Tensor product

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In mathematics, the tensor product, denoted by $\otimes$ , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. In each case the significance of the symbol is the same: the most general bilinear operation. In some contexts, this product is also referred to as outer product. The term "tensor product" is also used in relation to monoidal categories.

Example:

$\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix} \otimes \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ \end{bmatrix} = \begin{bmatrix} a_{11} \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ \end{bmatrix} & a_{12} \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ \end{bmatrix} \\ & \\ a_{21} \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ \end{bmatrix} & a_{22} \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ \end{bmatrix} \\ \end{bmatrix} = \begin{bmatrix} a_{11} b_{11} & a_{11} b_{12} & a_{12} b_{11} & a_{12} b_{12} \\ a_{11} b_{21} & a_{11} b_{22} & a_{12} b_{21} & a_{12} b_{22} \\ a_{21} b_{11} & a_{21} b_{12} & a_{22} b_{11} & a_{22} b_{12} \\ a_{21} b_{21} & a_{21} b_{22} & a_{22} b_{21} & a_{22} b_{22} \\ \end{bmatrix}$

Resultant rank = 4, resultant dimension = 4×4.

Here rank denotes the tensor rank (number of requisite indices), while dimension counts the number of degrees of freedom in the resulting array; the matrix rank is 1.

A representative case is the Kronecker product of any two rectangular arrays, considered as matrices. A dyadic product is the special case of the tensor product between two vectors of the same dimension.

Tensor product of vector spaces

The tensor product V ? W of two vector spaces V and W over a field K can be defined by the method of generators and relations.

To construct V ? W, one begins with the set of ordered pairs in the Cartesian product V × W. For the purposes of this construction, regard this Cartesian product as a set rather than a vector space. The free vector space on V × W is defined by taking the vector space in which the elements of V × W are a basis. Symbolically,

$F(V\times W) = \left\{\sum_{i=1}^n \alpha_i e_{(v_i\times w_i)}\mid n\in\mathbb{N}, \alpha_i\in K, (v_i\times w_i)\in V\times W \right\},$

where we have used the symbol e_{(v × w)} to emphasize that these are taken to be linearly independent for distinct (v × w) ? V × W.

The tensor product arises by defining the following three equivalence relations in F(V × W):

$e_{(v_1+v_2)\times w} \sim e_{v_1\times w}+e_{v_2\times w}\!$
$e_{v\times (w_1+w_2)} \sim e_{v\times w_1}+e_{v\times w_2}\!$
$ce_{v\times w}\sim e_{(cv)\times w} \sim e_{v\times (cw)} \!$

where v, v_i, w, and w_i are vectors from V and W (respectively), and c is from the underlying field K. Denoting by R the space generated by these three equivalence relations, the tensor product of the two vector spaces V and W is then the quotient space

$V \otimes W = F(V \times W) / R$

It is also called the tensor product space of V and W and is a vector space (which can be verified by directly checking the vector space axioms). The tensor product of two elements v and w is the equivalence class (e_{(v × w)} + R) of e_{(v × w)} in V ? W, denoted v ? w. This notation can somewhat obscure the fact that tensors are always cosets: manipulations performed via the representatives (v,w) must always be checked that they do not depend on the particular choice of representative.

The space R is mapped to zero in V ? W, so that the above three equivalence relations become equalities in the tensor product space

$(v_1+v_2)\otimes w - (v_1\otimes w+v_2\otimes w) = 0$
$v\otimes (w_1+w_2) - (v\otimes w_1+v\otimes w_2) = 0$
$cv\otimes w=v\otimes cw=c(v\otimes w).$

Given bases {v_i} and {w_i} for V and W respectively, the tensors {v_i ? w_j} form a basis for V ? W (generally ordered so that v_i ? w_j+1 comes before v_i+1 ? w_j). The dimension of the tensor product therefore is the product of dimensions of the original spaces; for instance R^m ? Rⁿ will have dimension mn.

