [[field (mathematics)|field Template:Mvar]].

One considers first a vector space Template:Mvar that has the Cartesian product $V\times W$ as a basis. That is, the basis elements of Template:Mvar are the pairs $(v,w)$ with $v\in V$ and Template:Tmath. To get such a vector space, one can define it as the vector space of the functions $V\times W\to F$ that have a finite number of nonzero values and identifying $(v,w)$ with the function that takes the value Template:Math on $(v,w)$ and Template:Math otherwise.

Let Template:Mvar be the linear subspace of Template:Mvar that is spanned by the relations that the tensor product must satisfy. More precisely, Template:Mvar is spanned by the elements of one of the forms:

$\begin{array}{cc}(v_1+v_2,w)& -(v_1,w)-(v_2,w),\\ (v,w_1+w_2)& -(v,w_1)-(v,w_2),\\ (sv,w)& -s(v,w),\\ (v,sw)& -s(v,w),\end{array}$

where Template:Tmath, $w,w_1,w_2\in W$ and Template:Tmath.

Then, the tensor product is defined as the quotient space:

$V\otimes W=L/R,$

and the image of $(v,w)$ in this quotient is denoted Template:Tmath.

It is straightforward to prove that the result of this construction satisfies the universal property considered below. (A very similar construction can be used to define the tensor product of modules.)

Universal property

Universal property of tensor product: if Template:Math is bilinear, there is a unique linear map Template:Math that makes the diagram commutative (that is, Template:Math).

In this section, the universal property satisfied by the tensor product is described. As for every universal property, two objects that satisfy the property are related by a unique isomorphism. It follows that this is a (non-constructive) way to define the tensor product of two vector spaces. In this context, the preceding constructions of tensor products may be viewed as proofs of existence of the tensor product so defined.

A consequence of this approach is that every property of the tensor product can be deduced from the universal property, and that, in practice, one may forget the method that has been used to prove its existence.

The "universal-property definition" of the tensor product of two vector spaces is the following (recall that a bilinear map is a function that is separately linear in each of its arguments):

The tensor product of two vector spaces Template:Mvar and Template:Mvar is a vector space denoted as Template:Tmath, together with a bilinear map $\phi :(v,w)\mapsto v\otimes w$ from $V\times W$ to Template:Tmath, such that, for every bilinear map Template:Tmath, there is a unique linear map Template:Tmath, such that $h=\tilde{h}\circ \phi$ (that is, $h(v,w)=\tilde{h}(v\otimes w)$ for every $v\in V$ and Template:Tmath).

Linearly disjoint

Like the universal property above, the following characterization may also be used to determine whether or not a given vector space and given bilinear map form a tensor product.Template:Sfn

For example, it follows immediately that if Template:Tmath and Template:Tmath, where $m$ and $n$ are positive integers, then one may set $Z={ℂ}^{mn}$ and define the bilinear map as $$\begin{array}{cc}T:{ℂ}^m\times {ℂ}^n& \to {ℂ}^{mn}\\ (x,y)=((x_1,\dots ,x_m),(y_1,\dots ,y_n))& \mapsto (x_i{y}_j{)}_{\stackrel{i=1,\dots ,m}{j=1,\dots ,n}}\end{array}$$ to form the tensor product of $X$ and Template:Tmath.Template:Sfn Often, this map $T$ is denoted by $\otimes$ so that $x\otimes y=T(x,y).$

As another example, suppose that ${ℂ}^S$ is the vector space of all complex-valued functions on a set $S$ with addition and scalar multiplication defined pointwise (meaning that $f+g$ is the map $s\mapsto f(s)+g(s)$ and $cf$ is the map Template:Tmath). Let $S$ and $T$ be any sets and for any $f\in {ℂ}^S$ and Template:Tmath, let $f\otimes g\in {ℂ}^{S\times T}$ denote the function defined by Template:Tmath. If $X\subseteq {ℂ}^S$ and $Y\subseteq {ℂ}^T$ are vector subspaces then the vector subspace $Z:=\mathrm{span}\{f\otimes g:f\in X,g\in Y\}$ of ${ℂ}^{S\times T}$ together with the bilinear map: $$\begin{array}{cccccc}& & X\times Y& & \to & Z\\ & & (f,g)& & \mapsto & f\otimes g\end{array}$$ form a tensor product of $X$ and Template:Tmath.Template:Sfn

Properties

Dimension

If Template:Math and Template:Math are vector spaces of finite dimension, then $V\otimes W$ is finite-dimensional, and its dimension is the product of the dimensions of Template:Math and Template:Math.

