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- In mathematics, the tensor product, denoted by <math>\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. 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. In set-builder notation, F(V\times W) \left\{\sum_{i1}^n \alpha_i e_{(v_i, w_i)}\mid n\in\mathbb{N}, \alpha_i\in K, (v_i, 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): <math>e_{(v_1+v_2,w)} \sim e_{(v_1,w)}+e_{(v_2,w)}\! <math>e_{(v,w_1+w_2)} \sim e_{(v,w_1)}+e_{(v,w_2)}\! <math>ce_{(v,w)}\sim e_{(cv,w)} \sim e_{(v,cw)} \! where v, vi, w, and wi 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 + 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: <math>(v_1 + v_2) \otimes w v_1 \otimes w + v_2 \otimes w; <math>v \otimes (w_1 + w_2) v \otimes w_1 + v \otimes w_2; <math>cv \otimes w v \otimes cw c(v \otimes w). Given bases {vi} and {wi} for V and W respectively, the tensors {vi ⊗ wj} form a basis for V ⊗ W (generally ordered so that vi ⊗ wj+1 comes before vi+1 ⊗ wj). The dimension of the tensor product therefore is the product of dimensions of the original spaces; for instance R ⊗ R 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. 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. 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 <math>V \otimes V^*), it agrees with matrix rank. 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 ψ. The tensor product construction V ⊗ W, together with the natural bilinear 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. As with any universal property, this characterizes the tensor product uniquely up to unique isomorphism. In the words of category theory, the tensor product, or more precisely the map <math> \varphi, is the terminal object in the category of bilinear maps from V × W. 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 ψ 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. 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 Vi, i ∈ I, is defined to be universal with respect to multilinear mappings of the direct product <math>\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} \overset{\mathrm{def}}{} \underbrace{V\otimes\cdots\otimes V}_{n}. A permutation σ 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 τσ is called the braiding map associated to the permutation σ. Tensor product of two tensors A tensor on V 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 r}\otimes V^{*\otimes s}\\ & & 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 ∈ Tm, and G ∈ Tn), 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}}. 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 Ts 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 Uβ, 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. See also: Classical treatment of tensors Kronecker product of two matrices 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 <math>U and <math>V this is: U \otimes V \begin{bmatrix} u_{1,1}V & u_{1,2}V & \cdots \\ u_{2,1}V & u_{2,2}V \\ \vdots & & \ddots \end{bmatrix} \begin{bmatrix} u_{1,1}v_{1,1} & u_{1,1}v_{1,2} & \cdots & u_{1,2}v_{1,1} & u_{1,2}v_{1,2} & \cdots \\ u_{1,1}v_{2,1} & u_{1,1}v_{2,2} & & u_{1,2}v_{2,1} & u_{1,2}v_{2,2} \\ \vdots & & \ddots \\ u_{2,1}v_{1,1} & u_{2,1}v_{1,2} \\ u_{2,1}v_{2,1} & u_{2,1}v_{2,2} \\ \vdots \end{bmatrix}. For example, take the tensor product of two three-dimensional square matrices: \begin{bmatrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \\ \end{bmatrix} \otimes \begin{bmatrix} b_{1,1} & b_{1,2} \\ b_{2,1} & b_{2,2} \\ \end{bmatrix} \begin{bmatrix} a_{1,1} \begin{bmatrix} b_{1,1} & b_{1,2} \\ b_{2,1} & b_{2,2} \\ \end{bmatrix} & a_{1,2} \begin{bmatrix} b_{1,1} & b_{1,2} \\ b_{2,1} & b_{2,2} \\ \end{bmatrix} \\ a_{2,1} \begin{bmatrix} b_{1,1} & b_{1,2} \\ b_{2,1} & b_{2,2} \\ \end{bmatrix} & a_{2,2} \begin{bmatrix} b_{1,1} & b_{1,2} \\ b_{2,1} & b_{2,2} \\ \end{bmatrix} \\ \end{bmatrix} \begin{bmatrix} a_{1,1} b_{1,1} & a_{1,1} b_{1,2} & a_{1,2} b_{1,1} & a_{1,2} b_{1,2} \\ a_{1,1} b_{2,1} & a_{1,1} b_{2,2} & a_{1,2} b_{2,1} & a_{1,2} b_{2,2} \\ a_{2,1} b_{1,1} & a_{2,1} b_{1,2} & a_{2,2} b_{1,1} & a_{2,2} b_{1,2} \\ a_{2,1} b_{2,1} & a_{2,1} b_{2,2} & a_{2,2} b_{2,1} & a_{2,2} b_{2,2} \\ \end{bmatrix}. Resultant rank 4, resultant dimension 16. Here rank denotes the tensor rank (number of requisite indices), while the matrix rank counts the number of degrees of freedom in the resulting array. 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 multilinear maps Given multilinear maps <math>f (x_1,\dots,x_k) and <math>g (x_1,\dots, x_m) their tensor product is the multilinear function (f \otimes g) (x_1,\dots,x_{k+m}) f(x_1,\dots,x_k) g(x_{k+1},\dots,x_{k+m}). 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 <math> (V \otimes W)^\star (the dual space of <math>V \otimes W, containing all linear functionals on that space) is naturally identified with the space of all bilinear functionals on <math>V \times W. In other words, every bilinear functional is a functional on the tensor product, and vice versa. Whenever <math>V and <math>W are finite dimensional, there is a natural isomorphism between <math> V^\star \otimes W^\star and <math>(V \otimes W)^\star, whereas for vector spaces of arbitrary dimension we only have an inclusion <math>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 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 <math>\otimes_R refers to a tensor product of modules over a ring R. 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 <math>\circ . \times (for example <math>A \circ . \times B or <math>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, and/or may not support higher-order functions such as the Jacobian derivative. See also Tensor product of R-algebras Tensor product of fields Tensor product of graphs Tensor product of Hilbert spaces Tensor product of line bundles Tensor product of modules Tensor product of quadratic forms Topological tensor product Dyadic product Notes References . . Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed. ), New York: Springer-Verlag, MR1878556, ISBN 978-0-387-95385-4 .
- Das Tensorprodukt ist ein sehr vielseitiger Begriff der Mathematik: in der linearen Algebra und der Differentialgeometrie dient es der Beschreibung von multilinearen Abbildungen, in der kommutativen Algebra und algebraischen Geometrie entspricht es einerseits der Einschränkung geometrischer Strukturen auf Teilmengen, andererseits dem kartesischen Produkt geometrischer Objekte. Dieser Artikel beschreibt die mathematischen und koordinatenfreien Aspekte des Tensorproduktes. Für einzelne Tensoren und Koordinatendarstellungen siehe Tensor.
- En matemáticas, el producto tensorial, denotado por <math>\otimes</math>, se puede aplicar en diversos contextos a vectores, matrices, tensores y espacios vectoriales. En cada caso la significación del símbolo es la misma: la operación bilineal más general. Un caso representativo es producto de Kronecker de cualesquiera dos matrices. ejemplo: <math>\begin{bmatrix}a_1 \\ a_2 \\ a_3\end{bmatrix} \otimes \begin{bmatrix}b_1 & b_2 & b_3 & b_4\end{bmatrix} = \begin{bmatrix}a_1b_1 & a_1b_2 & a_1b_3 & a_1b_4 \\ a_2b_1 & a_2b_2 & a_2b_3 & a_2b_4 \\ a_3b_1 & a_3b_2 & a_3b_3 & a_3b_4 \end{bmatrix}</math> rango resultante = 2, dimensión resultante = 3x4. Aquí el rango denota el número de índices indispensables, mientras que la dimensión cuenta el número de grados de libertad en el arreglo que resulta.
