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:

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

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.

1 Tensor product of two tensors
2 Tensor product of vector spaces
3 Universal property of the tensor product
4 Tensor product of Hilbert spaces
5 Relation with the dual space
6 Types of tensors, e.g., alternating
7 Over more general rings
8 Tensor product for computer programmers
- 8.1 Array programming languages
9 Notes
10 See also

[edit] Tensor product of two tensors

There is a general formula for the components of a product of two (or more) tensors. For example, if U and V are two covariant tensors of rank m and n (respectively), then the components of their tensor product are given by

$(U\otimes V)_{i_1i_2...i_{m+n}} = U_{i_{1}i_{2}...i_{m}}V_{i_{m+1}i_{m+2}i_{m+3}...i_{m+n}}$ .^[1]

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 U consumes the first rank(U) indices, and the factor V consumes the next rank(V) indices, so

$\mathrm{rank}( U \otimes V )=\mathrm{rank}(U)+\mathrm{rank}(V)$

[edit] Example

Let U be a tensor of type (1,1) with components U^α_β, and let V be a tensor of type (1,0) with components V^γ. 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.

[edit] 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:

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

[edit] 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

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

[edit] Tensor product of vector spaces

The tensor product $V \otimes W$ of two vector spaces V and W over a field $K$ has a formal definition by the method of generators and relations.

To construct $V \otimes W$ , 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 $(v\times w)\in V\times W$ .

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,w_i are vectors from V and W (respectively), and c is from the underlying field K. Denoting by $\sim$ the space generated by these three equivalence relations, the definition of the operator $\otimes$ is then the quotient space

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

The equivalence class of (v×w) is called the tensor product of v and w, denoted $v \otimes w$ . The space $\sim$ is mapped to the kernel, so that the above three equivalence relations become

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

The resulting space $V \otimes W$ is a vector space, which can be verified by directly checking the vector space axioms. It is called the tensor product space of V and W. Given bases ${v i}$ and ${w i}$ for V and W respectively, the tensors of the form $v_i \otimes w_j$ forms a basis for $V \otimes W$ . The dimension of the tensor product therefore is the product of dimensions of the original spaces; for instance $\mathbb{R}^m \otimes \mathbb{R}^n$ will have dimension $m n$ .

[edit] Universal property of the tensor product

The tensor product is characterized by a universal property. Consider the problem of embedding the Cartesian product V × W into a vector space X via a bilinear map φ. The tensor product construction V ⊗ W, together with the natural embedding map φ : 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, ψ), where X is a vector space, and ψ a bilinear mapping V × W → X, there exists a unique linear map

$T : V \otimes W \rightarrow X$

such that

$\psi = T \circ \varphi.$

Assuming this universal property, it can be readily verified that the tensor product is unique up to isomorphism.

An immediate consequence is the identification of

$B(V \times W, X),$

the bilinear maps from V × W to X and the linear maps

$L(V \otimes W, X).$

The natural isomorphism maps ψ to T.

[edit] 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.

[edit] 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.

[edit] Properties

If H₁ and H₂ have orthonormal bases {φ_k} and {ψ_l}, respectively, then {φ_k ⊗ ψ_l} is an orthonormal basis for H₁ ⊗ H₂.

[edit] Examples and applications

The following examples show how tensor products arise naturally.

Given two measure spaces X and Y, with measures μ and ν 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 × ν. 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) ⊗ φ ∈ L²(X) ⊗ H to f(x)φ ∈ 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 φ(x₁, x₂). A more intricate example is provided by the Fock spaces, which describe a variable number of particles.

[edit] 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.

[edit] 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.

[edit] Over more general rings

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

[edit] Tensor product for computer programmers

[edit] 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).

[edit] Notes

^ 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.

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