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Encyclopedia > Tensor product

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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. ... In mathematics, a monoidal category (or tensor category) is a 2-category with one object (a 2-monoid). ...


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_{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. In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ...


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. In mathematics, the Kronecker product, denoted by , is an operation on two matrices of arbitrary size resulting in a block matrix. ... In mathematics, in particular multilinear algebra, the dyadic product of a column vector and a row vector is the tensor product of the vectors. ...

Contents

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 In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... It has been suggested that Covariant transformation be merged into this article or section. ...

(Uotimes 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)

Example

Let U be a tensor of type (1,1) with components Uαβ, and let V be a tensor of type (1,0) with components Vγ. 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.


See also: Classical treatment of tensors The following is a component-based classical treatment of tensors. ...


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: In mathematics, the Kronecker product, denoted by , is an operation on two matrices of arbitrary size resulting in a block matrix. ... In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks. ...

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(x1,...xk) and g(x1,...xm) their tensor product is the multilinear function In linear algebra, a multilinear map is a mathematical function of several vector variables that is linear in each variable. ...

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

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. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...


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, In mathematics, the Cartesian product is a direct product of sets. ... This article is about sets in mathematics. ...

F(Vtimes W) = left{sum_{i=1}^n alpha_i e_{(v_itimes w_i)}mid ninmathbb{N}, alpha_iin K, (v_itimes w_i)in Vtimes W right},

where we have used the symbol e(v × w) to emphasize that these are taken to be linearly independent for distinct (vtimes w)in Vtimes W.


The tensor product arises by defining the following three equivalence relations in F(V×W): In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...

  • e_{(v_1+v_2)times w} sim e_{v_1times w}+e_{v_2times w}!
  • e_{vtimes (w_1+w_2)} sim e_{vtimes w_1}+e_{vtimes w_2}!
  • ce_{vtimes w}sim e_{(cv)times w} sim e_{vtimes (cw)} !

where v,vi,w,wi 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 In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given 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 In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

  • (v_1+v_2)otimes w - (v_1otimes w+v_2otimes w) = 0
  • votimes (w_1+w_2) - (votimes w_1+votimes w_2) = 0
  • 0votimes w=votimes 0w=0(votimes 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 {vi} and {wi} 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 mn.


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 In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

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. In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...


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. Image File history File links No higher resolution available. ... In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. ... In operator theory, a positive definite kernel is a generalization of a positive matrix. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...


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,cdotrangle_1 and langle cdot,cdotrangle_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 In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

langlephi_1otimesphi_2,psi_1otimespsi_2rangle = langlephi_1,psi_1rangle_1 , langlephi_2,psi_2rangle_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. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...


Properties

If H1 and H2 have orthonormal bases {φk} and {ψl}, respectively, then {φk ⊗ ψl} is an orthonormal basis for H1 ⊗ H2. In mathematics, an orthonormal basis of an inner product space V(i. ...


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. In mathematics, a measure is a function that assigns a number, e. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...


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. For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... The Fock space is an algebraic system (Hilbert space) used in quantum mechanics to describe quantum states with a variable or unknown number of identical 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. In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra. ... In mathematics, the term functional is applied to certain functions. ...


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^starsubset (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. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...


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. In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In the mathematics of the nineteenth century, an important role was played by the algebraic forms that generalise quadratic forms to degrees 3 and more, also known as quantics. ...


Over more general rings

The notation otimes_R refers to a tensor product of modules over a ring R. In mathematics, the tensor product construction may be carried out, not only for vector spaces (see tensor product), but for any pair of modules over a commutative ring, with result a third module, and for a pair of a left-module and a right-module over any ring, with result... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...


Tensor product for computer programmers

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). Array programming languages (also known as vector or multidimensional languages) generalize operations on scalars to apply transparently to vectors, matrices, and higher dimensional arrays. ... APL (for A Programming Language) is an array programming language based on a notation invented in 1957 by Kenneth E. Iverson while at Harvard University. ... The J programming language, developed in the early 1990s by Ken Iverson and Roger Hui, is a synthesis of APL (also by Iverson) and the FP and FL functional programming languages created by John Backus (of FORTRAN, ALGOL, and BNF fame). ...


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). In mathematics, the derivative of a function is one of the two central concepts of calculus. ...


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). Not to be confused with Matlab Upazila in Chandpur District, Bangladesh. ... In mathematics and computer science, higher-order functions are functions which can take other functions as arguments, and may also return functions as results. ... For the French Revolution faction, see Jacobin. ... Fortran (previously FORTRAN[1]) is a general-purpose[2], procedural,[3] imperative programming language that is especially suited to numeric computation and scientific computing. ...


Notes

  1. ^ 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.

It has been suggested that this article or section be merged into Covariant transformation. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

See also

Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. ... In mathematics, the tensor product construction may be carried out, not only for vector spaces (see tensor product), but for any pair of modules over a commutative ring, with result a third module, and for a pair of a left-module and a right-module over any ring, with result... In mathematics, there is a construction in abstract algebra of the tensor product of commutative rings; which puts a ring structure on the tensor product as abelian groups of two commutative rings R and S. This structure on RZS then makes it a coproduct in the category of commutative rings. ... In mathematics, the theory of fields in abstract algebra lacks a direct product: the direct product of two fields, considered as ring is never itself a field. ... In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In graph theory, the tensor product or categorical product G H of graphs G and H is a graph such that the vertex set of G H is the cartesian product V(G) V(H); and any two vertices (u,u) and (v,v) are adjacent in G H if... In mathematics, in particular multilinear algebra, the dyadic product of a column vector and a row vector is the tensor product of the vectors. ...

  Results from FactBites:
 
Tensor product - Wikipedia, the free encyclopedia (1132 words)
It means that if a pair of tensors are juxtaposed (placed side-by-side) then they combine by mere aggregation to form a new tensor which can be subsequently called the tensor product of the pair of juxtaposed tensors.
The tensor product inherits all the indices of its factors.
In J the tensor product is the dyadic form of */ (for example a */ b or a */ b */ c).
PlanetMath: tensor product (vector spaces) (594 words)
Historically, the tensor product was called the outer product, and has its origins in the absolute differential calculus (the theory of manifolds).
This relationship is the source of the terminology ``contravariant tensor'' and ``contravariant index'', and I surmise that it is this very medieval pit of darkness and confusion that spawned the present-day notion of ``contravariant functor''.
This is version 7 of tensor product (vector spaces), born on 2002-02-18, modified 2005-08-02.
  More results at FactBites »


 
 

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