Elements of V ? W are sometimes referred to as tensors, although this term refers to many other related concepts as well.^[1] An element of V ? W of the form v ? w is called a pure or simple tensor. In general, an element of the tensor product space is not a pure tensor, but rather a finite linear combination of pure tensors. To wit, if v₁ and v₂ are linearly independent, and w₁ and w₂ are also linearly independent, then v₁ ? w₁ + v₂ ? w₂ cannot be written as a pure tensor.

Characterization by a universal property

The tensor product is characterized by a universal property. Consider the problem of mapping the Cartesian product V × W into a vector space X via a bilinear map ?. The tensor product construction V ? W, together with the natural bilinear embedding map f : V × W ? V ? W given by

$\varphi (u,w)= u \otimes w ,$

is the "universal" solution to this problem in the following sense. For any other such pair (X, ?), where X is a vector space, and ? a bilinear mapping V × W ? X, there exists a unique linear map

$T : V \otimes W \rightarrow X$

such that

$\psi = T \circ \varphi.$

As with any universal property, this characterizes the tensor product uniquely up to unique isomorphism.

An immediate consequence is the identification of the bilinear maps from V × W to X

$L^2(V \times W, X),$

and the linear maps

$L(V \otimes W, X)$

obtained by sending ? to T.

As a functor

The tensor product also operates on linear maps between vector spaces. Specifically, given two linear maps S : V ? X and T : W ? Y between vector spaces, the tensor product of the two linear maps S and T is a linear map

$S\otimes T:V\otimes W\rightarrow X\otimes Y$

defined by

$(S\otimes T)(v\otimes w)=S(v)\otimes T(w).$

In this way, the tensor product becomes a bifunctor from the category of vector spaces to itself, covariant in both arguments.

The Kronecker product of two matrices is the matrix of the tensor product of the two corresponding linear maps under a standard choice of bases of the tensor products (see the article on Kronecker products).

More than two vector spaces

The construction and the universal property of the tensor product can be extended to allow for more than two vector spaces. For example, suppose that V₁, V₂, and V₃ are three vector spaces. The tensor product V₁?V₂?V₃ is defined along with a trilinear mapping from the direct product

$\varphi : V_1\times V_2\times V_3 \to V_1\otimes V_2\otimes V_3$

so that, any trilinear map F from the direct product to a vector space W

$F:V_1\times V_2\times V_3\to W$

factors uniquely as

$F = L\circ\varphi$

where L is a linear map. The tensor product is uniquely characterized by this property, up to a unique isomorphism.

This construction is related to repeated tensor products of two spaces. For example, if V₁, V₂, and V₃ are three vector spaces, then there are (natural) isomorphisms

$V_1\otimes V_2\otimes V_3\cong V_1\otimes(V_2\otimes V_3)\cong (V_1\otimes V_2)\otimes V_3.$

More generally, the tensor product of an arbitrary indexed family V_i, i ? I, is defined to be universal with respect to multilinear mappings of the direct product $\scriptstyle{\prod_{i\in I} V_i}.$

Tensor powers and braiding

Let n be a non-negative integer. The nth tensor power of the vector space V is the n-fold tensor product of V with itself. That is

$V^{\otimes n} \stackrel{def}{=} \underbrace{V\otimes\cdots\otimes V}_{n}.$

A permutation s of the set {1,2,…,n} determines a mapping of the nth Cartesian power of V

$\sigma : V^n\to V^n$

defined by

$\sigma(v_1,v_2,\dots,v_n) = (v_{\sigma 1}, v_{\sigma 2},\dots,v_{\sigma n}).$

Let

$\varphi:V^n \to V^{\otimes n}$

be the natural multilinear embedding of the Cartesian power of V into the tensor power of V. Then, by the universal property, there is a unique isomorphism

$\tau_\sigma : V^{\otimes n} \to V^{\otimes n}$

such that

$\varphi\circ\sigma = \tau_\sigma\circ\varphi.$

The isomorphism t_s is called the braiding map associated to the permutation s.

