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A representative case is the Kronecker Product of any two rectangular arrays, considered as matrices.

Example:

:\begin{bmatrix}b_1 \ b_2 \ b_3 \ b_4\end{bmatrix} \otimes \begin{bmatrix}a_1 & a_2 & a_3\end{bmatrix} = \begin{bmatrix}a_1b_1 & a_2b_1 & a_3b_1 \ a_1b_2 & a_2b_2 & a_3b_2 \ a_1b_3 & a_2b_3 & a_3b_3 \ a_1b_4 & a_2b_4 & a_3b_4\end{bmatrix}

Resultant rank = 2, resultant dimension = 4×3 = 12.

Here rank denotes the number of requisite indices, while dimension counts the number of degrees of freedom in the resulting array.


TENSOR PRODUCT OF TWO TENSORS


What does the general formula mean? 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 number of independent components multiplies

There is a general formula for the product of two (or more) Tensor s, as

:V \otimes U = V_{\left i_1,i_2,i_3,...i_n ight }U_{\left j_1,j_2,j_3,...j_m ight }.

We are assuming here '' Orthogonal '' tensors, with no distinction of covariant and contravariant indices, for simplicity.

The parameters introduced above work out like this:
:\mathrm{rank}( U \otimes V )=\mathrm{rank}(U)+\mathrm{rank}(V)

:\mathrm{dim}( U \otimes V )=\mathrm{dim}(U) \mathrm{dim}(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^
u {}_\sigma = (V \otimes U)^{\mu
u} {}_\sigma .

The tensor product inherits all the indices of its factors.

See also: 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:

:U \otimes V
= \begin{bmatrix} u_{11}V & u_{12}V & \cdots \
u_{21}V & u_{22}V \
dots & & \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} \
dots & & \ddots \
u_{21}v_{11} & u_{21}v_{12} \
u_{21}v_{21} & u_{21}v_{22} \
dots
\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 VECTOR SPACES


The tensor product V \otimes W of two Vector Space s ''V'' and ''W'' has a formal definition by the method of ''generators and relations''. The equivalence class under these relations (given below) of (v,w) is called a ''tensor'' and is denoted by v \otimes w. By construction, one can prove several identities between tensors and form an algebra of tensors.

To construct V \otimes W, take the vector space generated by V imes W and apply (factor out the subspace generated by) the following multilinear relations:

  • (v_1+v_2)\otimes w=v_1\otimes w+v_2\otimes w

  • v\otimes (w_1+w_2)=v\otimes w_1+v\otimes w_2

  • cv\otimes w=v\otimes cw=c(v\otimes w)


where v,v_i,w,w_i are vectors from the appropriate spaces, and c is from the underlying field.

We can then derive the identity

:0v\otimes w=v\otimes 0w=0(v\otimes w)=0,

the zero in V \otimes W.

The resulting tensor product V \otimes W is itself a vector space, which can be verified by directly checking the vector space axioms.
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 mn.


UNIVERSAL PROPERTY OF TENSOR PRODUCT


The space of all bilinear maps from V imes W to \mathbb R is naturally isomorphic to the space of all linear maps from V \otimes W to \mathbb R. This is built into the construction; V\otimes W has all and only the relations that are necessary to ensure that a homomorphism from V\otimes W to \mathbb R will be linear.

The tensor product in fact satisfies the Universal Property of being a Fibered Coproduct .


TENSOR PRODUCT OF HILBERT SPACES


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


Definition


Let ''H''1 and ''H''2 be two Hilbert spaces with inner products \langle \cdot,\cdot angle_1 and \langle \cdot,\cdot angle_2, respectively. Construct the tensor product of ''H''1 and ''H''2 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 angle = \langle\phi_1,\psi_1 angle_1 \, \langle\phi_2,\psi_2 angle_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. Finally, take the Completion under this inner product. The result is the tensor product of ''H''1 and ''H''2 as Hilbert spaces.


Properties


If ''H''1 and ''H''2 have Orthonormal Bases {φ''k''} and {ψ''l''}, respectively, then {φ''k'' ⊗ ψ''l''} is an orthonormal basis for ''H''1 ⊗ ''H''2.


Examples and applications


The following examples show how tensor products arise naturally.

Given two Measure Space s ''X'' and ''Y'', with measures μ and ν respectively, one may look at L 2(''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 Combination s 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 ''H''1, and another particle is described by ''H''2, then the system consisting of both particles is described by the tensor product of ''H''1 and ''H''2. 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(''x''1, ''x''2). A more intricate example is provided by the Fock Space s, which describe a variable number of particles.


RELATION WITH THE DUAL SPACE


Note that the space (V \otimes W)^\star (the Dual Space of V \otimes W containing all linear Functional s on that space)
corresponds naturally to the space of all
bilinear functionals on V imes W. In other words, every bilinear functional is a functional
on the tensor product, and vice versa.
There is a natural Isomorphism between
V^\star \otimes W^\star and (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 Space s of the tensor space, which are frequently useful. See Wedge Product for the first major example. Another would be the treatment of Algebraic Form s as symmetric tensors.


OVER MORE GENERAL RINGS

''See Tensor Product Of Modules Over A Ring ''


TENSOR PRODUCT FOR COMPUTER PROGRAMMERS



Array Programming Languages


  • / (for example '''a ---/ b''' or ''' a ---/ b ---/ c''').


  • /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).


SEE ALSO