Tensor product

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In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. 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. The variant of is used in control theory.

Tensor product of vector spaces

The tensor product of two vector spaces V and W over a field K is another vector space over K. It is denoted VK W, or VW when the underlying field K is understood.

Prerequisite: the free vector space

The definition of requires the notion of the free vector space F(S) on some set S. The elements of the vector space F(S) are formal sums of elements of S with coefficients in a given field K. A formal sum is an expression written in the form of a sum in which no actual arithmetic operations can be carried out. For example 2a + 3b is a formal sum, and {{ safesubst:#invoke:Unsubst||$B=1/1 − x}} is a formal sum with no restrictions on values of x (versus the usual case where |x| < 1 must hold for a geometric series to converge), since no "plugging in" will actually be performed. For the set of all formal sums of elements of S with coefficients in K to be a vector space, we need to define addition and scalar multiplication. The terms of a formal sum can be written in any order and the addition of formal sums is associative.

Addition of formal sums is defined as follows: if mn, aism + ajsn cannot be simplified. If m = n, then aism + ajsn = (ai + aj) ⋅ sm.

Scalar multiplication of formal sums is defined as follows: If k is in the field K, then k(a1s1 + a2s2 + ... + ansn) = ka1s1 + ka2s2 + ... + kansn.

The dimension of the vector space equals the number of elements in S.


Given two vector spaces V and W over a field K, the tensor product U of V and W, denoted as U = VW is defined as the vector space whose elements and operations are constructed as follows:

From the cartesian product V × W, the free vector space F(V × W) over K is formed. The vectors of VW are then defined to be the equivalence classes of F(V × W) under the following equivalence relations:

The operations of VW, i.e. the map of vector addition +: U × UU and scalar multiplication ⋅: K × UU are defined to be the respective operations +F and F from F(V × W), acting on any representatives

in the involved equivalence classes outputting the one equivalence class of the result.

One can show that the result is actually independent of which representatives of the involved classes have been chosen. In other words, the operations are well-defined.

In short, the tensor product VW is defined as the quotient space F(V × W)/N over the subspace N of F(V × W), which expresses the equivalence relations described above:

Notation and examples

Given bases {vi} and {wi} for V and W respectively, the tensors {viwj} form a basis for VW. The dimension of the tensor product therefore is the product of dimensions of the original spaces; for instance RmRn will have dimension mn.

Elements of VW are sometimes referred to as tensors, although this term refers to many other related concepts as well.[1] An element of VW of the form vw 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. That is to say, if v1 and v2 are linearly independent, and w1 and w2 are also linearly independent, then v1w1 + v2w2 cannot be written as a pure tensor. The number of simple tensors required to express an element of a tensor product is called the tensor rank (not to be confused with tensor order, which is the number of spaces one has taken the product of, in this case 2; in notation, the number of indices), and for linear operators or matrices, thought of as (1, 1) tensors (elements of the space VV), it agrees with matrix rank.

Tensor product of linear maps

The tensor product also operates on linear maps between vector spaces. Specifically, given two linear maps S : VX and T : WY between vector spaces, the tensor product of the two linear maps S and T is a linear map

defined by

In this way, the tensor product becomes a bifunctor from the category of vector spaces to itself, covariant in both arguments.[2]

If S and T are both injective, surjective, or continuous then ST is, respectively, injective, surjective, continuous.

By choosing bases of all vector spaces involved, the linear maps S and T can be represented by matrices. Then, the matrix describing the tensor product ST is the Kronecker product of the two matrices. For example, if V, X, W, and Y above are all two-dimensional and bases have been fixed for all of them, and S and T are given by the matrices and , respectively, then the tensor product of these two matrices is