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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 V \otimes W of two vector spaces and (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),\ v\in V, w\in W to an element of V \otimes W denoted v \otimes w. An element of the form v \otimes w is called the tensor product of and . 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 and , a basis of V \otimes W 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 V\times W into another vector space factors uniquely through a linear map V\otimes W\to Z (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 B_V and B_W. The ''tensor product'' V \otimes W of and is a vector space which has as a basis the set of all v\otimes w with v\in B_V and w \in B_W. 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 the functions from the Cartesian product B_V \times B_W to that have a finite number of nonzero values. The pointwise operations make V \otimes W a vector space. The function that maps (v,w) \in B_V \times B_W to and the other elements of B_V \times B_W to is denoted v\otimes w. The set \ is straightforwardly a basis of V \otimes W, which is called the ''tensor product'' of the bases B_V and B_W. The ''tensor product of two vectors'' is defined from their decomposition on the bases. More precisely, if x=\sum_ x_b\,b \in V \quad \text\quad y=\sum_ y_c\,c \in W are vectors decomposed on their respective bases, then the tensor product of and is \begin x\otimes y&=\left(\sum_ x_b\,b\right) \otimes \left(\sum_ y_c\,c\right)\\ &=\sum_\sum_ x_b y_c\, b\otimes c. \end If arranged into a rectangular array, the coordinate vector of x\otimes y 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 (x,y)\mapsto x\otimes y is a bilinear map from V\times W to V\otimes W. 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 V\times W as a basis. That is, the basis elements of are the pairs (v,w) with v\in V and w\in W. 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 on (v,w) 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 :\begin (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 where v, v_1, v_2\in V, w, w_1, w_2 \in W and s\in F. 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 v\otimes 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.)


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 V\otimes W, together with a bilinear map \colon (v,w)\mapsto v\otimes w from V\times W to V\otimes W, such that, for every bilinear map h\colon V\times W\to Z, there is a ''unique'' linear map \tilde h\colon V\otimes W\to Z, such that h=\tilde h \circ (that is, h(v, w)= \tilde h(v\otimes w) for every v\in V and w\in W).


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 m and n are positive integers then Z := \Complex^ and the bilinear map T : \Complex^m \times \Complex^n \to \Complex^ defined by sending (x, y) = \left(\left(x_1, \ldots, x_m\right), \left(y_1, \ldots, y_n\right)\right) to \left(x_i y_j\right)_ form a tensor product of X := \Complex^m and Y := \Complex^n. Often, this map T will be denoted by \,\otimes\, so that x \otimes y \;:=\; T(x, y) denotes this bilinear map's value at (x, y) \in X \times Y. As another example, suppose that \Complex^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 c f is the map s \mapsto c f(s)). Let S and T be any sets and for any f \in \Complex^S and g \in \Complex^T, let f \otimes g \in \Complex^ denote the function defined by (s, t) \mapsto f(s) g(t). If X \subseteq \Complex^S and Y \subseteq \Complex^T are vector subspaces then the vector subspace Z := \operatorname \left\ of \Complex^ together with the bilinear map \begin \;&& X \times Y &&\;\to \;& Z \\ .3ex && (f, g) &&\;\mapsto\;& f \otimes g \\ \end form a tensor product of X and Y.


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 V\otimes W is finite-dimensional, and its dimension is the product of the dimensions of and . This results from the fact that a basis of V\otimes W 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 U, V, W, there is a canonical isomorphism :(U\otimes V)\otimes W\cong U\otimes (V\otimes W), that maps (u\otimes v)\otimes w to u\otimes (v \otimes w). 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 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 : V \otimes W \cong W\otimes V, that maps v \otimes w to w \otimes v. On the other hand, even when V=W, the tensor product of vectors is not commutative; that is v\otimes w \neq w \otimes v, in general. The map x\otimes y \mapsto y\otimes x from V\otimes V to itself induces a linear automorphism that is called a . More generally and as usual (see tensor algebra), let denote V^ the tensor product of copies of the vector space . For every permutation of the first positive integers, the map :x_1\otimes \cdots \otimes x_n \mapsto x_\otimes \cdots \otimes x_ induces a linear automorphism of V^\to V^, which is called a braiding map.