This results from the fact that a basis of $V\otimes W$ is formed by taking all tensor products of a basis element of Template:Math and a basis element of Template:Math.

Associativity

The tensor product is associative in the sense that, given three vector spaces Template:Tmath, there is a canonical isomorphism:

$(U\otimes V)\otimes W\cong U\otimes (V\otimes W),$

that maps $(u\otimes v)\otimes w$ to Template:Tmath.

This allows omitting parentheses in the tensor product of more than two vector spaces or vectors.

Commutativity as vector space operation

The tensor product of two vector spaces $V$ and $W$ is commutative in the sense that there is a canonical isomorphism:

$V\otimes W\cong W\otimes V,$

that maps $v\otimes w$ to Template:Tmath.

On the other hand, even when Template:Tmath, the tensor product of vectors is not commutative; that is Template:Tmath, in general.

Template:Anchor The map $x\otimes y\mapsto y\otimes x$ from $V\otimes V$ to itself induces a linear automorphism that is called a Template:Vanchor. More generally and as usual (see tensor algebra), let $V^{\otimes n}$ denote the tensor product of Template:Mvar copies of the vector space Template:Mvar. For every permutation Template:Mvar of the first Template:Mvar positive integers, the map:

$x_1\otimes \cdots \otimes x_n\mapsto x_{s(1)}\otimes \cdots \otimes x_{s(n)}$

induces a linear automorphism of Template:Tmath, which is called a braiding map. This determines a right action of the symmetric group $S_n$ on $V^{\otimes n}$.

Tensor product of linear maps

Template:Redirect Given a linear map Template:Tmath, and a vector space Template:Mvar, the tensor product:

$f\otimes W:U\otimes W\to V\otimes W$

is the unique linear map such that:

$(f\otimes W)(u\otimes w)=f(u)\otimes w.$

The tensor product $W\otimes f$ is defined similarly.

Given two linear maps $f:U\to V$ and Template:Tmath, their tensor product:

$f\otimes g:U\otimes W\to V\otimes Z$

is the unique linear map that satisfies:

$(f\otimes g)(u\otimes w)=f(u)\otimes g(w).$

One has:

$f\otimes g=(f\otimes Z)\circ (U\otimes g)=(V\otimes g)\circ (f\otimes W).$

In terms of category theory, this means that the tensor product is a bifunctor from the category of vector spaces to itself.

If Template:Mvar and Template:Mvar are both injective or surjective, then the same is true for all above defined linear maps. In particular, the tensor product with a vector space is an exact functor; this means that every exact sequence is mapped to an exact sequence (tensor products of modules do not transform injections into injections, but they are right exact functors).

By choosing bases of all vector spaces involved, the linear maps Template:Math and Template:Math can be represented by matrices. Then, depending on how the tensor $v\otimes w$ is vectorized, the matrix describing the tensor product $f\otimes g$ is the Kronecker product of the two matrices. For example, if Template:Math, and Template:Math above are all two-dimensional and bases have been fixed for all of them, and Template:Math and Template:Math are given by the matrices: $$A=\left[\begin{array}{cc}{a}_{1,1}& a_{1,2}\\ a_{2,1}& a_{2,2}\end{array}\right],B=\left[\begin{array}{cc}{b}_{1,1}& b_{1,2}\\ b_{2,1}& b_{2,2}\end{array}\right],$$ respectively, then the tensor product of these two matrices is: $\begin{array}{cc}\left[\begin{array}{cc}{a}_{1,1}& a_{1,2}\\ a_{2,1}& a_{2,2}\end{array}\right]\otimes \left[\begin{array}{cc}{b}_{1,1}& b_{1,2}\\ b_{2,1}& b_{2,2}\end{array}\right]& =\left[\begin{array}{cc}{a}_{1,1}\left[\begin{array}{cc}{b}_{1,1}& b_{1,2}\\ b_{2,1}& b_{2,2}\end{array}\right]& a_{1,2}\left[\begin{array}{cc}{b}_{1,1}& b_{1,2}\\ b_{2,1}& b_{2,2}\end{array}\right]\\ a_{2,1}\left[\begin{array}{cc}{b}_{1,1}& b_{1,2}\\ b_{2,1}& b_{2,2}\end{array}\right]& a_{2,2}\left[\begin{array}{cc}{b}_{1,1}& b_{1,2}\\ b_{2,1}& b_{2,2}\end{array}\right]\end{array}\right]\\ & =\left[\begin{array}{cccc}{a}_{1,1}{b}_{1,1}& a_{1,1}{b}_{1,2}& a_{1,2}{b}_{1,1}& a_{1,2}{b}_{1,2}\\ a_{1,1}{b}_{2,1}& a_{1,1}{b}_{2,2}& a_{1,2}{b}_{2,1}& a_{1,2}{b}_{2,2}\\ a_{2,1}{b}_{1,1}& a_{2,1}{b}_{1,2}& a_{2,2}{b}_{1,1}& a_{2,2}{b}_{1,2}\\ a_{2,1}{b}_{2,1}& a_{2,1}{b}_{2,2}& a_{2,2}{b}_{2,1}& a_{2,2}{b}_{2,2}\end{array}\right].\end{array}$

The resultant rank is at most 4, and thus the resultant dimension is 4. Template:Em here denotes the tensor rank i.e. the number of requisite indices (while the matrix rank counts the number of degrees of freedom in the resulting array).Template:Clarification needed Template:Tmath.

A dyadic product is the special case of the tensor product between two vectors of the same dimension.

General tensors

Template:See also For non-negative integers Template:Math and Template:Math a type $(r,s)$ tensor on a vector space Template:Math is an element of: $$T_s^r(V)=\underset{r}{\underbrace{V\otimes \cdots \otimes V}}\otimes \underset{s}{\underbrace{V^{\ast }\otimes \cdots \otimes V^{\ast }}}=V^{\otimes r}\otimes {\left(V^{\ast }\right)}^{\otimes s}.$$ Here $V^{\ast }$ is the dual vector space (which consists of all linear maps Template:Math from Template:Math to the ground field Template:Math).

There is a product map, called the Template:Em:Template:Refn $$T_s^r(V){\otimes }_K{T}_{s^{\prime }}^{r^{\prime }}(V)\to T_{s+s^{\prime }}^{r+r^{\prime }}(V).$$

It is defined by grouping all occurring "factors" Template:Math together: writing $v_i$ for an element of Template:Math and $f_i$ for an element of the dual space: $$(v_1\otimes f_1)\otimes (v{\text{'}}_1)=v_1\otimes v{\text{'}}_1\otimes f_1.$$

If Template:Math is finite dimensional, then picking a basis of Template:Math and the corresponding dual basis of $V^{\ast }$ naturally induces a basis of $T_s^r(V)$ (this basis is described in the article on Kronecker products). In terms of these bases, the components of a (tensor) product of two (or more) tensors can be computed. For example, if Template:Math and Template:Math are two covariant tensors of orders Template:Math and Template:Math respectively (i.e. $F\in T_m^0$ and Template:Tmath), then the components of their tensor product are given by: $$(F\otimes G{)}_{i_1{i}_2\cdots i_{m+n}}=F_{i_1{i}_2\cdots i_m}{G}_{i_{m+1}{i}_{m+2}{i}_{m+3}\cdots i_{m+n}}.$$

Thus, the components of the tensor product of two tensors are the ordinary product of the components of each tensor. Another example: let Template:Math be a tensor of type Template:Math with components Template:Tmath, and let Template:Math be a tensor of type $(1,0)$ with components Template:Tmath. Then: $${(U\otimes V)}^{\alpha }{}_{\beta }{}^{\gamma }=U^{\alpha }{}_{\beta }{V}^{\gamma }$$ and: $$(V\otimes U{)}^{\mu \nu }{}_{\sigma }=V^{\mu }{U}^{\nu }{}_{\sigma }.$$

Tensors equipped with their product operation form an algebra, called the tensor algebra, which is graded by the order of a tensor.

Evaluation map and tensor contraction

For tensors of type Template:Math there is a canonical evaluation map: $$V\otimes V^{\ast }\to K$$ defined by its action on pure tensors: $$v\otimes f\mapsto f(v).$$

More generally, for tensors of type Template:Tmath, with Template:Math, there is a map, called tensor contraction: $$T_s^r(V)\to T_{s-1}^{r-1}(V).$$ (The copies of $V$ and $V^{\ast }$ on which this map is to be applied must be specified.)

