Product (mathematics)

Template:Calculation results

In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors. For example, 21 is the product of 3 and 7 (the result of multiplication), and $x\cdot (2+x)$ is the product of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2+x)} (indicating that the two factors should be multiplied together). When one factor is an integer, the product is called a multiple.

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

Originally, a product was and is still the result of the multiplication of two or more numbers. For example, $15$ is the product of $3$ and $5$ . The fundamental theorem of arithmetic states that every composite number is a product of prime numbers, that is unique up to the order of the factors.

With the introduction of mathematical notation and variables at the end of the 15th century, it became common to consider the multiplication of numbers that are either unspecified (coefficients and parameters), or to be found (unknowns). These multiplications that cannot be effectively performed are called products. For example, in the linear equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax+b=0,} the term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax} denotes the product of the coefficient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and the unknown Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x.}

Later and essentially from the 19th century on, new binary operations have been introduced, which do not involve numbers at all, and have been called products; for example, the dot product. Most of this article is devoted to such non-numerical products.

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).^[1] For example, the expression Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \prod_{i=1}^{6}i^2} is another way of writing Template:Tmath.^[2]

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z/N\Z} can be added:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a + N\Z) + (b + N\Z) = a + b + N\Z}

and multiplied:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a + N\Z) \cdot (b + N\Z) = a \cdot b + N\Z}

Convolution

File:Convolucion Funcion Pi.gif

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\limits_{-\infty}^\infty |f(t)|\,\mathrm{d}t < \infty\qquad\mbox{and}\qquad \int\limits_{-\infty}^\infty |g(t)|\,\mathrm{d}t < \infty, }

then the integral

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f*g) (t) \;:= \int\limits_{-\infty}^\infty f(\tau)\cdot g(t - \tau)\,\mathrm{d}\tau }

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \biggl(\sum_{i=0}^n a_i X^i\biggr) \cdot \biggl(\sum_{j=0}^m b_j X^j\biggr) = \sum_{k=0}^{n+m} c_k X^k }

with

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_k = \sum_{i+j=k} a_i \cdot b_j }

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R \times V \rightarrow V} .

Scalar product

A scalar product is a bi-linear map:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot : V \times V \rightarrow \R }

with the following conditions, that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v \cdot v > 0} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \not= v \in V} .

From the scalar product, one can define a norm by letting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|v\| := \sqrt{v \cdot v} } .

The scalar product also allows one to define an angle between two vectors:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos\angle(v, w) = \frac{v \cdot w}{\|v\| \cdot \|w\|}}

In Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -dimensional Euclidean space, the standard scalar product (called the dot product) is given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \biggl(\sum_{i=1}^n \alpha_i e_i\biggr) \cdot \biggl(\sum_{i=1}^n \beta_i e_i\biggr) = \sum_{i=1}^n \alpha_i\,\beta_i}

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^{[lower-alpha 1]} determinant:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u \times v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ \end{vmatrix}}

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^[3]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t_1 x_1 + t_2 x_2) = t_1 f(x_1) + t_2 f(x_2), \forall x_1, x_2 \in V, \forall t_1, t_2 \in \mathbb{F}.}

If one only considers finite dimensional vector spaces, then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(\mathbf{v}) = f\left(v_i \mathbf{b_V}^i\right) = v_i f\left(\mathbf{b_V}^i\right) = {f^i}_j v_i \mathbf{b_W}^j,}

in which b_V and b_W denote the bases of V and W, and v_i denotes the component of v on b_Vⁱ, 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \circ f(\mathbf{v}) = g\left({f^i}_j v_i \mathbf{b_W}^j\right) = {g^j}_k {f^i}_j v_i \mathbf{b_U}^k.}

Or in matrix form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \circ f(\mathbf{v}) = \mathbf{G} \mathbf{F} \mathbf{v},}

in which the i-row, j-column element of F, denoted by F_ij, is f^j_i, and G_ij=g^j_i.

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 with real-valued entries, Template:Tmath in Template:Tmath and Template:Tmath in Template:Tmath (the number of columns of Template:Tmath must match the number of rows of Template:Tmath), their product is a matrix Template:Tmath in Template:Tmath whose entries are given by a sum of pairwise products of the entries in the corresponding row of Template:Tmath and column of Template:Tmath:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ij} = \sum_{k=1}^r a_{ik} b_{kj} = a_{i1} b_{1j} + a_{i2} b_{2j} + \cdots + a_{ir} b_{rj}}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal U = \{u_1, \ldots, u_r\}} be a basis of U, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal V = \{v_1, \ldots, v_s\}} be a basis of V and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal W = \{w_1, \ldots, w_t\}} be a basis of W. In terms of this basis, let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = M^{\mathcal U}_{\mathcal V}(f) \in \R^{s\times r}} be the matrix representing f : U → V and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = M^{\mathcal V}_{\mathcal W}(g) \in \R^{r\times t}} be the matrix representing g : V → W. Then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B\cdot A = M^{\mathcal U}_{\mathcal W} (g \circ f) \in \R^{s\times t}}

is the matrix representing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \circ f : U \rightarrow W} .

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V \otimes W(v, m) = V(v) W(w), \forall v \in V^*, \forall w \in W^*,}

where V^* and W^* denote the dual spaces of V and W.^[4]

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 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.^[5]

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:

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^[1]
the Wick product of random variables
the cap, cup, Massey and slant product in algebraic topology
the smash product and wedge sum (sometimes called the wedge product) in homotopy

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:

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, 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.

Notes

↑ 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.

References

↑ ^1.0 ^1.1 Weisstein, Eric W. "Product". mathworld.wolfram.com. Retrieved 2020-08-16.
↑ "Summation and Product Notation". math.illinoisstate.edu. Retrieved 2020-08-16.
↑ Clarke, Francis (2013). Functional analysis, calculus of variations and optimal control. Dordrecht: Springer. pp. 9–10. ISBN 978-1447148203.
↑ Boothby, William M. (1986). An introduction to differentiable manifolds and Riemannian geometry (2nd ed.). Orlando: Academic Press. p. 200. ISBN 0080874398.
↑ Moschovakis, Yiannis (2006). Notes on set theory (2nd ed.). New York: Springer. p. 13. ISBN 0387316094.

Bibliography

Template:Jarchow Locally Convex Spaces

[3] 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.

[:0-1] 1.0 ^1.1 Weisstein, Eric W. "Product". mathworld.wolfram.com. Retrieved 2020-08-16.

[2] "Summation and Product Notation". math.illinoisstate.edu. Retrieved 2020-08-16.

[4] Clarke, Francis (2013). Functional analysis, calculus of variations and optimal control. Dordrecht: Springer. pp. 9–10. ISBN 978-1447148203.

[5] Boothby, William M. (1986). An introduction to differentiable manifolds and Riemannian geometry (2nd ed.). Orlando: Academic Press. p. 200. ISBN 0080874398.

[6] Moschovakis, Yiannis (2006). Notes on set theory (2nd ed.). New York: Springer. p. 13. ISBN 0387316094.

[1]

[2]

[lower-alpha 1]

[3]

[4]

[5]

Product (mathematics)

Contents

Product of two numbers

Product of a sequence

Commutative rings

Residue classes of integers

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

See also

Notes

References

Bibliography

Navigation menu

Product (mathematics)

Product of two numbers

Product of a sequence

Commutative rings

Residue classes of integers

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

See also

Notes

References

Bibliography

Navigation menu

Search