- On appelle produit tensoriel, ou produit de Kronecker, le produit de chaque composante d'un tenseur par chaque composante d'un autre tenseur. Le produit d'un tenseur d'ordre <math>p</math> avec un tenseur d'ordre <math>q</math> est un tenseur d'ordre <math>p+q</math>. Le produit tensoriel n'est pas commutatif mais pseudo-commutatif.
- Il prodotto tensoriale V ⊗ W di due spazi vettoriali V, W, è uno spazio vettoriale che contiene i prodotti di vettori di V e W in senso universale. Si può pensare ad una applicazione bilineare V × W → L come a un prodotto ⋅ tra vettori di V e W a valori in un terzo spazio vettoriale L (sebbene non necessario, è utile vederla in questo modo). Dato un altro spazio M e un omomorfismo φ : L → M, si ha che φ (v · w) è un prodotto a valori in M. Si può dimostrare che esiste un "prodotto universale" ⊗ a valori in un certo spazio V ⊗ W con la proprietà che tutti i possibili prodotti su V × W si possono ottenere, in modo unico, trasformando linearmente il codominio V ⊗ W. Se v e w sono rispettivamente elementi di V e W si denota con v ⊗ w il prodotto di v e w in V ⊗ W. Per dimostrarne l'esistenza lo si costruisce come spazio quoziente dello spazio vettoriale libero su V×W imponendo le relazioni ovvie per far si che la proiezione dopo l'immersione sia bilineare. Prendendo spazi quozienti del prodotto tensoriale si possono aggiungere proprietà a ⊗ ottenendo, ad esempio, il prodotto universale simmetrico (basta imporre la relazione v ⊗ w - w ⊗ v = 0, cioè prendere il quoziente / K dove K è il sottospazio generato da tutti gli elementi del tipo v ⊗ w - w ⊗ v) o antisimmetrico (imponendo v ⊗ w + w ⊗ v = 0). Queste costruzioni sono fondamentali in svariati campi. Partendo con degli R - moduli M, N (strutture che generalizzano gli spazi vettoriali prendendo gli scalari in un anello invece che in un campo), e supponendo R commutativo per semplicità, si può dare la stessa definizione che per il caso degli spazi vettoriali di M ⊗R N (con i moduli di solito si specifica rispetto a quale anello li si considera mettendo un pedice a &otimes). Anche la dimostrazione dell'esistenza rimane la stessa. Nonostante le similitudini iniziali il prodotto tensoriale tra moduli può riservare delle sorprese (ad esempio Z / ⊗ Z / = 0 se m ed n sono coprimi).
- テンソル積（テンソルせき、tensor product）は、線型代数学で重線型性を扱うための線型化を担う概念で、既知のベクトル空間・加群から新たなベクトル空間を作り出す操作の一つである。テンソル積を繰り返して得られるテンソル空間は物理的なテンソルを数学的に定式化する。テンソル空間に種々の積を入れてさまざまな多重線型代数・クリフォード代数が定式化されるが、その基本となる演算がテンソル積である。
- In de lineaire algebra is het tensorproduct een mechanisme om twee vectorruimten te combineren tot een nieuwe vectorruimte. De nieuwe vectorruimte biedt op natuurlijke wijze een domein aan willekeurige bilineaire afbeeldingen die uitgaan van het cartesisch product van de twee vectorruimten. Alle betrokken vectorruimten hebben hetzelfde scalairen-lichaam. Voorbeeld \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} Definitie Als V en W vectorruimten zijn over een commutatief lichaam K, dan noteren we het tensorproduct als V\otimes W of uitdrukkelijker V\otimes_K W. Er bestaat een canonieke bilineaire injectie V\times W\to V\otimes W die eveneens met het symbool <math>\otimes \, genoteerd wordt. We spreken van het tensorproduct van twee vectoren. De elementen van de vorm <math>v\otimes w brengen de