On the other hand, if $V$ is Template:Em, there is a canonical map in the other direction (called the coevaluation map): $$\{\begin{array}{c}K\to V\otimes V^{\ast }\\ \lambda \mapsto \sum _i\lambda v_i\otimes v_i^{\ast }\end{array}$$ where $v_1,\dots ,v_n$ is any basis of Template:Tmath, and $v_i^{\ast }$ is its dual basis. This map does not depend on the choice of basis.

The interplay of evaluation and coevaluation can be used to characterize finite-dimensional vector spaces without referring to bases.

Adjoint representation

The tensor product $T_s^r(V)$ may be naturally viewed as a module for the Lie algebra $\mathrm{E}\mathrm{n}\mathrm{d}(V)$ by means of the diagonal action: for simplicity let us assume Template:Tmath, then, for each Template:Tmath, $$u(a\otimes b)=u(a)\otimes b-a\otimes u^{\ast }(b),$$ where $u^{\ast }\in \mathrm{E}\mathrm{n}\mathrm{d}\left(V^{\ast }\right)$ is the transpose of Template:Math, that is, in terms of the obvious pairing on Template:Tmath, $$⟨u(a),b⟩=⟨a,u^{\ast }(b)⟩.$$

There is a canonical isomorphism $T_1^1(V)\to \mathrm{E}\mathrm{n}\mathrm{d}(V)$ given by: $$(a\otimes b)(x)=⟨x,b⟩a.$$

Under this isomorphism, every Template:Math in $\mathrm{E}\mathrm{n}\mathrm{d}(V)$ may be first viewed as an endomorphism of $T_1^1(V)$ and then viewed as an endomorphism of Template:Tmath. In fact it is the adjoint representation Template:Math of Template:Tmath.

Linear maps as tensors

Given two finite dimensional vector spaces Template:Math, Template:Math over the same field Template:Math, denote the dual space of Template:Math as Template:Math, and the Template:Math-vector space of all linear maps from Template:Math to Template:Math as Template:Math. There is an isomorphism: $$U^{\ast }\otimes V\cong \mathrm{H}\mathrm{o}\mathrm{m}(U,V),$$ defined by an action of the pure tensor $f\otimes v\in U^{\ast }\otimes V$ on an element of Template:Tmath, $$(f\otimes v)(u)=f(u)v.$$

Its "inverse" can be defined using a basis $\{u_i\}$ and its dual basis $\{u_i^{\ast }\}$ as in the section "Evaluation map and tensor contraction" above: $$\{\begin{array}{c}\mathrm{H}\mathrm{o}\mathrm{m}(U,V)\to U^{\ast }\otimes V\\ F\mapsto \sum _i{u}_i^{\ast }\otimes F(u_i).\end{array}$$

This result implies: $$\dim (U\otimes V)=\dim (U)\dim (V),$$ which automatically gives the important fact that $\{u_i\otimes v_j\}$ forms a basis of $U\otimes V$ where $\{u_i\},\{v_j\}$ are bases of Template:Math and Template:Math.

Furthermore, given three vector spaces Template:Math, Template:Math, Template:Math the tensor product is linked to the vector space of all linear maps, as follows: $$\mathrm{H}\mathrm{o}\mathrm{m}(U\otimes V,W)\cong \mathrm{H}\mathrm{o}\mathrm{m}(U,\mathrm{H}\mathrm{o}\mathrm{m}(V,W)).$$ This is an example of adjoint functors: the tensor product is "left adjoint" to Hom.

Tensor products of modules over a ring

Template:Main The tensor product of two modules Template:Math and Template:Math over a commutative ring Template:Math is defined in exactly the same way as the tensor product of vector spaces over a field: $$A{\otimes }_{R}B:=F(A\times B)/G,$$ where now $F(A\times B)$ is the [[Free module|free Template:Math-module]] generated by the cartesian product and Template:Math is the Template:Math-module generated by these relations.

More generally, the tensor product can be defined even if the ring is non-commutative. In this case Template:Math has to be a right-Template:Math-module and Template:Math is a left-Template:Math-module, and instead of the last two relations above, the relation: $$(ar,b)\sim (a,rb)$$ is imposed. If Template:Math is non-commutative, this is no longer an Template:Math-module, but just an abelian group.

