In mathematics, the tensor product
V ⊗ W
V x W → V ⊗ W
(v,w), v\inV,w\inW
V ⊗ W
An element of the form
v ⊗ w
V ⊗ W
V ⊗ W
V ⊗ W
V ⊗ W
The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from
V x W
V ⊗ W\toZ
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 (so like an vector field but with tensors instead of vectors), 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.
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, 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.
BV
The tensor product
V ⊗ W
v ⊗ w
v\inBV
V ⊗ W
BV x BW
V ⊗ W
(v,w)
BV x BW
The set
\{v ⊗ w\midv\inBV,w\inBW\}
BV
We can equivalently define
V ⊗ W
V x W
(x,y)\inV x W
x
y
BV
BW
xv
yw
B
Hence, we see that the value of
B
(x,y)\inV x W
v ⊗ w
BV x BW
v ⊗ w:V x W\toF
Then we can express any bilinear form
B
v ⊗ w
Hom(V,W;F)
B
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\inV
y\inW
This definition is quite clearly derived from the coefficients of
B(v,w)
B(x,y)
BV
{ ⊗ }:(x,y)\mapstox ⊗ y
V x W
V ⊗ W
If arranged into a rectangular array, the coordinate vector of
x ⊗ y
x
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.
A construction of the tensor product that is basis independent can be obtained in the following way.
Let and be two vector spaces over a field .
V x W
(v,w)
v\inV
V x W\toF
(v,w)
(v,w)
Let be the linear subspace of that is spanned by the relations that the tensor product must satisfy. More precisely, is spanned by the elements of one of the forms:
\begin{align} (v1+v2,w)&-(v1,w)-(v2,w),\\ (v,w1+w2)&-(v,w1)-(v,w2),\\ (sv,w)&-s(v,w),\\ (v,sw)&-s(v,w), \end{align}
w,w1,w2\inW
Then, the tensor product is defined as the quotient space:
V ⊗ W=L/R,
(v,w)
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.)
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 and is a vector space denoted as, together with a bilinear map
{ ⊗ }:(v,w)\mapstov ⊗ w
V x W
h=\tildeh\circ{ ⊗ }
h(v,w)=\tildeh(v ⊗ w)
v\inV
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.
For example, it follows immediately that if
m
n
Z:=\Complexmn
T:\Complexm x \Complexn\to\Complexmn
(x,y)=\left(\left(x1,\ldots,xm\right),\left(y1,\ldots,yn\right)\right)
\left(xiyj\right)\stackrel{i=1,\ldots,m{j=1,\ldots,n}}
X:=\Complexm
T
⊗
x ⊗ y := T(x,y)
As another example, suppose that
\ComplexS
S
f+g
s\mapstof(s)+g(s)
cf
S
T
f\in\ComplexS
f ⊗ g\in\ComplexS
X\subseteq\ComplexS
Y\subseteq\ComplexT
Z:=\operatorname{span}\left\{f ⊗ g:f\inX,g\inY\right\}
\ComplexS
X
If and are vectors spaces of finite dimension, then
V ⊗ W
This results from the fact that a basis of
V ⊗ W
The tensor product is associative in the sense that, given three vector spaces, there is a canonical isomorphism:
(U ⊗ V) ⊗ W\congU ⊗ (V ⊗ W),
(u ⊗ v) ⊗ w
This allows omitting parentheses in the tensor product of more than two vector spaces or vectors.
The tensor product of two vector spaces
V
W
V ⊗ W\congW ⊗ V,
v ⊗ w
On the other hand, even when, the tensor product of vectors is not commutative; that is, in general.
The map
x ⊗ y\mapstoy ⊗ x
V ⊗ V
V ⊗
x1 ⊗ … ⊗ xn\mapstoxs(1) ⊗ … ⊗ xs(n)
Given a linear map, and a vector space, the tensor product:
f ⊗ W:U ⊗ W\toV ⊗ W
(f ⊗ W)(u ⊗ w)=f(u) ⊗ w.
W ⊗ f
Given two linear maps
f:U\toV
f ⊗ g:U ⊗ W\toV ⊗ Z
(f ⊗ g)(u ⊗ w)=f(u) ⊗ g(w).
One has:
f ⊗ g=(f ⊗ Z)\circ(U ⊗ g)=(V ⊗ g)\circ(f ⊗ W).
In terms of category theory, this means that the tensor product is a bifunctor from the category of vector spaces to itself.[1]
If and 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 and can be represented by matrices. Then, depending on how the tensor
v ⊗ w
f ⊗ g
The resultant rank is at most 4, and thus the resultant dimension is 4. 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). .
A dyadic product is the special case of the tensor product between two vectors of the same dimension.
