In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the tensor product
of two
vector spaces and (over the same
field) is a vector space to which is associated a
bilinear map that maps a pair
to an element of
denoted
An element of the form
is called the tensor product of and . An element of
is a
tensor, and the tensor product of two vectors is sometimes called an ''elementary tensor'' or a ''decomposable tensor''. The elementary tensors span
in the sense that every element of
is a sum of elementary tensors. If
bases are given for and , a basis of
is formed by all tensor products of a basis element of and a basis element of .
The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from
into another vector space factors uniquely through a
linear map (see
Universal property).
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
vector field of tensors, one at each point of the
space-time manifold, and each belonging to the tensor product with itself of the
cotangent space at the point.
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 , below. As for every universal property, all
objects
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ai ...
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 and be two
vector spaces over a
field , with respective
bases and
The ''tensor product''
of and is a vector space which has as a basis the set of all
with
and
This definition can be formalized in the following way (this formalization is rarely used in practice, as the preceding informal definition is generally sufficient):
is the set of the
functions from the
Cartesian product to that have a finite number of nonzero values. The
pointwise operations make
a vector space. The function that maps
to and the other elements of
to is denoted
The set
is straightforwardly a basis of
which is called the ''tensor product'' of the bases
and
The ''tensor product of two vectors'' is defined from their decomposition on the bases. More precisely, if
are vectors decomposed on their respective bases, then the tensor product of and is
If arranged into a rectangular array, the
coordinate vector of
is the
outer product of the coordinate vectors of and . Therefore, the tensor product is a generalization of the outer product.
It is straightforward to verify that the map
is a bilinear map from
to
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 and be two
vector spaces over a
field .
One considers first a vector space that has the
Cartesian product as a
basis. That is, the basis elements of are the
pairs with
and
To get such a vector space, one can define it as the vector space of the
functions that have a finite number of nonzero values, and identifying
with the function that takes the value on
and otherwise.
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
:
where
and
Then, the tensor product is defined as the
quotient space
:
and the image of
in this quotient is denoted
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
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
from
to
such that, for every bilinear map
there is a ''unique'' linear map
such that
(that is,
for every
and
).
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.
For example, it follows immediately that if
and
are positive integers then
and the bilinear map
defined by sending
to
form a tensor product of
and
Often, this map
will be denoted by
so that
denotes this bilinear map's value at
As another example, suppose that
is the vector space of all complex-valued functions on a set
with addition and scalar multiplication defined pointwise (meaning that
is the map
and
is the map
). Let
and
be any sets and for any
and
let
denote the function defined by
If
and
are vector subspaces then the vector subspace
of
together with the bilinear map
form a tensor product of
and
Properties
Dimension
If and are vectors spaces of finite
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
, then
is finite-dimensional, and its dimension is the product of the dimensions of and .
This results from the fact that a basis of
is formed by taking all tensor products of a basis element of and a basis element of .
Associativity
The tensor product is
associative in the sense that, given three vector spaces
there is a canonical isomorphism
:
that maps
to
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
and
is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
in the sense that there is a canonical isomorphism
:
that maps
to
On the other hand, even when
the tensor product of vectors is not commutative; that is
in general.
The map
from
to itself induces a linear
automorphism that is called a .
More generally and as usual (see
tensor algebra), let denote
the tensor product of copies of the vector space . For every
permutation of the first positive integers, the map
:
induces a linear automorphism of
which is called a braiding map.
Tensor product of linear maps
Given a linear map
and a vector space , the ''tensor product''
:
is the unique linear map such that
:
The tensor product
is defined similarly.
Given two linear maps
and
their tensor product
:
is the unique linear map that satisfies
:
One has
:
In terms of
category theory, this means that the tensor product is a
bifunctor from the
category of vector spaces to itself.
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
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Mu ...
; 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 functor
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Mu ...
s).
By choosing bases of all vector spaces involved, the linear maps and can be represented by
matrices. Then, depending on how the tensor
is vectorized, the matrix describing the tensor product
is the
Kronecker product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to ...
of the two matrices. For example, if , and above are all two-dimensional and bases have been fixed for all of them, and and are given by the matrices
respectively, then the tensor product of these two matrices is
The resultant rank is at most 4, and thus the resultant dimension is 4. Note that 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). Note
A
dyadic product is the special case of the tensor product between two vectors of the same dimension.
General tensors
For non-negative integers and a type
tensor on a vector space is an element of
Here
is the
dual vector space (which consists of all
linear maps from to the ground field ).
There is a product map, called the
It is defined by grouping all occurring "factors" together: writing
for an element of and
for an element of the dual space,
Picking a basis of and the corresponding
dual basis of
naturally induces a basis for
(this basis is described in the
article on Kronecker products). In terms of these bases, the
components
Circuit Component may refer to:
•Are devices that perform functions when they are connected in a circuit.
In engineering, science, and technology Generic systems
*System components, an entity with discrete structure, such as an assemb ...
of a (tensor) product of two (or more)
tensors can be computed. For example, if and are two
covariant tensors of orders and respectively (i.e.
and
), then the components of their tensor product are given by
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
with components
Then
and
Tensors equipped with their product operation form an
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, called the
tensor algebra.
Evaluation map and tensor contraction
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
and
on which this map is to be applied must be specified.)
