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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 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, 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 matrices, one can also define products on many different algebraic structures. An overview of these different kinds of products is given here.

Contents

Product of two numbersEdit

Product of two natural numbersEdit

 
3 by 4 is 12

Placing several stones into a rectangular pattern with   rows and   columns gives

 

stones.

Product of two integersEdit

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:

  • Minus times Minus gives Plus
  • Minus times Plus gives Minus
  • Plus times Minus gives Minus
  • Plus times Plus gives Plus

Product of two fractionsEdit

Two fractions can be multiplied by multiplying their numerators and denominators:

 

Product of two real numbersEdit

For a rigorous definition of the product of two real numbers see Construction of the real numbers.

Product of two complex numbersEdit

Two complex numbers can be multiplied by the distributive law and the fact that  , as follows:

 

Geometric meaning of complex multiplicationEdit

 
A complex number in polar coordinates.

Complex numbers can be written in polar coordinates:

 

Furthermore,

 , from which we obtain:
 

The geometric meaning is that we multiply the magnitudes and add the angles.

Product of two quaternionsEdit

The product of two quaternions can be found in the article on quaternions. However, it is interesting to note that in this case,   and   are in general different.

Product of sequencesEdit

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 ringsEdit

Commutative rings have a product operation.

Residue classes of integersEdit

Residue classes in the rings   can be added:

 

and multiplied:

 

Rings of functionsEdit

Functions to the real numbers can be added or multiplied by adding or multiplying their outputs:

 
 

ConvolutionEdit

 
The convolution of the square wave with itself gives the triangular function

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 ringsEdit

The product of two polynomials is given by the following:

 

with

 

Products in linear algebraEdit

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 multiplicationEdit

By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map  .

Scalar productEdit

A scalar product is a bilinear 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) is given by:

 

Cross product in 3-dimensional spaceEdit

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[a] determinant:

 

Composition of linear mappingsEdit

A linear mapping can be defined as a function f between two vector spaces V and W with underlying field F, satisfying[1]

 

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 matricesEdit

Given two matrices

  and  

their product is given by

 

Composition of linear functions as matrix productEdit

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 spacesEdit

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.[2]

For infinite-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 productEdit

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 algebraEdit

Other kinds of products in linear algebra include:

Cartesian productEdit

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.[3]

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 productEdit

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 structuresEdit

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 theoryEdit

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:

Other productsEdit

  • 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 alsoEdit

NotesEdit

  1. ^ Here, “formal" means that this notation has the form of a determinant, but does not strictly adhere to the definition; it is a mnemonic used to remember the expansion of the cross product.

ReferencesEdit

  1. ^ Clarke, Francis (2013). Functional analysis, calculus of variations and optimal control. Dordrecht: Springer. pp. 9–10. ISBN 1447148207. 
  2. ^ Boothby, William M. (1986). An introduction to differentiable manifolds and Riemannian geometry (2nd ed.). Orlando: Academic Press. p. 200. ISBN 0080874398. 
  3. ^ Moschovakis, Yiannis (2006). Notes on set theory (2nd ed.). New York: Springer. p. 13. ISBN 0387316094. 

External linksEdit