See also: Tensor. For non-negative integers and a type
(r,s)
V*
There is a product map, called the :
It is defined by grouping all occurring "factors" together: writing
vi
fi
If is finite dimensional, then picking a basis of and the corresponding dual basis of
V*
r(V) | |
T | |
s |
F\in
0 | |
T | |
m |
Thus, the components of the tensor product of two tensors are the ordinary product of the components of each tensor. Another example: let be a tensor of type with components, and let be a tensor of type
(1,0)
Tensors equipped with their product operation form an algebra, called the tensor algebra.
For tensors of type there is a canonical evaluation map:defined by its action on pure tensors:
More generally, for tensors of type, with, there is a map, called tensor contraction:(The copies of
V
V*
On the other hand, if
V
v1,\ldots,vn
* | |
v | |
i |
The interplay of evaluation and coevaluation can be used to characterize finite-dimensional vector spaces without referring to bases.[4]
The tensor product
r | |
T | |
s(V) |
End(V)
u*\inEnd\left(V*\right)
There is a canonical isomorphism
1 | |
T | |
1(V) |
\toEnd(V)
Under this isomorphism, every in
End(V)
1 | |
T | |
1(V) |
Given two finite dimensional vector spaces, over the same field, denote the dual space of as, and the -vector space of all linear maps from to as . There is an isomorphism:defined by an action of the pure tensor
f ⊗ v\inU* ⊗ V
Its "inverse" can be defined using a basis
\{ui\}
* | |
\{u | |
i\} |
This result implies:which automatically gives the important fact that
\{ui ⊗ vj\}
U ⊗ V
\{ui\},\{vj\}
Furthermore, given three vector spaces,, the tensor product is linked to the vector space of all linear maps, as follows:This is an example of adjoint functors: the tensor product is "left adjoint" to Hom.
See main article: Tensor product of modules. The tensor product of two modules and over a commutative ring is defined in exactly the same way as the tensor product of vector spaces over a field:where now
F(A x B)
More generally, the tensor product can be defined even if the ring is non-commutative. In this case has to be a right--module and is a left--module, and instead of the last two relations above, the relation:is imposed. If is non-commutative, this is no longer an -module, but just an abelian group.
The universal property also carries over, slightly modified: the map
\varphi:A x B\toA ⊗ RB
(a,b)\mapstoa ⊗ b
The first two properties make a bilinear map of the abelian group . For any middle linear map
\psi
A ⊗ RB
\varphi
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:where
F(A x B)
A x B
F(A x B)
The universal property can be stated as follows. Let G be an abelian group with a map
q:A x B\toG
Then there is a unique map
\overline{q}:A ⊗ B\toG
\overline{q}(a ⊗ b)=q(a,b)
a\inA
Furthermore, we can give
A ⊗ RB
A ⊗ RB
A ⊗ RB
A ⊗ RB
ra:=ar
A ⊗ RB
For vector spaces, the tensor product
V ⊗ W
More generally, given a presentation of some -module, that is, a number of generators
mi\inM,i\inI
Here, and the map
NJ\toNI
n\inN
NJ
aijn
See main article: Tensor product of algebras. Let be a commutative ring. The tensor product of -modules applies, in particular, if and are -algebras. In this case, the tensor product
A ⊗ RB
A particular example is when and are fields containing a common subfield . The tensor product of fields is closely related to Galois theory: if, say,, where is some irreducible polynomial with coefficients in, the tensor product can be calculated as:where now is interpreted as the same polynomial, but with its coefficients regarded as elements of . In the larger field, the polynomial may become reducible, which brings in Galois theory. For example, if is a Galois extension of, then:is isomorphic (as an -algebra) to the .
A
K
\psi:Pn-1\toPn-1
A
\psi
A
A
n
A
K
A=
(a | |
i1i2 … id |
)
d
n x n x … x n
(a | |
i1i2 … id |
)
K
A\in(Kn) ⊗
Kn\toKn
Pn-1\toPn-1
Thus each of the
n
\psi
\psii
d-1
A
2 x 2
See main article: Topological tensor product and Tensor product of Hilbert spaces.
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;[7] the morphisms for that property must be restricted to Hilbert–Schmidt operators.[8]
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.
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).
See main article: 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.
See main article: Tensor product of quadratic forms.
Given two multilinear forms
f(x1,...,xk)
g(x1,...,xm)
V
K
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.
See main article: Sheaf of modules.
See also: Tensor product bundle.
See main article: Tensor product of fields.
See main article: Tensor product of graphs. 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.
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.
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, the exterior product
V\wedgeV
When the underlying field of does not have characteristic 2, then this definition is equivalent to:
The image of
v1 ⊗ v2
v1\wedgev2
V ⊗ ... ⊗ V
The symmetric algebra is constructed in a similar manner, from the symmetric product:
More generally:
That is, in the symmetric algebra two adjacent vectors (and therefore all of them) can be interchanged. The resulting objects are called symmetric tensors.
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).