In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. Thus, for instance, 6 is the product of 2 and 3 (the result of multiplication), and
Contents
- Product of two natural numbers
- Product of two integers
- Product of two fractions
- Product of two real numbers
- Product of two complex numbers
- Geometric meaning of complex multiplication
- Product of two quaternions
- Product of sequences
- Commutative rings
- Residue classes of integers
- Rings of functions
- Convolution
- Polynomial rings
- Products in linear algebra
- Scalar multiplication
- Scalar product
- Cross product in 3 dimensional space
- Composition of linear mappings
- Product of two matrices
- Composition of linear functions as matrix product
- Tensor product of vector spaces
- The class of all objects with a tensor product
- Other products in linear algebra
- Cartesian product
- Empty product
- Products over other algebraic structures
- Products in category theory
- Other products
- References
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, and multiplication in other algebras is in general non-commutative.
There are many different kinds of products in mathematics: besides being able to multiply just numbers, polynomials or matricies, one can also define products on many different algebraic structures. An overview of these different kinds of products is given here.
Product of two natural numbers
Placing several stones into a rectangular pattern with
stones.
Product of two integers
Integers allow positive and negative numbers. The two numbers are multiplied just like natural numbers, except we need an additional rule for the signs:
In words, we have:
Product of two fractions
Two fractions can be multiplied by multiplying their numerators and denominators:
Product of two real numbers
For a rigorous definition of the product of two real numbers see Construction of the real numbers.
Product of two complex numbers
Two complex numbers can be multiplied by the distributive law and the fact that
Geometric meaning of complex multiplication
Complex numbers can be written in polar coordinates:
Furthermore,
The geometric meaning is that we multiply the magnitudes and add the angles.
Product of two quaternions
The product of two quaternions can be found in the article on quaternions. However, it is interesting to note that in this case,
Product of sequences
The product operator for the product of a sequence is denoted by the capital Greek letter Pi ∏ (in analogy to the use of the capital Sigma ∑ as summation symbol). 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
and multiplied:
Rings of functions
Functions to the real numbers can be added or multiplied by adding or multiplying their outputs:
Convolution
Two functions from the reals to itself can be multiplied in another way, called the 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) but have very different meanings. Others have very different names (outer product, tensor product, Kronecker product) but convey essentially the same idea. A brief overview of these is given here.
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 bilinear map:
with the following conditions, that
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
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:
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 bV andbW denote the bases of V and W, and vi denotes the component of v on bVi, 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 Fij, is fji, and Gij=gji.
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
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 und W. Let
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 inifinite-dimensional vector spaces, one also has the:
The tensor product, outer product and Kronecker product 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 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. 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 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:
Cartesian product
In set theory, a Cartesian product is a mathematical operation which returns a set (or product set) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
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. 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, set theory, computer programming and category theory.
Products over other algebraic structures
Products over other kinds of algebraic structures include:
A few of the above products are examples of the general notion of an internal product in a monoidal category; 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), which describes how to combine two objects of some kind to create an object, possibly of a different kind. But also, in category theory, one has: