TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the tensor product $V \otimes W$ of two
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s and (over the same
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
) is a vector space that can be thought of as the ''space of all
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

s'' that can be built from vectors from its constituent spaces using an additional operation that can be considered as a generalization and abstraction of the
outer product In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
. Because of the connection with tensors, which are the elements of a tensor product, tensor products find uses in many areas of application including in physics and engineering, though the full theoretical mechanics of them described below may not be commonly cited there. For example, in
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
, the
gravitational field In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

is described through the
metric tensor In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
, which is a field (in the sense of physics) of tensors, one at each point in the
space-time In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

, and each of which lives in the tensor self-product of
tangent space In , the tangent space of a generalizes to higher dimensions the notion of tangent planes to surfaces in three dimensions and tangent lines to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can ...
s $T_x M$ at its point of residence on the manifold (such a collection of tensor products attached to another space is called a tensor bundle).

# Tensors in finite dimensions, and the outer product

The concept of tensor product generalizes the idea of forming tensors from vectors using the outer product, which is an operation that can be defined in finite-dimensional vector spaces using
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
: given two vectors $\mathbf \in V$ and $\mathbf \in W$ written in terms of components, i.e. $\mathbf = \begin v_1 \\ v_2 \\ \vdots \\ v_n \end$ and $\mathbf = \begin w_1 \\ w_2 \\ \vdots \\ w_m \end$ their
outer product In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
or is given by the $n \times m$ matrix $\mathbf \otimes \mathbf = \begin v_1 w_1 && v_1 w_2 && \cdots && v_1 w_m \\ v_2 w_1 && v_2 w_2 && \cdots && v_2 w_m \\ \vdots && \vdots && \ddots && \vdots \\ v_n w_1 && v_n w_2 && \cdots && v_n w_m\end$ or, in terms of elements, the $ij$-th component is $(\mathbf \otimes \mathbf)_ = v_i w_j.$ The matrix formed this way corresponds naturally to a tensor $T$, where such is understood as a multilinear functional on $V^* \times W^*,$ by sandwiching it with
matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

between a vector and its
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
, or transpose: $T(\mathbf^T, \mathbf^T) := \mathbf^T (\mathbf \otimes \mathbf) \mathbf$ It is important to note that the tensor, as written, takes two vectors—this is an important point that will be dealt with later. In the case of finite dimensions, there is not a strong distinction between a space and its dual, however, it does matter in infinite dimensions and, moreover, getting the regular-vs-dual part right is essential to ensuring that the idea of tensors being developed here corresponds correctly to other senses in which they are viewed, such as in terms of transformations, which is common in physics. The tensors constructed this way generate a vector space themselves when we add and scale them in the natural componentwise fashion and, in fact, multilinear functionals of the type given can be written as some sum of outer products, which we may call pure tensors or simple tensors. This is sufficient to define the tensor product when we can write vectors and transformations in terms of matrices, however, to get a fully general operation, a more abstract approach will be required. Especially, we would like to isolate the "essential features" of the tensor product without having to specify a particular basis for its construction, and that is what we will do in the following sections.

# Abstracting the tensor product

To achieve that aim, the most natural way to proceed is to try and isolate an essential characterizing property, which will describe, out of all possible vector spaces we could build from ''V'' and ''W'', the one which (up to
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

) is their tensor product, and which will apply without consideration of any arbitrary choices such as a choice of basis. And the way to do this is to flip the tensor concept "inside out"—instead of viewing the tensors as objects that vectors in the manner of a bilinear map, we will view them instead as objects to a bilinear map. The trick is in recognizing that the Kronecker product "" regarding which vectors went into it: the ratios of vector components can be derived from $\frac = \frac = \frac$ and from those ratios, the individual components themselves recovered (up to a constant factor). As a result, a single Kronecker outer product can be used in lieu of the pair $\left(v, w\right)$ of vectors that formed it, and conversely. Most importantly, this means we can write any map $f : V \times W \to Z,$ for any third vector space ''Z'', as a map $f_ : V \otimes W \to Z$ where $f_(\mathbf \otimes \mathbf) := f(\mathbf, \mathbf)$ The , then, is that if we have the combining operation $\,\otimes,\,$ and we are given any bilinear map of the form mentioned, there is such $f_$ that meets this requirement. This is not hard to see if we expand in terms of bases, but the more important point of it is that it can be used as a way to characterize the tensor product—that is, we can use it to the tensor product axiomatically with no reference to such. However, before we can do that, we first need to show that the tensor product exists and is unique for vector spaces ''V'' and ''W'' and, to do that, we need a construction.

