In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a product is the result of
multiplication, or an
expression that identifies
objects
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ai ...
(numbers or
variables) to be multiplied, called ''factors''. For example, 30 is the product of 6 and 5 (the result of multiplication), and
is the product of
and
(indicating that the two factors should be multiplied together).
The order in which
real or
complex numbers are multiplied has no bearing on the product; this is known as the ''
commutative law'' of multiplication. When
matrices or members of various other
associative algebras are multiplied, the product usually depends on the order of the factors.
Matrix multiplication, for example, is non-commutative, and so is multiplication in other algebras in general as well.
There are many different kinds of products in mathematics: besides being able to multiply just numbers, polynomials or matrices, one can also define products on many different
algebraic structures.
Product of two numbers
Product of a sequence
The product operator for the
product of a sequence is denoted by the capital Greek letter
pi Π (in analogy to the use of the capital Sigma
Σ as
summation symbol).
For example, the expression
is another way of writing
.
The product of a sequence consisting of only one number is just that number itself; the product of no factors at all is known as the
empty product, and is equal to 1.
Commutative rings
Commutative rings have a product operation.
Residue classes of integers
Residue classes in the rings
can be added:
:
and multiplied:
:
Convolution
Two functions from the reals to itself can be multiplied in another way, called the
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
.
If
:
then the integral
:
is well defined and is called the convolution.
Under the
Fourier transform, convolution becomes point-wise function multiplication.
Polynomial rings
The product of two polynomials is given by the following:
:
with
:
Products in linear algebra
There are many different kinds of products in linear algebra. Some of these have confusingly similar names (
outer product,
exterior product) with very different meanings, while others have very different names (outer product, tensor product, Kronecker product) and yet convey essentially the same idea. A brief overview of these is given in the following sections.
Scalar multiplication
By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map
.
Scalar product
A
scalar product is a bi-linear map:
:
with the following conditions, that
for all
.
From the scalar product, one can define a
norm by letting
.
The scalar product also allows one to define an angle between two vectors:
:
In
-dimensional Euclidean space, the standard scalar product (called the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
) is given by:
:
Cross product in 3-dimensional space
The
cross product of two vectors in 3-dimensions is a vector perpendicular to the two factors, with length equal to the area of the parallelogram spanned by the two factors.
The cross product can also be expressed as the
formal determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
:
:
Composition of linear mappings
A linear mapping can be defined as a function ''f'' between two vector spaces ''V'' and ''W'' with underlying field F, satisfying
:
If one only considers finite dimensional vector spaces, then
:
in which b
V and b
W denote the
bases of ''V'' and ''W'', and ''v
i'' denotes the
component
Circuit Component may refer to:
•Are devices that perform functions when they are connected in a circuit.
In engineering, science, and technology Generic systems
* System components, an entity with discrete structure, such as an assem ...
of v on b
V''i'', and
Einstein summation convention is applied.
Now we consider the composition of two linear mappings between finite dimensional vector spaces. Let the linear mapping ''f'' map ''V'' to ''W'', and let the linear mapping ''g'' map ''W'' to ''U''. Then one can get
:
Or in matrix form:
:
in which the ''i''-row, ''j''-column element of F, denoted by ''F
ij'', is ''f
ji'', and ''G
ij=g
ji''.
The composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication.
Product of two matrices
Given two matrices
:
and
their product is given by
:
Composition of linear functions as matrix product
There is a relationship between the composition of linear functions and the product of two matrices. To see this, let r = dim(U), s = dim(V) and t = dim(W) be the (finite)
dimensions of vector spaces U, V and W. Let
be a
basis of U,
be a basis of V and
be a basis of W. In terms of this basis, let
be the matrix representing f : U → V and
be the matrix representing g : V → W. Then
:
is the matrix representing
.
In other words: the matrix product is the description in coordinates of the composition of linear functions.
Tensor product of vector spaces
Given two finite dimensional vector spaces ''V'' and ''W'', the tensor product of them can be defined as a (2,0)-tensor satisfying:
:
where ''V
*'' and ''W
*'' denote the
dual spaces of ''V'' and ''W''.