Tensor product of linear maps

Given a linear map f\colon U\to V, and a vector space , the ''tensor product'' :f\otimes W\colon 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\colon U\to V and g\colon W\to Z, their tensor product :f\otimes g\colon 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 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 v \otimes w is vectorized, the matrix describing the tensor product S \otimes T 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 A=\begin a_ & a_ \\ a_ & a_ \\ \end, \qquad B=\begin b_ & b_ \\ b_ & b_ \\ \end, respectively, then the tensor product of these two matrices is \begin a_ & a_ \\ a_ & a_ \\ \end \otimes \begin b_ & b_ \\ b_ & b_ \\ \end = \begin a_ \begin b_ & b_ \\ b_ & b_ \\ \end & a_ \begin b_ & b_ \\ b_ & b_ \\ \end \\ pt a_ \begin b_ & b_ \\ b_ & b_ \\ \end & a_ \begin b_ & b_ \\ b_ & b_ \\ \end \\ \end = \begin a_ b_ & a_ b_ & a_ b_ & a_ b_ \\ a_ b_ & a_ b_ & a_ b_ & a_ b_ \\ a_ b_ & a_ b_ & a_ b_ & a_ b_ \\ a_ b_ & a_ b_ & a_ b_ & a_ b_ \\ \end. 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 \operatorname A \otimes B = \operatorname A \times \operatorname B. 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 (r, s) tensor on a vector space is an element of T^r_s(V) = \underbrace_r \otimes \underbrace_s = V^ \otimes \left(V^*\right)^. Here V^* is the dual vector space (which consists of all linear maps from to the ground field ). There is a product map, called the T^r_s (V) \otimes_K T^_ (V) \to T^_(V). It is defined by grouping all occurring "factors" together: writing v_i for an element of and f_i for an element of the dual space, (v_1 \otimes f_1) \otimes (v'_1) = v_1 \otimes v'_1 \otimes f_1. Picking a basis of and the corresponding dual basis of V^* naturally induces a basis for T_s^r(V) (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. F \in T_m^0 and G \in T_n^0), then the components of their tensor product are given by (F \otimes G)_ = F_ G_. 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 U^_, and let be a tensor of type (1, 0) with components V^. Then \left(U \otimes V\right)^\alpha _\beta ^\gamma = U^\alpha _\beta V^\gamma and (V \otimes U)^ _\sigma = V^\mu U^\nu _\sigma. 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 V \otimes V^* \to K defined by its action on pure tensors: v \otimes f \mapsto f(v). More generally, for tensors of type (r, s), with , there is a map, called tensor contraction, T^r_s (V) \to T^_(V). (The copies of V and V^* on which this map is to be applied must be specified.) On the other hand, if V is , there is a canonical map in the other direction (called the coevaluation map) \begin K \to V \otimes V^* \\ \lambda \mapsto \sum_i \lambda v_i \otimes v^*_i \end where v_1, \ldots, v_n is any basis of V, and v_i^* 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^r_s(V) may be naturally viewed as a module for the Lie algebra \mathrm(V) by means of the diagonal action: for simplicity let us assume r = s = 1, then, for each u \in \mathrm(V), u(a \otimes b) = u(a) \otimes b - a \otimes u^*(b), where u^* \in \mathrm\left(V^*\right) is the transpose of , that is, in terms of the obvious pairing on V \otimes V^*, \langle u(a), b \rangle = \langle a, u^*(b) \rangle. There is a canonical isomorphism T^1_1(V) \to \mathrm(V) given by (a \otimes b)(x) = \langle x, b \rangle a. Under this isomorphism, every in \mathrm(V) may be first viewed as an endomorphism of T^1_1(V) and then viewed as an endomorphism of \mathrm(V). 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 \mathrm(V).


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, U^* \otimes V \cong \mathrm(U, V), defined by an action of the pure tensor f \otimes v \in U^*\otimes V on an element of U, (f \otimes v)(u) = f(u) v. Its "inverse" can be defined using a basis \ and its dual basis \ as in the section " Evaluation map and tensor contraction" above: \begin \mathrm (U,V) \to U^* \otimes V \\ F \mapsto \sum_i u^*_i \otimes F(u_i). \end This result implies \dim(U \otimes V) = \dim(U)\dim(V), which automatically gives the important fact that \ forms a basis for U \otimes V 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: \mathrm (U \otimes V, W) \cong \mathrm (U, \mathrm(V, W)). 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: A \otimes_R B := F (A \times B) / G where now F(A \times B) 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 (ar,b)\sim (a,rb) 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 \varphi : 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 \phi(a + a', b) &= \phi(a, b) + \phi(a', b) \\ \phi(a, b + b') &= \phi(a, b) + \phi(a, b') \\ \phi(ar, b) &= \phi(a, rb) \end 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 ...
A \times B. For any middle linear map \psi of A \times B, a unique group homomorphism of A \otimes_R B satisfies \psi = f \circ \varphi, 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 &\forall a, a_1, a_2 \in A, \forall b, b_1, b_2 \in B, \text 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 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 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 Then there is a unique map \overline:A\otimes B \to G such that \overline(a\otimes b) = q(a,b) for all a \in A and b \in B. Furthermore, we can give A \otimes_R B a module structure under some extra conditions: # If ''A'' is a (''S'',''R'')-bimodule, then A \otimes_R B is a left ''S''-module where s(a\otimes b):=(sa)\otimes b. # If ''B'' is a (''R'',''S'')-bimodule, then A \otimes_R B is a right ''S''-module where (a\otimes b)s:=a\otimes (bs). # 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. # If ''R'' is a commutative ring, then ''A'' and ''B'' are (''R'',''R'')-bimodules where ra:=ar and br:=rb. 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 of immediately determine a basis of V \otimes W, 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 M \otimes_\mathbf \mathbf/n\mathbf = M/nM. More generally, given a presentation of some -module , that is, a number of generators m_i \in M, i \in I together with relations \sum_ a_ m_i = 0,\qquad a_ \in R, the tensor product can be computed as the following cokernel: M \otimes_R N = \operatorname \left(N^J \to N^I\right) Here N^J = \oplus_ N, and the map N^J \to N^I is determined by sending some n \in N in the th copy of N^J to a_ n (in N^I). Colloquially, this may be rephrased by saying that a presentation of gives rise to a presentation of M \otimes_R N. 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 M_1 \to M_2, 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 , 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 A \otimes_R B is an -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 \otimes_R R \cong R , y 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 A \otimes_R B \cong B / f(x) 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 A \otimes_R A \cong A / f(x) is isomorphic (as an -algebra) to the A^.


Eigenconfigurations of tensors

Square matrices A with entries in a field K represent linear maps of vector spaces, say K^n \to K^n, and thus linear maps \psi : \mathbb^ \to \mathbb^ of projective spaces over K. If A 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 \psi is well-defined everywhere, and the eigenvectors of A correspond to the fixed points of \psi. The ''eigenconfiguration'' of A consists of n points in \mathbb^, provided A is generic and K is algebraically closed. The fixed points of nonlinear maps are the eigenvectors of tensors. Let A = (a_) be a d-dimensional tensor of format n \times n \times \cdots \times n with entries (a_) lying in an algebraically closed field K of characteristic zero. Such a tensor A \in (K^)^ defines polynomial maps K^n \to K^n and \mathbb^ \to \mathbb^ with coordinates \psi_i(x_1, \ldots, x_n) = \sum_^n \sum_^n \cdots \sum_^n a_ x_ x_\cdots x_ \;\; \mbox i = 1, \ldots, n Thus each of the n coordinates of \psi 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; ...
\psi_i of degree d-1 in \mathbf = \left(x_1, \ldots, x_n\right). The eigenvectors of A are the solutions of the constraint \mbox \begin x_1 & x_2 & \cdots & x_n \\ \psi_1(\mathbf) & \psi_2(\mathbf) & \cdots & \psi_n(\mathbf) \end \leq 1 and the eigenconfiguration is given by the variety of the 2 \times 2 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 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_) = f(x_1,\dots,x_k) g(x_,\dots,x_). 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 V \wedge V is defined as V \wedge V := V \otimes V/\. Note that when the underlying field of does not have characteristic 2, then this definition is equivalent to V \wedge V := V \otimes V / \. The image of v_1 \otimes v_2 in the exterior product is usually denoted v_1 \wedge v_2 and satisfies, by construction, v_1 \wedge v_2 = - v_2 \wedge v_1. Similar constructions are possible for V \otimes \dots \otimes V ( factors), giving rise to \Lambda^n V, 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 V \odot V := V \otimes V / \. More generally \operatorname^n V := \underbrace_n / (\dots \otimes v_i \otimes v_ \otimes \dots - \dots \otimes v_ \otimes v_ \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). 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

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Notes


References

* * * * * * * * * * {{DEFAULTSORT:Tensor Product Operations on vectors Operations on structures Bilinear maps