# The constructive tensor product

## The free vector space

To perform such a construction, the first step we will consider involves introducing something called a "" over a given set. The thrust behind this idea basically consists of what we said in the first section above: since a generic tensor $T$ can be written by the double sum $T = \sum_^ \sum_^ v_ (\mathbf_i \otimes \mathbf_j)$ the most natural way to approach this problem is somehow to figure out how we can "forget" about the specific choice of bases $\mathbf$ and $\mathbf$ that are used here. In mathematics, the way we "forget" about representational details of something is to establish an identification that tells us that two different things that are to be considered representations of the same thing are in fact such, i.e. which, given those says either "yes, they are" or "no, they aren't", and then "lump together" all representations as constituting the "thing represented" without reference to any one in particular by packaging them all together into a single set. In formal terms, we first build an
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, and then take the
quotient set Set, The Set, or SET may refer to: Science, technology, and mathematics Mathematics * Set (mathematics), a collection of distinct elements or members * Category of sets, the category whose objects and morphisms are sets and total functions, respe ...
by that relation. But before we can do that, we first need to develop what we are going to take the equivalence relation over. The way we do that is to approach this the other way around, from the "bottom up": since we are not guaranteed a, at least constructible, basis when starting from arbitrary vector spaces, we might instead try to start by guaranteeing we have one—that is, we will start first by considering a "basis", on its own, as given, and then building the vector space on top. To that end, we accomplish the following: suppose that $B$ is some set, which we could call an ''abstract basis set''. Now consider all ''formal'' expressions of the form $\mathbf = a_1 \beta_1 + a_2 \beta_2 + \cdots + a_n \beta_n$ of arbitrary, but finite, length $n$ and for which $a_j$ are scalars and $\beta_j$ are members of $B.$ Intuitively, this is a linear combination of the basis vectors in the usual sense of expanding an element of a vector space. We call this a "formal expression" because technically it is illegal to multiply $a_j \beta_j$ since there is no defined multiplication operation by default on an arbitrary set and arbitrary field of scalars. Instead, we will "pretend" (similar to defining the
imaginary number An imaginary number is a complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
s) that this refers to something, and then will go about manipulating it according to the rules we expect for a vector space, e.g. the sum of two such strings using the same sequence of members of $B$ is $(a_1 \beta_1 + a_2 \beta_2 + \cdots + a_n \beta_n) + (b_1 \beta_1 + b_2 \beta_2 + \cdots + b_n \beta_n) = (a_1 + b_1) \beta_1 + (a_2 + b_2) \beta_2 + \cdots + (a_n + b_n) \beta_n$ where we have used the
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
,
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, and distributive laws to rearrange the first sum into the second. Continuing this way for scalar multiples and all different-length combinations of vectors allows us to build up a vector addition and scalar multiplication on this set of formal expressions, and we call it the free vector space over $B,$ writing $F\left(B\right).$ Note that the elements of $B,$ considered as length-one formal expressions with coefficient 1 out front, form a
Hamel basis In mathematics, a Set (mathematics), set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred ...
for this space. The tensor product expression is then abstracted by considering that if $\beta_j$ and $\gamma_j$ represent "abstract basis vectors" from two sets $B$ and $G,$ i.e. that "$\beta_j = \mathbf_j$" and "$\gamma_j = \mathbf_j$", then pairs of these in the Cartesian product $B \times G,$ i.e. $\left(\beta_i, \gamma_j\right)$ are taken as standing for the tensor products $\mathbf_i \otimes \mathbf_j.$ (Note that the tensor products in the expression are in some sense "atomic", i.e. additions and scalar multiplications do not split them up into anything else, so we can replace them with something different without altering the mathematical structure.) With such an identification, we can thus define the tensor product of two free vector spaces $F\left(B\right)$ and $F\left(G\right)$ as being something (yet to be decided) that is isomorphic to $F\left(B \times G\right).$

