Tensor product
In mathematics, the tensor product of two vector spaces and 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 , i.e. the bilinear map is associated to a unique linear map from the tensor product to.
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, 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 and be two vector spaces over a field, with respective bases and.The tensor product of and is a vector space that has as a basis the set of all with and. This definition can be formalized in the following way : 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 then straightforwardly a basis of, which is called the tensor product of the bases and.
We can equivalently define to be the set of bilinear forms on that are nonzero at only a finite number of elements of. To see this, given and a bilinear form, we can decompose and in the bases and as:
where only a finite number of 's and 's are nonzero, and find by the bilinearity of that:
Hence, we see that the value of for any is uniquely and totally determined by the values that it takes on. This lets us extend the maps defined on as before into bilinear maps , by letting:
Then we can express any bilinear form as a formal linear combination of the maps according to:
making these maps similar to a Schauder basis for the vector space of all bilinear forms on. To instead have it be a proper Hamel basis, it only remains to add the requirement that is nonzero at an only a finite number of elements of, 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 and as before:
This definition is quite clearly derived from the coefficients of in the expansion by bilinearity of using the bases and, as done above. It is then straightforward to verify that with this definition, the map is a bilinear map from to satisfying the universal property that any construction of the tensor product satisfies.
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, 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 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.
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 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 :
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, where and are positive integers, then one may set and define the bilinear map as
to form the tensor product of and. Often, this map is denoted by so that
As another example, suppose that is the vector space of all complex-valued functions on a set with addition and scalar multiplication defined pointwise. 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 vector spaces of finite dimension, 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 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, 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; this means that every exact sequence is mapped to an exact sequence.
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 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. here denotes the tensor rank i.e. the number of requisite indices..
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.
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:
If is finite dimensional, then picking a basis of and the corresponding dual basis of naturally induces a basis of . In terms of these bases, the components of a product of two tensors can be computed. For example, if and are two covariant tensors of orders and respectively, 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, called the tensor algebra.