For infinite-dimensional vector spaces, one also has the:
*
Tensor product of Hilbert spaces
*
Topological tensor product In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see Tensor product of Hi ...
.
The tensor product,
outer product and
Kronecker product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to ...
all convey the same general idea. The differences between these are that the Kronecker product is just a tensor product of matrices, with respect to a previously-fixed basis, whereas the tensor product is usually given in its
intrinsic definition. The outer product is simply the Kronecker product, limited to vectors (instead of matrices).
The class of all objects with a tensor product
In general, whenever one has two mathematical
objects
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ai ...
that can be combined in a way that behaves like a linear algebra tensor product, then this can be most generally understood as the
internal product of a
monoidal category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left ...
. That is, the monoidal category captures precisely the meaning of a tensor product; it captures exactly the notion of why it is that tensor products behave the way they do. More precisely, a monoidal category is the
class
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differently ...
of all things (of a given
type) that have a tensor product.
Other products in linear algebra
Other kinds of products in linear algebra include:
*
Hadamard product
*
Kronecker product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to ...
* The product of
tensors:
**
Wedge product or exterior product
**
Interior product
**
Outer product
**
Tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
Cartesian product
In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, a
Cartesian product is a
mathematical operation which returns a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
(or product set) from multiple sets. That is, for sets ''A'' and ''B'', the Cartesian product is the set of all
ordered pairs —where and .
The class of all things (of a given
type) that have Cartesian products is called a
Cartesian category. Many of these are
Cartesian closed categories
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mat ...
. Sets are an example of such objects.
Empty product
The
empty product on numbers and most
algebraic structures has the value of 1 (the identity element of multiplication), just like the
empty sum has the value of 0 (the identity element of addition). However, the concept of the empty product is more general, and requires special treatment in
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
,
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
,
computer programming
Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as anal ...
and
category theory.
Products over other algebraic structures
Products over other kinds of
algebraic structures include:
* the
Cartesian product of sets
* the
direct product of groups, and also the
semidirect product,
knit product and
wreath product
* the
free product of groups
* the
product of rings
* the
product of ideals
* the
product of topological spaces
* the
Wick product of
random variables
* the
cap,
cup
A cup is an open-top used to hold hot or cold liquids for pouring or drinking; while mainly used for drinking, it also can be used to store solids for pouring (e.g., sugar, flour, grains, salt). Cups may be made of glass, metal, china, cl ...
,
Massey
Massey may refer to:
Places
Canada
* Massey, Ontario
* Massey Island, Nunavut
New Zealand
* Massey, New Zealand, an Auckland suburb
United States
* Massey, Alabama
* Massey, Iowa
* Massey, Maryland
People
* Massey (surname)
Educatio ...
and
slant product
In algebraic topology the cap product is a method of adjoining a chain of degree ''p'' with a cochain of degree ''q'', such that ''q'' ≤ ''p'', to form a composite chain of degree ''p'' − ''q''. It was introduced by Eduard Čech in 1936, ...
in algebraic topology
* the
smash product
In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) (''X,'' ''x''0) and (''Y'', ''y''0) is the quotient of the product space ''X'' × ''Y'' under the ...
and
wedge sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the q ...
(sometimes called the wedge product) in
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
A few of the above products are examples of the general notion of an
internal product in a
monoidal category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left ...
; the rest are describable by the general notion of a
product in category theory.
Products in category theory
All of the previous examples are special cases or examples of the general notion of a product. For the general treatment of the concept of a product, see
product (category theory)
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rin ...
, which describes how to combine two
objects
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ai ...
of some kind to create an object, possibly of a different kind. But also, in category theory, one has:
* the
fiber product or pullback,
* the
product category, a category that is the product of categories.
* the
ultraproduct, in
model theory.
* the
internal product of a
monoidal category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left ...
, which captures the essence of a tensor product.
Other products
* A function's
product integral (as a continuous equivalent to the product of a sequence or as the multiplicative version of the normal/standard/additive integral. The product integral is also known as "continuous product" or "multiplical".
*
Complex multiplication, a theory of elliptic curves.
See also
*
*
Indefinite product
*
Infinite product
*
*
Notes
References
Bibliography
*
{{DEFAULTSORT:Product (Mathematics)
Multiplication