## The equivalence relation

The above definition will work for any vector space in which we specify a basis, since we can just rebuild it as the free vector space over that basis: the above construction exactly mirrors how you represent vectors via the Hamel basis construction by design. In effect, we haven't gained anything ... until we do this. Now, we are not assuming access to bases for vector spaces $V$ and $W$ that we want to form the tensor product $V \otimes W$ of. Instead, we will take of $V$ and $W$ as "basis" to build up the tensors. This is the next best thing and the one thing we are to be able to do, regardless of any concerns in finding a specific basis; this corresponds to adding together arbitrary outer products $\mathbf \otimes \mathbf$ of arbitrary vectors. The only difference here is that if we use the free vector space construction and form the obvious $F\left(V\right) \otimes F\left(W\right) = F\left(V \times W\right),$ it will have many redundant versions of what should be the same tensor; going back to our basis case if we consider the example where $V = W = \R^2$ in the standard basis, we may consider that the tensor formed by the vectors $\mathbf = \begin 0 & 3 \end^\mathsf$ and $\mathbf = \begin 5 & -3 \end^\mathsf,$ i.e. $T := \mathbf \otimes \mathbf = \begin 0 & 0 \\ 15 & -9 \end,$ could be represented by other sums, such as the sum using individual basic tensors $\mathbf_i \otimes \mathbf_j,$ e.g. $T = 0\left(\mathbf_1 \otimes \mathbf_1\right) + 0\left(\mathbf_1 \otimes \mathbf_2\right) + 15\left(\mathbf_2 \otimes \mathbf_1\right) - 9\left(\mathbf_2 \otimes \mathbf_2\right).$ These, while equal expressions in the concrete case, would correspond to distinct elements of the free vector space $F\left(V \times W\right),$ namely $T = (x, y)$ in the first case and $T = 0(e_1, e_1) + 0(e_1, e_2) + 15(e_2, e_1) - 9(e_2, e_2)$ in the second case. Thus we must condense them—this is where the equivalence relation comes into play. The trick to building it is to note that given any vector $\mathbf$ in a vector space, it is always possible to represent it as the sum of two other vectors $\mathbf$ and $\mathbf$ not equal to the original. If nothing else, let $\mathbf$ be any vector and then take $\mathbf := \mathbf - \mathbf$—which also shows that if we are given one vector and then a second vector, we can write the first vector in terms of the second together with a suitable third vector (indeed in many ways—just consider scalar multiples of the second vector in the same subtraction.). This is useful to us because the outer product satisfies the following linearity properties, which can be proven by simple algebra on the corresponding matrix expressions: $\begin (\mathbf \otimes \mathbf)^\mathsf &= (\mathbf \otimes \mathbf) \\ (\mathbf + \mathbf) \otimes \mathbf &= \mathbf \otimes \mathbf + \mathbf \otimes \mathbf \\ \mathbf \otimes (\mathbf + \mathbf) &= \mathbf \otimes \mathbf + \mathbf \otimes \mathbf \\ c (\mathbf \otimes \mathbf) &= (c\mathbf) \otimes \mathbf = \mathbf \otimes (c\mathbf) \end$ If we want to relate the outer product $\mathbf \otimes \mathbf$ to, say, $\mathbf \otimes \mathbf,$ we can use the first relation above together with a suitable expression of $\mathbf$ as a sum of some vector and some scalar multiple of $\mathbf.$ Equality between two concrete tensors is then obtained if using the above rules will permit us to rearrange one sum of outer products into the other by suitably decomposing vectors—regardless of if we have a set of actual basis vectors. Applying that to our example above, we see that of course we have $\begin \mathbf &= 0 \mathbf_1 + 3 \mathbf_2 \\ \mathbf &= 5 \mathbf_1 - 3 \mathbf_2 \end$ for which substitution in $T = \mathbf \otimes \mathbf$ gives us $T = \left(0 \mathbf_1 + 3 \mathbf_2) \otimes (5 \mathbf_1 - 3 \mathbf_2\right)$ and judicious use of the distributivity properties lets us rearrange to the desired form. Likewise, there is a corresponding "mirror" manipulation in terms of the free vector space elements $\left(x, y\right)$ and $\left(e_1, e_1\right),$ $\left(e_1, e_2\right),$ etc., and this finally leads us to the formal definition of the tensor product.

## Putting all the construction together

The abstract tensor product of two vector spaces $V$ and $W$ over a common base field $K$ is the quotient vector space $V \otimes W := F(V \times W)/$ where $\sim$ is the
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of generated by assuming that, for each $\left(v, w\right)$ and $\left(v\text{'}, w\text{'}\right)$ taken as formal expressions in the free vector space $F\left(V \times W\right),$ the following hold: ;''Identity'': $\left(v, w\right) \sim \left(v, w\right).$ ;''Symmetry'': $\left(v, w\right) \sim \left(v\text{'}, w\text{'}\right)$ implies $\left(v\text{'}, w\text{'}\right) \sim \left(v, w\right).$ ;''Transitivity'': $\left(v, w\right) \sim \left(v\text{'}, w\text{'}\right)$ and $\left(v\text{'}, w\text{'}\right) \sim \left(v\text{'}\text{'}, w\text{'}\text{'}\right)$ implies $\left(v, w\right) \sim \left(v\text{'}\text{'}, w\text{'}\text{'}\right).$ ;''Distributivity'': $\left(v, w\right) + \left(v\text{'}, w\right) \sim \left(v + v\text{'}, w\right)$ and $\left(v, w\right) + \left(v, w\text{'}\right) \sim \left(v, w + w\text{'}\right).$ ;''Scalar multiples'': $c\left(v, w\right) \sim \left(v, cw\right)$ and $c\left(v, w\right) \sim \left(cv, w\right).$ and then testing equivalence of generic formal expressions through suitable manipulations based thereupon. Arithmetic is defined on the tensor product by choosing representative elements, applying the arithmetical rules, and finally taking the equivalence class. Moreover, given any two vectors $v \in V$ and $w \in W,$ the equivalence class

# Properties

## Notation

Elements of $V \otimes W$ are often referred to as ''tensors'', although this term refers to many other related concepts as well. If belongs to and belongs to , then the equivalence class of is denoted by $v \otimes w,$ which is called the tensor product of with . In physics and engineering, this use of the $\,\otimes\,$ symbol refers specifically to the
outer product In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
operation; the result of the outer product $v \otimes w$ is one of the standard ways of representing the equivalence class $v \otimes w.$ An element of $V \otimes W$ that can be written in the form $v \otimes w$ is called a or . In general, an element of the tensor product space is not a pure tensor, but rather a finite linear combination of pure tensors. For example, if $v_1$ and $v_2$ are
linearly independent In the theory of vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...
, and $w_1$ and $w_2$ are also linearly independent, then $v_1 \otimes w_1 + v_2 \otimes w_2$ 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 In mathematics, a tensor is an algebraic object that describes a (multilinear mapping, multilinear) relationship between sets of algebraic objects related to a vector space. Objects that tensors may map between include Vector (mathematics and ph ...
(not to be confused with
tensor order In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, 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 tensors (elements of the space $V \otimes V^*$), it agrees with
matrix rank In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
.