The universal property also carries over, slightly modified: the map $\phi :A\times B\to A{\otimes }_{R}B$ defined by $(a,b)\mapsto a\otimes b$ is a middle linear map (referred to as "the canonical middle linear map"); that is, it satisfies: $$\begin{array}{cc}\phi (a+a^{\prime },b)& =\phi (a,b)+\phi (a^{\prime },b)\\ \phi (a,b+b^{\prime })& =\phi (a,b)+\phi (a,b^{\prime })\\ \phi (ar,b)& =\phi (a,rb)\end{array}$$

The first two properties make Template:Math a bilinear map of the abelian group Template:Tmath. For any middle linear map $\psi$ of Template:Tmath, a unique group homomorphism Template:Math of $A{\otimes }_{R}B$ satisfies Template:Tmath, and this property determines $\phi$ within group isomorphism. See the main article for details.

Tensor product of modules over a non-commutative ring

Let A be a right R-module and B be a left R-module. Then the tensor product of A and B is an abelian group defined by: $$A{\otimes }_{R}B:=F(A\times B)/G$$ where $F(A\times B)$ is a free abelian group over $A\times B$ and G is the subgroup of $F(A\times B)$ generated by relations: $$\begin{array}{cc}& \mathrm{\forall }a,a_1,a_2\in A,\mathrm{\forall }b,b_1,b_2\in B,\text{ for all }r\in R:\\ & (a_1,b)+(a_2,b)-(a_1+a_2,b),\\ & (a,b_1)+(a,b_2)-(a,b_1+b_2),\\ & (ar,b)-(a,rb).\end{array}$$

The universal property can be stated as follows. Let G be an abelian group with a map $q:A\times B\to G$ that is bilinear, in the sense that: $$\begin{array}{cc}q(a_1+a_2,b)& =q(a_1,b)+q(a_2,b),\\ q(a,b_1+b_2)& =q(a,b_1)+q(a,b_2),\\ q(ar,b)& =q(a,rb).\end{array}$$

Then there is a unique map $\bar{q}:A\otimes B\to G$ such that $\bar{q}(a\otimes b)=q(a,b)$ for all $a\in A$ and Template:Tmath.

Furthermore, we can give $A{\otimes }_{R}B$ a module structure under some extra conditions:

1. If A is a (S,R)-bimodule, then $A{\otimes }_{R}B$ is a left S-module, where Template:Tmath.
2. If B is a (R,S)-bimodule, then $A{\otimes }_{R}B$ is a right S-module, where Template:Tmath.
3. If A is a (S,R)-bimodule and B is a (R,T)-bimodule, then $A{\otimes }_{R}B$ is a (S,T)-bimodule, where the left and right actions are defined in the same way as the previous two examples.
4. If R is a commutative ring, then A and B are (R,R)-bimodules where $ra:=ar$ and Template:Tmath. By 3), we can conclude $A{\otimes }_{R}B$ is a (R,R)-bimodule.

Computing the tensor product

For vector spaces, the tensor product $V\otimes W$ is quickly computed since bases of Template:Math of Template:Math immediately determine a basis of Template:Tmath, as was mentioned above. For modules over a general (commutative) ring, not every module is free. For example, Template:Math is not a free abelian group (Template:Math-module). The tensor product with Template:Math is given by: $$M{\otimes }_{𝐙}𝐙/n𝐙=M/nM.$$

More generally, given a presentation of some Template:Math-module Template:Math, that is, a number of generators $m_i\in M,i\in I$ together with relations: $$\sum _{j\in J}{a}_{ji}{m}_i=0,a_{ij}\in R,$$ the tensor product can be computed as the following cokernel: $$M{\otimes }_{R}N=\mathrm{coker}(N^J\to N^I)$$

Here Template:Tmath, and the map $N^J\to N^I$ is determined by sending some $n\in N$ in the Template:Mathth copy of $N^J$ to $a_{ij}n$ (in Template:Tmath). Colloquially, this may be rephrased by saying that a presentation of Template:Math gives rise to a presentation of Template:Tmath. This is referred to by saying that the tensor product is a right exact functor. It is not in general left exact, that is, given an injective map of Template:Math-modules Template:Tmath, the tensor product: $$M_1{\otimes }_{R}N\to M_2{\otimes }_{R}N$$ is not usually injective. For example, tensoring the (injective) map given by multiplication with Template:Math, Template:Math with Template:Math yields the zero map Template:Math, which is not injective. Higher Tor functors measure the defect of the tensor product being not left exact. All higher Tor functors are assembled in the derived tensor product.