Tensor product of two tensors

A "tensor on V" (generally abbreviated to simply a "tensor") is an element of a vector space of the form

$\begin{matrix} T^r_s(V) & = & \underbrace{ V\otimes \dots \otimes V} & \otimes & \underbrace{ V^*\otimes \dots \otimes V^*} & = & V^{\otimes m}\otimes V^{\otimes n}\\ & & r & & s \end{matrix}$

for non-negative integers r and s. There is a general formula for the components of a (tensor) product of two (or more) tensors. For example, if F and G are two covariant tensors of rank m and n (respectively) (i.e. F ? T_m⁰, and G ? T_n⁰), then the components of their tensor product are given by

$(F\otimes G)_{i_1i_2...i_{m+n}} = F_{i_{1}i_{2}...i_{m}}G_{i_{m+1}i_{m+2}i_{m+3}...i_{m+n}}$ .^[2]

In this example, it is assumed that there is a chosen basis B of the vector space V, and the basis on any tensor space T_s^r is tacitly assumed to be the standard one (this basis is described in the article on Kronecker products). Thus, the components of the tensor product of two tensors are the ordinary product of the components of each tensor.

Note that in the tensor product, the factor F consumes the first rank(F) indices, and the factor G consumes the next rank(G) indices, so

$\mathrm{rank}( F \otimes G )=\mathrm{rank}(F)+\mathrm{rank}(G)$

Example

Let U be a tensor of type (1,1) with components U^a_ß, and let V be a tensor of type (1,0) with components V^?. Then

$U^\alpha {}_\beta V^\gamma = (U \otimes V)^\alpha {}_\beta {}^\gamma$

and

$V^\mu U^\nu {}_\sigma = (V \otimes U)^{\mu \nu} {}_\sigma$ .

The tensor product inherits all the indices of its factors.

Kronecker product of two matrices

Main article: Kronecker product

With matrices this operation is usually called the Kronecker product, a term used to make clear that the result has a particular block structure imposed upon it, in which each element of the first matrix is replaced by the second matrix, scaled by that element. For matrices $U$ and $V$ this is:

$U \otimes V = \begin{bmatrix} u_{11}V & u_{12}V & \cdots \\ u_{21}V & u_{22}V \\ \vdots & & \ddots \end{bmatrix} = \begin{bmatrix} u_{11}v_{11} & u_{11}v_{12} & \cdots & u_{12}v_{11} & u_{12}v_{12} & \cdots \\ u_{11}v_{21} & u_{11}v_{22} & & u_{12}v_{21} & u_{12}v_{22} \\ \vdots & & \ddots \\ u_{21}v_{11} & u_{21}v_{12} \\ u_{21}v_{21} & u_{21}v_{22} \\ \vdots \end{bmatrix}$ .

Tensor product of multilinear maps

Given multilinear maps $f (x 1,... x k)$ and $g (x 1,... x m)$ their tensor product is the multilinear function

$(f \otimes g) (x_1,...,x_{k+m})=f(x_1,...,x_k)g(x_{k+1},...,x_{k+m})$

Tensor product of Hilbert spaces

It has been suggested that this section be split into a new article. (Discuss)

Main article: Tensor product of Hilbert spaces

Further information: Positive definite kernel#Direct sum and tensor product

The tensor product of two Hilbert spaces is another Hilbert space, which is defined as described below.

Definition

The discussion so far has been purely algebraic. In light of the extra structure on Hilbert spaces, one would like to introduce an inner product, and therefore a topology, on the tensor product that arise naturally from those of the factors. Let H₁ and H₂ be two Hilbert spaces with inner products $\langle \cdot,\cdot\rangle_1$ and $\langle \cdot,\cdot\rangle_2$ , respectively. Construct the tensor product of H₁ and H₂ as vector spaces as explained above. We can turn this vector space tensor product into an inner product space by defining

$\langle\phi_1\otimes\phi_2,\psi_1\otimes\psi_2\rangle = \langle\phi_1,\psi_1\rangle_1 \, \langle\phi_2,\psi_2\rangle_2 \quad \mbox{for all } \phi_1,\psi_1 \in H_1 \mbox{ and } \phi_2,\psi_2 \in H_2$

and extending by linearity. That this inner product is the natural one is justified by the identification of scalar-valued bilinear maps on H₁ × H₂ and linear functionals on their vector space tensor product. Finally, take the completion under this inner product. The resulting Hilbert space is the tensor product of H₁ and H₂.