## Dimension

Given bases $\left\$ and $\left\$ for and respectively, the tensors $\left\$ form a basis for $V \otimes W.$ Therefore, if and are finite-dimensional, the dimension of the tensor product is the product of dimensions of the original spaces; for instance $\R^m \otimes \R^n$ is isomorphic to $\R^.$

## Tensor product of linear maps

The tensor product also operates on
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s between vector spaces. Specifically, given two linear maps $S : V \to X$ and $T : W \to Y$ between vector spaces, the and is a linear map $S \otimes T : V \otimes W \to X \otimes Y$ defined by $(S \otimes T)(v \otimes w) = S(v) \otimes T(w).$ In this way, the tensor product becomes a
bifunctor In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
from the category of vector spaces to itself, covariant in both arguments. If and are both
injective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
,
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
or (in the case that , , , and are
normed vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s or
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space (an Abstra ...
s)
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
, then $S \otimes T$ is injective, surjective or continuous, respectively. By choosing bases of all vector spaces involved, the linear maps and can be represented by
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
. Then, depending on how the tensor $v \otimes w$ is vectorized, the matrix describing the tensor product $S \otimes T$ is the
Kronecker product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of the two matrices. For example, if , and above are all two-dimensional and bases have been fixed for all of them, and and are given by the matrices $A=\begin a_ & a_ \\ a_ & a_ \\ \end, \qquad B=\begin b_ & b_ \\ b_ & b_ \\ \end,$ respectively, then the tensor product of these two matrices is $\begin a_ & a_ \\ a_ & a_ \\ \end \otimes \begin b_ & b_ \\ b_ & b_ \\ \end = \begin a_ \begin b_ & b_ \\ b_ & b_ \\ \end & a_ \begin b_ & b_ \\ b_ & b_ \\ \end \\$ a_ \begin b_ & b_ \\ b_ & b_ \\ \end & a_ \begin b_ & b_ \\ b_ & b_ \\ \end \\ \end = \begin a_ b_ & a_ b_ & a_ b_ & a_ b_ \\ a_ b_ & a_ b_ & a_ b_ & a_ b_ \\ a_ b_ & a_ b_ & a_ b_ & a_ b_ \\ a_ b_ & a_ b_ & a_ b_ & a_ b_ \\ \end. The resultant rank is at most 4, and thus the resultant dimension is 4. Note that here denotes the
tensor rank In mathematics, a tensor is an algebraic object that describes a (multilinear mapping, multilinear) relationship between sets of algebraic objects related to a vector space. Objects that tensors may map between include Vector (mathematics and ph ...
i.e. the number of requisite indices (while the
matrix rank In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
counts the number of degrees of freedom in the resulting array). Note $\operatorname A \otimes B = \operatorname A \times \operatorname B.$ A
dyadic product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
is the special case of the tensor product between two vectors of the same dimension.

## Universal property

In the context of vector spaces, the tensor product $V \otimes W$ and the associated bilinear map $\varphi: V \times W \to V \otimes W$ are characterized up to isomorphism by a
universal property In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
regarding
bilinear map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. (Recall that a bilinear map is a function that is ''separately'' linear in each of its arguments.) Informally, $\varphi$ is the most general bilinear map out of $V \times W.$ This characterization can simplify proofs about the tensor product. For example, the tensor product is symmetric, meaning there is a
canonical isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
: $V \otimes W \cong W \otimes V.$ To construct, say, a map from $V \otimes W$ to $W \otimes V,$ it suffices to give a bilinear map $h: V \times W \to W \otimes V$ that maps $\left(v,w\right)$ to $w \otimes v.$ Then the universal property of $V \otimes W$ means $h$ factors into a map $\tilde:V \otimes W \to W \otimes V.$ A map $\tilde:W \otimes V \to V \otimes W$ in the opposite direction is similarly defined, and one checks that the two linear maps $\tilde$ and $\tilde$ are
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
to one another by again using their universal properties. The universal property is extremely useful in showing that a map to a tensor product is injective. For example, suppose we want to show that $\R \otimes \R$ is isomorphic to $\R.$ Since all simple tensors are of the form $a \otimes b = \left(ab\right) \otimes 1,$ and hence all elements of the tensor product are of the form $x \otimes 1$ by additivity in the first coordinate, we have a natural candidate for an isomorphism $\R \rightarrow \R \otimes \R$ given by mapping $x$ to $x \otimes 1,$ and this map is trivially surjective. Showing injectivity directly would involve somehow showing that there are no non-trivial relationships between $x \otimes 1$ and $y \otimes 1$ for $x \neq y,$ which seems daunting. However, we know that there is a bilinear map $\R \times \R \rightarrow \R$ given by multiplying the coordinates together, and the universal property of the tensor product then furnishes a map of vector spaces $\R \otimes \R \rightarrow \R$ which maps $x \otimes 1$ to $x,$ and hence is an inverse of the previously constructed homomorphism, immediately implying the desired result. Note that, a priori, it is not even clear that this inverse map is well-defined, but the universal property and associated bilinear map together imply this is the case. Similar reasoning can be used to show that the tensor product is associative, that is, there are natural isomorphisms $V_1 \otimes \left(V_2\otimes V_3\right) \cong \left(V_1\otimes V_2\right) \otimes V_3.$ Therefore, it is customary to omit the parentheses and write $V_1 \otimes V_2 \otimes V_3$, so the ''ijk-th'' component of $\mathbf \otimes \mathbf \otimes \mathbf$ is $(\mathbf \otimes \mathbf\otimes \mathbf)_ = u_i v_j w_k,$ similar to the first
example Example may refer to: * ''exempli gratia Notes and references Notes References Sources * * * Further reading * * {{Latin phrases E ...'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain na ...
on this page. The category of vector spaces with tensor product is an example of a
symmetric monoidal categoryIn category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...
. The universal-property definition of a tensor product is valid in more categories than just the category of vector spaces. Instead of using multilinear (bilinear) maps, the general tensor product definition uses multimorphisms.

## Tensor powers and braiding

Let be a non-negative integer. The th tensor power of the vector space is the -fold tensor product of with itself. That is $V^ \;\overset\; \underbrace_.$ A
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

$\sigma$ of the set $\$ determines a mapping of the th Cartesian power of as follows: $\begin \sigma : V^n \to V^n \\ \sigma\left(v_1,v_2,\ldots,v_n\right) = \left(v_, v_,\ldots,v_\right) \end$ Let $\varphi : V^n \to V^$ be the natural multilinear embedding of the Cartesian power of into the tensor power of . Then, by the universal property, there is a unique isomorphism $\tau_\sigma : V^ \to V^$ such that $\varphi\circ\sigma = \tau_\sigma\circ\varphi.$ The isomorphism $\tau_$ is called the braiding map associated to the permutation $\sigma.$

# Product of tensors

For non-negative integers and a type $\left(r, s\right)$
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

on a vector space is an element of $T^r_s(V) = \underbrace_r \otimes \underbrace_s = V^ \otimes \left(V^*\right)^.$ Here $V^*$ is the
dual vector space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
(which consists of all
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s from to the ground field ). There is a product map, called the $T^r_s (V) \otimes_K T^_ (V) \to T^_(V).$ It is defined by grouping all occurring "factors" together: writing $v_i$ for an element of and $f_i$ for an element of the dual space, $(v_1 \otimes f_1) \otimes (v'_1) = v_1 \otimes v'_1 \otimes f_1.$ Picking a basis of and the corresponding
dual basis In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
of $V^*$ naturally induces a basis for $T_s^r\left(V\right)$ (this basis is described in the article on Kronecker products). In terms of these bases, the
components Component may refer to: In engineering, science, and technology Generic systems *System A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounde ...
of a (tensor) product of two (or more)
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

s can be computed. For example, if and are two covariant tensors of orders and respectively (i.e. $F \in T_m^0$ and $G \in T_n^0$), then the components of their tensor product are given by $(F \otimes G)_ = F_ G_.$ Thus, the components of the tensor product of two tensors are the ordinary product of the components of each tensor. Another example: let be a tensor of type with components $U^_,$ and let be a tensor of type $\left(1, 0\right)$ with components $V^.$ Then $\left(U \otimes V\right)^\alpha _\beta ^\gamma = U^\alpha _\beta V^\gamma$ and $(V \otimes U)^ _\sigma = V^\mu U^\nu _\sigma.$ Tensors equipped with their product operation form an
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...
, called the
tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', ...
.

## Evaluation map and tensor contraction

For tensors of type there is a canonical evaluation map $V \otimes V^* \to K$ defined by its action on pure tensors: $v \otimes f \mapsto f(v).$ More generally, for tensors of type $\left(r, s\right),$ with , there is a map, called
tensor contraction In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual vector space, dual. In components, it is expressed as a sum of products of scalar compo ...
, $T^r_s (V) \to T^_(V).$ (The copies of $V$ and $V^*$ on which this map is to be applied must be specified.) On the other hand, if $V$ is , there is a canonical map in the other direction (called the coevaluation map) $\begin K \to V \otimes V^* \\ \lambda \mapsto \sum_i \lambda v_i \otimes v^*_i \end$ where $v_1, \ldots, v_n$ is any basis of $V,$ and $v_i^*$ is its
dual basis In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
. This map does not depend on the choice of basis. The interplay of evaluation and coevaluation can be used to characterize finite-dimensional vector spaces without referring to bases.

The tensor product $T^r_s\left(V\right)$ may be naturally viewed as a module for the
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
$\mathrm\left(V\right)$ by means of the diagonal action: for simplicity let us assume $r = s = 1,$ then, for each $u \in \mathrm\left(V\right),$ $u(a \otimes b) = u(a) \otimes b - a \otimes u^*(b),$ where $u^* \in \mathrm\left\left(V^*\right\right)$ is the
transpose In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...

of , that is, in terms of the obvious pairing on $V \otimes V^*,$ $\langle u(a), b \rangle = \langle a, u^*(b) \rangle.$ There is a canonical isomorphism $T^1_1\left(V\right) \to \mathrm\left(V\right)$ given by $(a \otimes b)(x) = \langle x, b \rangle a.$ Under this isomorphism, every in $\mathrm\left(V\right)$ may be first viewed as an endomorphism of $T^1_1\left(V\right)$ and then viewed as an endomorphism of $\mathrm\left(V\right).$ In fact it is the
adjoint representation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of $\mathrm\left(V\right).$

# Relation of tensor product to Hom

Given two finite dimensional vector spaces , over the same field , denote the
dual space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
of as , and the -vector space of all linear maps from to as . There is an isomorphism, $U^* \otimes V \cong \mathrm(U, V),$ defined by an action of the pure tensor $f \otimes v \in U^*\otimes V$ on an element of $U,$ $(f \otimes v)(u) = f(u) v.$ Its "inverse" can be defined using a basis $\$ and its dual basis $\$ as in the section " Evaluation map and tensor contraction" above: $\begin \mathrm (U,V) \to U^* \otimes V \\ F \mapsto \sum_i u^*_i \otimes F(u_i). \end$ This result implies $\dim(U \otimes V) = \dim(U)\dim(V),$ which automatically gives the important fact that $\$ forms a basis for $U \otimes V$ where $\, \$ are bases of and . Furthermore, given three vector spaces , , the tensor product is linked to the vector space of ''all'' linear maps, as follows: $\mathrm (U \otimes V, W) \cong \mathrm (U, \mathrm(V, W)).$ This is an example of
adjoint functor In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s: the tensor product is "left adjoint" to Hom.

# Tensor products of modules over a ring

The tensor product of two
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a syst ...
and over a ''
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
'' ring is defined in exactly the same way as the tensor product of vector spaces over a field: $A \otimes_R B := F (A \times B) / G$ where now $F\left(A \times B\right)$ is the generated by the cartesian product and is the -module generated by the same relations as above. More generally, the tensor product can be defined even if the ring is
non-commutative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. In this case has to be a right--module and is a left--module, and instead of the last two relations above, the relation $(ar,b)\sim (a,rb)$ is imposed. If is non-commutative, this is no longer an -module, but just an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
. The universal property also carries over, slightly modified: the map $\varphi : A \times B \to A \otimes_R B$ defined by $\left(a, b\right) \mapsto a \otimes b$ is a middle linear map (referred to as "the canonical middle linear map".); that is, it satisfies: $\begin \phi(a + a', b) &= \phi(a, b) + \phi(a', b) \\ \phi(a, b + b') &= \phi(a, b) + \phi(a, b') \\ \phi(ar, b) &= \phi(a, rb) \end$ The first two properties make a bilinear map of the
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
$A \times B.$ For any middle linear map $\psi$ of $A \times B,$ a unique group homomorphism of $A \otimes_R B$ satisfies $\psi = f \circ \varphi,$ and this property determines $\phi$ within group isomorphism. See the for details.

## Tensor product of modules over a non-commutative ring

Let ''A'' be a right ''R''-module and ''B'' be a left ''R''-module. Then the tensor product of ''A'' and ''B'' is an abelian group defined by $A \otimes_R B := F (A \times B) / G$ where $F \left(A \times B\right)$ is a
free abelian group In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation that is associative, commutative, and invertible. A basis, also called ...
over $A \times B$ and G is the subgroup of $F \left(A \times B\right)$ generated by relations $\begin &\forall a, a_1, a_2 \in A, \forall b, b_1, b_2 \in B, \text r \in R:\\ &(a_1,b) + (a_2,b) - (a_1 + a_2,b),\\ &(a,b_1) + (a,b_2) - (a,b_1+b_2),\\ &(ar,b) - (a,rb).\\ \end$ The universal property can be stated as follows. Let ''G'' be an abelian group with a map $q:A\times B \to G$ that is bilinear, in the sense that $\begin q(a_1 + a_2, b) &= q(a_1, b) + q(a_2, b),\\ q(a, b_1 + b_2) &= q(a, b_1) + q(a, b_2),\\ q(ar, b) &= q(a, rb). \end$ Then there is a unique map $\overline:A\otimes B \to G$ such that $\overline\left(a\otimes b\right) = q\left(a,b\right)$ for all $a \in A$ and $b \in B.$ Furthermore, we can give $A \otimes_R B$ a module structure under some extra conditions: # If ''A'' is a (''S'',''R'')-bimodule, then $A \otimes_R B$ is a left ''S''-module where $s\left(a\otimes b\right):=\left(sa\right)\otimes b.$ # If ''B'' is a (''R'',''S'')-bimodule, then $A \otimes_R B$ is a right ''S''-module where $\left(a\otimes b\right)s:=a\otimes \left(bs\right).$ # If ''A'' is a (''S'',''R'')-bimodule and ''B'' is a (''R'',''T'')-bimodule, then $A \otimes_R B$ is a (''S'',''T'')-bimodule, where the left and right actions are defined in the same way as the previous two examples. # If ''R'' is a commutative ring, then ''A'' and ''B'' are (''R'',''R'')-bimodules where $ra:=ar$ and $br:=rb.$ By 3), we can conclude $A \otimes_R B$ is a (''R'',''R'')-bimodule.