Tensor product of algebras

Template:Main Let Template:Math be a commutative ring. The tensor product of Template:Math-modules applies, in particular, if Template:Math and Template:Math are [[Algebra (ring theory)|Template:Math-algebras]]. In this case, the tensor product $A{\otimes }_{R}B$ is an Template:Math-algebra itself by putting: $$(a_1\otimes b_1)\cdot (a_2\otimes b_2)=(a_1\cdot a_2)\otimes (b_1\cdot b_2).$$ For example: $$R[x]{\otimes }_{R}R[y]\cong R[x,y].$$

A particular example is when Template:Math and Template:Math are fields containing a common subfield Template:Math. The tensor product of fields is closely related to Galois theory: if, say, Template:Math, where Template:Math is some irreducible polynomial with coefficients in Template:Math, the tensor product can be calculated as: $$A{\otimes }_{R}B\cong B[x]/f(x)$$ where now Template:Math is interpreted as the same polynomial, but with its coefficients regarded as elements of Template:Math. In the larger field Template:Math, the polynomial may become reducible, which brings in Galois theory. For example, if Template:Math is a Galois extension of Template:Math, then: $$A{\otimes }_{R}A\cong A[x]/f(x)$$ is isomorphic (as an Template:Math-algebra) to Template:Tmath.

Eigenconfigurations of tensors

Square matrices $A$ with entries in a field $K$ represent linear maps of vector spaces, say Template:Tmath, and thus linear maps $\psi :{\mathbb{P}}^{n-1}\to {\mathbb{P}}^{n-1}$ of projective spaces over Template:Tmath. If $A$ is nonsingular then $\psi$ is well-defined everywhere, and the eigenvectors of $A$ correspond to the fixed points of Template:Tmath. The eigenconfiguration of $A$ consists of $n$ points in Template:Tmath, provided $A$ is generic and $K$ is algebraically closed. The fixed points of nonlinear maps are the eigenvectors of tensors. Let $A=(a_{i_1{i}_2\cdots i_d})$ be a $d$-dimensional tensor of format $n\times n\times \cdots \times n$ with entries $(a_{i_1{i}_2\cdots i_d})$ lying in an algebraically closed field $K$ of characteristic zero. Such a tensor $A\in (K^n{)}^{\otimes d}$ defines polynomial maps $K^n\to K^n$ and ${\mathbb{P}}^{n-1}\to {\mathbb{P}}^{n-1}$ with coordinates: $${\psi }_i(x_1,\dots ,x_n)=\sum _{j_2=1}^n\sum _{j_3=1}^n\cdots \sum _{j_d=1}^n{a}_{i{j}_2{j}_3\cdots j_d}{x}_{j_2}{x}_{j_3}\cdots x_{j_d}\text{for }i=1,\dots ,n$$

Thus each of the $n$ coordinates of $\psi$ is a homogeneous polynomial ${\psi }_i$ of degree $d-1$ in Template:Tmath. The eigenvectors of $A$ are the solutions of the constraint: $$\text{rank}\left(\begin{array}{cccc}{x}_1& x_2& \cdots & x_n\\ {\psi }_1(𝐱)& {\psi }_2(𝐱)& \cdots & {\psi }_n(𝐱)\end{array}\right)\le 1$$ and the eigenconfiguration is given by the variety of the $2\times 2$ minors of this matrix.

Other examples of tensor products

Topological tensor products

Hilbert spaces generalize finite-dimensional vector spaces to arbitrary dimensions. There is an analogous operation, also called the "tensor product," that makes Hilbert spaces a symmetric monoidal category. It is essentially constructed as the metric space completion of the algebraic tensor product discussed above. However, it does not satisfy the obvious analogue of the universal property defining tensor products; the morphisms for that property must be restricted to Hilbert–Schmidt operators.

In situations where the imposition of an inner product is inappropriate, one can still attempt to complete the algebraic tensor product, as a topological tensor product. However, such a construction is no longer uniquely specified: in many cases, there are multiple natural topologies on the algebraic tensor product.

Tensor product of graded vector spaces

Some vector spaces can be decomposed into direct sums of subspaces. In such cases, the tensor product of two spaces can be decomposed into sums of products of the subspaces (in analogy to the way that multiplication distributes over addition).