Properties

If H₁ and H₂ have orthonormal bases {f_k} and {?_l}, respectively, then {f_k ? ?_l} is an orthonormal basis for H₁ ? H₂.

Examples and applications

The following examples show how tensor products arise naturally.

Given two measure spaces X and Y, with measures µ and ? respectively, one may look at L²(X × Y), the space of functions on X × Y that are square integrable with respect to the product measure µ × ?. If f is a square integrable function on X, and g is a square integrable function on Y, then we can define a function h on X × Y by h(x,y) = f(x) g(y). The definition of the product measure ensures that all functions of this form are square integrable, so this defines a bilinear mapping L²(X) × L²(Y) ? L²(X × Y). Linear combinations of functions of the form f(x) g(y) are also in L²(X × Y). It turns out that the set of linear combinations is in fact dense in L²(X × Y), if L²(X) and L²(Y) are separable. This shows that L²(X) ? L²(Y) is isomorphic to L²(X × Y), and it also explains why we need to take the completion in the construction of the Hilbert space tensor product.

Similarly, we can show that L²(X; H), denoting the space of square integrable functions X ? H, is isomorphic to L²(X) ? H if this space is separable. The isomorphism maps f(x) ? f ? L²(X) ? H to f(x)f ? L²(X; H). We can combine this with the previous example and conclude that L²(X) ? L²(Y) and L²(X × Y) are both isomorphic to L²(X; L²(Y)).

Tensor products of Hilbert spaces arise often in quantum mechanics. If some particle is described by the Hilbert space H₁, and another particle is described by H₂, then the system consisting of both particles is described by the tensor product of H₁ and H₂. For example, the state space of a quantum harmonic oscillator is L²(R), so the state space of two oscillators is L²(R) ? L²(R), which is isomorphic to L²(R²). Therefore, the two-particle system is described by wave functions of the form f(x₁, x₂). A more intricate example is provided by the Fock spaces, which describe a variable number of particles.

Relation with the dual space

In the discussion on the universal property, replacing X by the underlying scalar field of V and W yields that the space $(V \otimes W)^\star$ (the dual space of $V \otimes W$ , containing all linear functionals on that space) is naturally identified with the space of all bilinear functionals on $V \times W$ . In other words, every bilinear functional is a functional on the tensor product, and vice versa.

Whenever $V$ and $W$ are finite dimensional, there is a natural isomorphism between $V^\star \otimes W^\star$ and $(V \otimes W)^\star$ , whereas for vector spaces of arbitrary dimension we only have an inclusion $V^\star \otimes W^\star\subset (V \otimes W)^\star$ . So, the tensors of the linear functionals are bilinear functionals. This gives us a new way to look at the space of bilinear functionals, as a tensor product itself.

Types of tensors, e.g., alternating

Linear subspaces of the bilinear operators (or in general, multilinear operators) determine natural quotient spaces of the tensor space, which are frequently useful. See wedge product for the first major example. Another would be the treatment of algebraic forms as symmetric tensors.

Over more general rings

The notation $\otimes_R$ refers to a tensor product of modules over a ring R.

Tensor product for computer programmers

It has been suggested that this section be split into a new article. (Discuss)

Array programming languages

Array programming languages may have this pattern built in. For example, in APL the tensor product is expressed as $\circ . \times$ (for example $A \circ . \times B$ or $A \circ . \times B \circ . \times C$ ). 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 -- the result is then 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).

Notes

^ See tensor or tensor (intrinsic definition).
^ Analogous formulas also hold for contravariant tensors, as well as tensors of mixed variance. Although in many cases such as when there is an inner product defined, the distinction is irrelevant.

References

Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9 .
Halmos, Paul (1974), Finite dimensional vector spaces, Springer, ISBN 0387900934 .
Lang, Serge (2002), Algebra, Springer-Verlag, ISBN 0-387-95385-X .
MacLane, S. & Birkhoff, G. (1999), Algebra, AMS Chelsea, ISBN 0-8218-1646-2 .

Categories: Abstract algebra | Multilinear algebra | Binary operations

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