## Computing the tensor product

For vector spaces, the tensor product $V \otimes W$ is quickly computed since bases of of immediately determine a basis of $V \otimes W,$ as was mentioned above. For modules over a general (commutative) ring, not every module is free. For example, is not a free abelian group (-module). The tensor product with is given by $M \otimes_\mathbf \mathbf/n\mathbf = M/nM.$ More generally, given a
presentation A presentation conveys information from a speaker to an audience An audience is a group of people who participate in a show or encounter a work of art A work of art, artwork, art piece, piece of art or art object is an artistic creation ...
of some -module , that is, a number of generators $m_i \in M, i \in I$ together with relations $\sum_ a_ m_i = 0,\qquad a_ \in R,$ the tensor product can be computed as the following
cokernel The cokernel of a linear mapping In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...

: $M \otimes_R N = \operatorname \left(N^J \to N^I\right)$ Here $N^J = \oplus_ N,$ and the map $N^J \to N^I$ is determined by sending some $n \in N$ in the th copy of $N^J$ to $a_ n$ (in $N^I$). Colloquially, this may be rephrased by saying that a presentation of gives rise to a presentation of $M \oplus_R N.$ This is referred to by saying that the tensor product is a
right exact functor In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. It is not in general left exact, that is, given an injective map of -modules $M_1 \to M_2,$ the tensor product $M_1 \otimes_R N \to M_2 \otimes_R N$ is not usually injective. For example, tensoring the (injective) map given by multiplication with , with yields the zero map , which is not injective. Higher
Tor functor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
s measure the defect of the tensor product being not left exact. All higher Tor functors are assembled in the
derived tensor productIn algebra, given a differential graded algebra ''A'' over a commutative ring In ring theory, a branch of abstract algebra, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutat ...
.

# Tensor product of algebras

Let be a commutative ring. The tensor product of -modules applies, in particular, if and are -algebras. In this case, the tensor product $A \otimes_R B$ is an -algebra itself by putting $(a_1 \otimes b_1) \cdot (a_2 \otimes b_2) = (a_1 \cdot a_2) \otimes (b_1 \cdot b_2).$ For example,
, y The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
A particular example is when and are fields containing a common subfield . The
tensor product of fields In mathematics, the tensor product of two fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument ...
is closely related to
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
: if, say, , where is some
irreducible polynomial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
with coefficients in , the tensor product can be calculated as where now is interpreted as the same polynomial, but with its coefficients regarded as elements of . In the larger field , the polynomial may become reducible, which brings in Galois theory. For example, if is a
Galois extension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
of , then is isomorphic (as an -algebra) to the $A^.$

# Eigenconfigurations of tensors

Square
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In , a matrix (plural matrices) is a array or table of s, s, or s, arranged in rows and columns, which is used to represent a or a property of such an object. Fo ...
$A$ with entries in a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
$K$ represent linear maps of
vector spaces In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, say $K^n \to K^n,$ and thus linear maps $\psi : \mathbb^ \to \mathbb^$ of
projective spaces In mathematics, the concept of a projective space originated from the visual effect of perspective (graphical), perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean sp ...
over $K.$ If $A$ is
nonsingularIn linear algebra, an ''n''-by-''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...
then $\psi$ is
well-defined In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
everywhere, and the
eigenvectors In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces ...
of $A$ correspond to the fixed points of $\psi.$ The ''eigenconfiguration'' of $A$ consists of $n$ points in $\mathbb^,$ provided $A$ is generic and $K$ is
algebraically closed In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. The fixed points of nonlinear maps are the eigenvectors of tensors. Let $A = \left(a_\right)$ be a $d$-dimensional tensor of format $n \times n \times \cdots \times n$ with entries $\left(a_\right)$ lying in an algebraically closed field $K$ of
characteristic Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
zero. Such a tensor $A \in \left(K^\right)^$ defines polynomial maps $K^n \to K^n$ and $\mathbb^ \to \mathbb^$ with coordinates $\psi_i(x_1, \ldots, x_n) = \sum_^n \sum_^n \cdots \sum_^n a_ x_ x_\cdots x_ \;\; \mbox i = 1, \ldots, n$ Thus each of the $n$ coordinates of $\psi$ is a
homogeneous polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
$\psi_i$ of degree $d-1$ in $\mathbf = \left\left(x_1, \ldots, x_n\right\right).$ The eigenvectors of $A$ are the solutions of the constraint $\mbox \begin x_1 & x_2 & \cdots & x_n \\ \psi_1(\mathbf) & \psi_2(\mathbf) & \cdots & \psi_n(\mathbf) \end \leq 1$ and the eigenconfiguration is given by the
variety Variety may refer to: Science and technology Mathematics * Algebraic variety, the set of solutions of a system of polynomial equations * Variety (universal algebra), classes of algebraic structures defined by equations in universal algebra Hort ...
of the $2 \times 2$
minors Minor may refer to: * Minor (law) In law Law is a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenc ...
of this matrix.