Tensor product of representations

Vector spaces endowed with an additional multiplicative structure are called algebras. The tensor product of such algebras is described by the Littlewood–Richardson rule.

Tensor product of algebraic fields

Given two fields that are algebraic extensions of a common base field, their tensor product over the base field is again algebraic over the base field. Specifically, it is the algebraic extension generated by the products of the generators of the original two fields.Template:Cn For example, the tensor product over the rational numbers of $\mathbb{Q}[\sqrt{2}]$ and $\mathbb{Q}[\sqrt{3}]$ is generated by $1\cdot 1=1$, $1\cdot \sqrt{2}=\sqrt{2}$, $1\cdot \sqrt{3}=\sqrt{3}$, and $\sqrt{2}\cdot \sqrt{3}=\sqrt{6}$, and can be denoted $\mathbb{Q}[\sqrt{2},\sqrt{3}]$.

Tensor product of multilinear forms

Given two multilinear forms $f(x_1,\dots ,x_k)$ and $g(x_1,\dots ,x_m)$ on a vector space $V$ over the field $K$ their tensor product is the multilinear form: $$(f\otimes g)(x_1,\dots ,x_{k+m})=f(x_1,\dots ,x_k)g(x_{k+1},\dots ,x_{k+m}).$$

This is a special case of the product of tensors if they are seen as multilinear maps (see also tensors as multilinear maps). Thus the components of the tensor product of multilinear forms can be computed by the Kronecker product.

Tensor product of graphs

Template:Main It should be mentioned that, though called "tensor product", this is not a tensor product of graphs in the above sense; actually it is the category-theoretic product in the category of graphs and graph homomorphisms. However it is actually the Kronecker tensor product of the adjacency matrices of the graphs. Compare also the section Tensor product of linear maps above.

Monoidal categories

The most general setting for the tensor product is the monoidal category. It captures the algebraic essence of tensoring, without making any specific reference to what is being tensored. Thus, all tensor products can be expressed as an application of the monoidal category to some particular setting, acting on some particular objects.

Quotient algebras

A number of important subspaces of the tensor algebra can be constructed as quotients: these include the exterior algebra, the symmetric algebra, the Clifford algebra, the Weyl algebra, and the universal enveloping algebra in general.

The exterior algebra is constructed from the exterior product. Given a vector space Template:Math, the exterior product $V\wedge V$ is defined as: $$V\wedge V:=V\otimes V/\{v\otimes v\mid v\in V\}.$$

When the underlying field of Template:Math does not have characteristic 2, then this definition is equivalent to: $$V\wedge V:=V\otimes V/\{v_1\otimes v_2+v_2\otimes v_1\mid (v_1,v_2)\in V^2\}.$$

The image of $v_1\otimes v_2$ in the exterior product is usually denoted $v_1\wedge v_2$ and satisfies, by construction, Template:Tmath. Similar constructions are possible for $V\otimes \dots \otimes V$ (Template:Math factors), giving rise to Template:Tmath, the Template:Mathth exterior power of Template:Math. The latter notion is the basis of [[differential form|differential Template:Math-forms]].

The symmetric algebra is constructed in a similar manner, from the symmetric product: $$V\odot V:=V\otimes V/\{v_1\otimes v_2-v_2\otimes v_1\mid (v_1,v_2)\in V^2\}.$$

More generally: $${\mathrm{Sym}}^{n}V:=\underset{n}{\underbrace{V\otimes \dots \otimes V}}/(\dots \otimes v_i\otimes v_{i+1}\otimes \dots -\dots \otimes v_{i+1}\otimes v_i\otimes \dots )$$

That is, in the symmetric algebra two adjacent vectors (and therefore all of them) can be interchanged. The resulting objects are called symmetric tensors.

Tensor product in programming

Array programming languages

Array programming languages may have this pattern built in. For example, in APL the tensor product is expressed as ○.× (for example A ○.× B or A ○.× B ○.× C). In J the tensor product is the dyadic form of */ (for example a */ b or a */ b */ c).

J's treatment also allows the representation of some tensor fields, as a and b may be functions instead of constants. This product of two functions is a derived function, and if a and b are differentiable, then a */ b is differentiable.

However, these kinds of notation are not universally present in array languages. Other array languages may require explicit treatment of indices (for example, MATLAB), and/or may not support higher-order functions such as the Jacobian derivative (for example, Fortran/APL).

See also

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References

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