vectorruimte <math>V\times W voort. Als B een basis is voor V, en C een basis voor W, dan is \{b\otimes c|b\in B,c\in C\} een basis voor <math>V\otimes W. In het bijzonder geldt voor eindigdimensionale vectorruimten: \dim V\otimes W\dim V. \dim W Dit in tegenstelling tot het cartesisch product van vectorruimten, waar \dim V\times W\dim V+\dim W. In de natuurkunde, en dan vooral in de analytische mechanica en de relativiteitstheorie, wordt veelvuldig gebruikgemaakt van tensoren. Het daar gehanteerde tensorbegrip komt overeen met het hierboven geschetste begrip, toegepast op een eindig aantal exemplaren van de raakbundel TM (en zijn duaal, de co-rakende bundel <math>T^{*}M) van een Riemann-variëteit M. In die context is een tensor van rang n een sectie van het tensorproduct van n dergelijke bundels. De coördinaten van een dergelijke sectie worden gegeven door een stel van \dim M\dim TM\dim T^{*}M^n functies die aan bepaalde transformatiewetten voldoen bij overgang naar een ander stel basisvectoren. Het onderscheid tussen "covariant" en "contravariant" slaat dan op het onderscheid tussen de rakende bundel en de co-rakende bundel. Er bestaat een veralgemening van het tensorproduct tot willekeurige modulen over een commutatieve ring R. Het universeel object dat canoniek alle <math>n-voudige tensorproducten van een vectorruimte V met zichzelf omvat voor n0,1,2,\ldots, noemt men de universele algebra of tensoralgebra over V.
- W matematyce iloczyn tensorowy, oznaczany przez <math>\otimes</math>, może być używany w kilku różnych kontekstach: wektorów, macierzy, tensorów oraz przestrzeni wektorowych. We wszystkich tych przypadkach znaczenie iloczynu tensorowego jest takie samo: oznacza on operator biliniowy. Niech <math>u</math> i <math>v</math> będą wektorami o wymiarach <math>m</math> i <math>n</math>, wówczas iloczyn tensorowy wektorów <math>u \otimes v</math> jest <math>mn</math>-wymiarowym wektorem: <math>u \otimes v = w_{} = u_i v_j</math>, gdzie <math> = n(i-1) + j</math>. Niech <math>M</math> i <math>N</math> będą macierzami o wymiarach <math>m \times m</math> i <math>n \times n</math>, wówczas iloczyn tensorowy macierzy <math>M \otimes N</math> jest macierzą o wymiarze <math>mn \times mn</math> postaci: <math>M \otimes N = O_{,<j,l>} = M_{ij} N_{kl}</math>. Zobacz też: iloczyn Kroneckera
- Em matemática, o produto tensoril, simbolizado por <math>\otimes</math>, pode ser aplicado em diferentes contextos a vetores, matrizes, tensores, espaços vetoriais, álgebras, espaços vetoriais topológicos, e módulos. Em cada caso o significado do símbolo é o mesmo: a operação bilinear mais geral. Em alguns contextos, este poduto é também referido como sendo produto externo. O termo "produto tensorial" é também usado em relação a categorias monoidais.
- Те́нзорное произведе́ние — одно из основных понятий линейной алгебры.
- 在数学中，张量积，记为 <math>\otimes</math>，可以应用于不同的上下文中如向量、矩阵、张量、向量空间、代数、拓扑向量空间和模。在各种情况下这个符号的意义是同样的: 最一般的双线性运算。在某些上下文中也叫做外积。 例子: \mathbf{b} \otimes \mathbf{a} \rightarrow \begin{bmatrix}b_1 \\ b_2 \\ b_3 \\ b_4\end{bmatrix} \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}</math> 结果的秩为 2, 结果的维数为 4×3 = 12. 这里的秩指示张量秩（所需指标数），而维数计算在结果数组(阵列)中自由度的数目；矩阵的秩是 1。 代表情况是任何两个被当作矩阵的矩形数组的克罗内克积。在同维数的两个向量之间的张量积的特殊情况是并矢积。
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