# Other examples of tensor products

## Tensor product of Hilbert spaces

Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s generalize finite-dimensional vector spaces to countably-infinite dimensions. The tensor product is still defined; it is the
tensor product of Hilbert spaces In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
.

## Topological tensor product

When the basis for a vector space is no longer countable, then the appropriate axiomatic formalization for the vector space is that of a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space (an Abstra ...
. The tensor product is still defined, it is the
topological tensor productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
.

## Tensor product of graded vector spaces

Some vector spaces can be decomposed into
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
s of subspaces. In such cases, the tensor product of two spaces can be decomposed into sums of products of the subspaces (in analogy to the way that multiplication distributes over addition).

## Tensor product of representations

Vector spaces endowed with an additional multiplicative structure are called
algebras In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. The tensor product of such algebras is described by the
Littlewood–Richardson ruleIn mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur polynomial, Schur functions as a linear combination of other Schur functions. These coefficients ar ...
.

## Tensor product of multilinear forms

Given two
multilinear form In abstract algebra and multilinear algebra, a multilinear form on a vector space V over a Field (mathematics), field K is a Map (mathematics), map :f\colon V^k \to K that is separately ''K''-linear in each of its ''k'' arguments. More generally ...
s $f\left(x_1,\dots,x_k\right)$ and $g \left(x_1,\dots, x_m\right)$ on a vector space $V$ over the field $K$ their tensor product is the multilinear form $(f \otimes g) (x_1,\dots,x_) = f(x_1,\dots,x_k) g(x_,\dots,x_).$ This is a special case of the product of tensors if they are seen as multilinear maps (see also Tensor#As multilinear maps, tensors as multilinear maps). Thus the components of the tensor product of multilinear forms can be computed by the
Kronecker product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
.

## Tensor product of graphs

It should be mentioned that, though called "tensor product", this is not a tensor product of graphs in the above sense; actually it is the Product (category theory), category-theoretic product in the category of graphs and graph homomorphisms. However it is actually the Kronecker product, Kronecker tensor product of the adjacency matrix, adjacency matrices of the graphs. Compare also the section tensor product#Tensor product of linear maps, Tensor product of linear maps above.

## Monoidal categories

The most general setting for the tensor product is the monoidal category. It captures the algebraic essence of tensoring, without making any specific reference to what is being tensored. Thus, all tensor products can be expressed as an application of the monoidal category to some particular setting, acting on some particular objects.

# Quotient algebras

A number of important subspaces of the
tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', ...
can be constructed as quotient space (linear algebra), quotients: these include the exterior algebra, the symmetric algebra, the Clifford algebra, the Weyl algebra, and the universal enveloping algebra in general. The exterior algebra is constructed from the exterior product. Given a vector space , the exterior product $V \wedge V$ is defined as $V \wedge V := V \otimes V/\.$ Note that when the underlying field of does not have characteristic 2, then this definition is equivalent to $V \wedge V := V \otimes V / \.$ The image of $v_1 \otimes v_2$ in the exterior product is usually denoted $v_1 \wedge v_2$ and satisfies, by construction, $v_1 \wedge v_2 = - v_2 \wedge v_1.$ Similar constructions are possible for $V \otimes \dots \otimes V$ ( factors), giving rise to $\Lambda^n V,$ the th exterior power of . The latter notion is the basis of differential form, differential -forms. The symmetric algebra is constructed in a similar manner, from the symmetric tensor#symmetric product, symmetric product $V \odot V := V \otimes V / \.$ More generally $\operatorname^n V := \underbrace_n / (\dots \otimes v_i \otimes v_ \otimes \dots - \dots \otimes v_ \otimes v_ \otimes \dots)$ That is, in the symmetric algebra two adjacent vectors (and therefore all of them) can be interchanged. The resulting objects are called symmetric tensors.

# Tensor product in programming

## Array programming languages

Array programming languages may have this pattern built in. For example, in APL programming language, APL the tensor product is expressed as ○.× (for example A ○.× B or A ○.× B ○.× C). In J programming language, 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. This product of two functions is a derived function, and if a and b are Differentiable function, 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 matrix and determinant, Jacobian derivative (for example, Fortran/APL).