On the other hand, if
is , there is a canonical map in the other direction (called the coevaluation map)
where
is any basis of
and
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
may be naturally viewed as a module for the
Lie algebra by means of the diagonal action: for simplicity let us assume
then, for each
where
is the
transpose of , that is, in terms of the obvious pairing on
There is a canonical isomorphism
given by
Under this isomorphism, every in
may be first viewed as an endomorphism of
and then viewed as an endomorphism of
In fact it is the
adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is ...
of
Linear maps as tensors
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
on an element of
Its "inverse" can be defined using a basis
and its dual basis
as in the section "
Evaluation map and tensor contraction" above:
This result implies
which automatically gives the important fact that
forms a basis for
where
are bases of and .
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.
Tensor products of modules over a ring
The tensor product of two
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
and over a ''
commutative''
ring is defined in exactly the same way as the tensor product of vector spaces over a field:
where now
is the
free -module generated by the cartesian product and is the -module generated by
the same relations as above.
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
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
.
The universal property also carries over, slightly modified: the map
defined by
is a
middle linear map (referred to as "the canonical middle linear map".); that is, it satisfies:
[
]
The first two properties make a bilinear map of the
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
For any middle linear map
of
a unique group homomorphism of
satisfies
and this property determines
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
where
is a
free abelian group over
and G is the subgroup of
generated by relations
The universal property can be stated as follows. Let ''G'' be an abelian group with a map
that is bilinear, in the sense that
Then there is a unique map
such that
for all
and
Furthermore, we can give
a module structure under some extra conditions:
# If ''A'' is a (''S'',''R'')-bimodule, then
is a left ''S''-module where
# If ''B'' is a (''R'',''S'')-bimodule, then
is a right ''S''-module where
# If ''A'' is a (''S'',''R'')-bimodule and ''B'' is a (''R'',''T'')-bimodule, then
is a (''S'',''T'')-bimodule, where the left and right actions are defined in the same way as the previous two examples.
# If ''R'' is a commutative ring, then ''A'' and ''B'' are (''R'',''R'')-bimodules where
and
By 3), we can conclude
is a (''R'',''R'')-bimodule.
Computing the tensor product
For vector spaces, the tensor product
is quickly computed since bases of of immediately determine a basis of
as was mentioned above. For modules over a general (commutative) ring, not every module is free. For example, is not a free abelian group (-module). The tensor product with is given by
More generally, given a
presentation of some -module , that is, a number of generators
together with relations
the tensor product can be computed as the following
cokernel:
Here
and the map
is determined by sending some
in the th copy of
to
(in
). Colloquially, this may be rephrased by saying that a presentation of gives rise to a presentation of
This is referred to by saying that the tensor product is a
right exact functor
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Mu ...
. It is not in general left exact, that is, given an injective map of -modules
the tensor product
is not usually injective. For example, tensoring the (injective) map given by multiplication with , with yields the zero map , which is not injective. Higher
Tor functor
In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to co ...
s 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
Let be a commutative ring. The tensor product of -modules applies, in particular, if and are
-algebras. In this case, the tensor product
is an -algebra itself by putting
For example,
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
Eigenconfigurations of tensors
Square
matrices with entries in a
field represent
linear maps of
vector spaces, say
and thus linear maps
of
projective spaces over
If
is
nonsingular
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplic ...
then
is
well-defined everywhere, and the
eigenvectors of
correspond to the fixed points of
The ''eigenconfiguration'' of
consists of
points in
provided
is generic and
is
algebraically closed. The fixed points of nonlinear maps are the eigenvectors of tensors. Let
be a
-dimensional tensor of format
with entries
lying in an algebraically closed field
of
characteristic zero. Such a tensor
defines
polynomial maps and
with coordinates
Thus each of the
coordinates of
is a
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
of degree
in
The eigenvectors of
are the solutions of the constraint
and the eigenconfiguration is given by the
variety of the
minors of this matrix.
Other examples of tensor products
Tensor product of Hilbert spaces
Hilbert spaces generalize finite-dimensional vector spaces to
countably-infinite dimensions. The tensor product is still defined; it is the
tensor product of Hilbert spaces.
Topological tensor product
When the basis for a vector space is no longer countable, then the appropriate axiomatic formalization for the vector space is that of a
topological vector space. The tensor product is still defined, it is the
topological tensor product In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see Tensor product of Hi ...
.
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 quadratic forms
Tensor product of multilinear forms
Given two
multilinear forms
and
on a vector space
over the field
their tensor product is the multilinear form
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
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to ...
.
Tensor product of sheaves of modules
Tensor product of line bundles
Tensor product of fields
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.
Monoidal categories
The most general setting for the tensor product is the
monoidal category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left ...
. 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 , the exterior product
is defined as
Note that when the underlying field of does not have characteristic 2, then this definition is equivalent to
The image of
in the exterior product is usually denoted
and satisfies, by construction,
Similar constructions are possible for
( factors), giving rise to
the th
exterior power of . The latter notion is the basis of
differential -forms.
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.
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
).
Note that 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
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
), and/or may not support
higher-order functions such as the
Jacobian derivative (for example,
Fortran/APL).
See also
*
*
*
*
*
*
Notes
References
*
*
*
*
*
*
*
*
*
*
{{DEFAULTSORT:Tensor Product
Operations on vectors
Operations on structures
Bilinear maps