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Product (mathematics)

From Wikipedia, the free encyclopedia
Mathematical form
Arithmetic operations
Addition (+)
term+termsummand+summandaddend+addendaugend+addend}={\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}sum{\displaystyle \scriptstyle {\text{sum}}}
Subtraction (−)
term−termminuend−subtrahend}={\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}difference{\displaystyle \scriptstyle {\text{difference}}}
Multiplication (×)
factor×factormultiplier×multiplicand}={\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}product{\displaystyle \scriptstyle {\text{product}}}
Division (÷)
dividenddivisornumeratordenominator}={\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}{fractionquotientratio{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}
Exponentiation
baseexponentbasepower}={\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}power{\displaystyle \scriptstyle {\text{power}}}
nth root (√)
radicanddegree={\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}root{\displaystyle \scriptstyle {\text{root}}}
Logarithm (log)
logbase⁡(anti-logarithm)={\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}logarithm{\displaystyle \scriptstyle {\text{logarithm}}}

Inmathematics, aproduct is the result ofmultiplication, or anexpression that identifiesobjects (numbers orvariables) to be multiplied, calledfactors. For example, 21 is the product of 3 and 7 (the result of multiplication), andx⋅(2+x){\displaystyle x\cdot (2+x)} is the product ofx{\displaystyle x} and(2+x){\displaystyle (2+x)} (indicating that the two factors should be multiplied together).When one factor is aninteger, the product is called amultiple.

The order in whichreal orcomplex numbers are multiplied has no bearing on the product; this is known as thecommutative law of multiplication. Whenmatrices or members of various otherassociative 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 differentalgebraic structures.

Product of two numbers

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Main article:Multiplication

Originally, a product was and is still the result of the multiplication of two or morenumbers. For example,15 is the product of3 and5. Thefundamental theorem of arithmetic states that everycomposite number is a product ofprime numbers, that is uniqueup to the order of the factors.

With the introduction ofmathematical notation andvariables at the end of the 15th century, it became common to consider the multiplication of numbers that are either unspecified (coefficients andparameters), or to be found (unknowns). These multiplications that cannot be effectively performed are calledproducts. For example, in thelinear equationax+b=0,{\displaystyle ax+b=0,} the termax{\displaystyle ax} denotes theproduct of the coefficienta{\displaystyle a} and the unknownx.{\displaystyle x.}

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

Product of a sequence

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See also:Multiplication § Product of a sequence

The product operator for theproduct of a sequence is denoted by the capital Greek letterpiΠ (in analogy to the use of the capital SigmaΣ assummation symbol).[1] For example, the expression∏i=16i2{\displaystyle \textstyle \prod _{i=1}^{6}i^{2}} is another way of writing⁠1⋅4⋅9⋅16⋅25⋅36{\displaystyle 1\cdot 4\cdot 9\cdot 16\cdot 25\cdot 36}⁠.[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 theempty product, and is equal to 1.

Commutative rings

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Commutative rings have a product operation.

Residue classes of integers

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Main article:Residue class

Residue classes in the ringsZ/NZ{\displaystyle \mathbb {Z} /N\mathbb {Z} } can be added:

(a+NZ)+(b+NZ)=a+b+NZ{\displaystyle (a+N\mathbb {Z} )+(b+N\mathbb {Z} )=a+b+N\mathbb {Z} }

and multiplied:

(a+NZ)⋅(b+NZ)=a⋅b+NZ{\displaystyle (a+N\mathbb {Z} )\cdot (b+N\mathbb {Z} )=a\cdot b+N\mathbb {Z} }

Convolution

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Main article:Convolution
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 theconvolution.

If

∫−∞∞|f(t)|dt<∞and∫−∞∞|g(t)|dt<∞,{\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

(f∗g)(t):=∫−∞∞f(τ)⋅g(t−τ)dτ{\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 theFourier transform, convolution becomes point-wise function multiplication.

Polynomial rings

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Main article:Polynomial ring

The product of two polynomials is given by the following:

(∑i=0naiXi)⋅(∑j=0mbjXj)=∑k=0n+mckXk{\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

ck=∑i+j=kai⋅bj{\displaystyle c_{k}=\sum _{i+j=k}a_{i}\cdot b_{j}}

Products in linear algebra

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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

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Main article:Scalar multiplication
Further information:Scaling (geometry)

By the very definition of a vector space, one can form the product of any scalar with any vector, giving a mapR×V→V{\displaystyle \mathbb {R} \times V\rightarrow V}.

Scalar product

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Main article:Scalar product

Ascalar product is a bi-linear map:

⋅:V×V→R{\displaystyle \cdot :V\times V\rightarrow \mathbb {R} }

with the following conditions, thatv⋅v>0{\displaystyle v\cdot v>0} for all0≠v∈V{\displaystyle 0\not =v\in V}.

From the scalar product, one can define anorm by letting‖v‖:=v⋅v{\displaystyle \|v\|:={\sqrt {v\cdot v}}}.

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

cos⁡∠(v,w)=v⋅w‖v‖⋅‖w‖{\displaystyle \cos \angle (v,w)={\frac {v\cdot w}{\|v\|\cdot \|w\|}}}

Inn{\displaystyle n}-dimensionalEuclidean space, the standard scalar product (called thedot product) is given by:

(∑i=1nαiei)⋅(∑i=1nβiei)=∑i=1nαiβi{\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

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Main article:Cross product

Thecross product of two vectors in 3-dimensions is a vector perpendicular to the two factors, with length equal to the area of theparallelogram spanned by the two factors.

The cross product can also be expressed as theformal[a]determinant:

u×v=|ijku1u2u3v1v2v3|{\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

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Main article:Function composition

A linear mapping can be defined as a functionf between two vector spacesV andW with underlying fieldF, satisfying[3]

f(t1x1+t2x2)=t1f(x1)+t2f(x2),∀x1,x2∈V,∀t1,t2∈F.{\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

f(v)=f(vibVi)=vif(bVi)=fijvibWj,{\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 whichbV andbW denote thebases ofV andW, andvi denotes thecomponent ofv onbVi, andEinstein summation convention is applied.

Now we consider the composition of two linear mappings between finite dimensional vector spaces. Let the linear mappingf mapV toW, and let the linear mappingg mapW toU. Then one can get

g∘f(v)=g(fijvibWj)=gjkfijvibUk.{\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:

g∘f(v)=GFv,{\displaystyle g\circ f(\mathbf {v} )=\mathbf {G} \mathbf {F} \mathbf {v} ,}

in which thei-row,j-column element ofF, denoted byFij, isfji, andGij=gji.

The composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication.

Product of two matrices

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Main article:Matrix product

Given twomatrices with real-valued entries,⁠A{\displaystyle A}⁠ in⁠Rs×r{\displaystyle \textstyle \mathbb {R} ^{s\times r}}⁠ and⁠B{\displaystyle B}⁠ in⁠Rr×t{\displaystyle \textstyle \mathbb {R} ^{r\times t}}⁠ (the number of columns of⁠A{\displaystyle A}⁠ must match the number of rows of⁠B{\displaystyle B}⁠), their product is a matrix⁠C=AB{\displaystyle C=AB}⁠ in⁠Rs×t{\displaystyle \textstyle \mathbb {R} ^{s\times t}}⁠ whose entries are given by a sum of pairwise products of the entries in the corresponding row of⁠A{\displaystyle A}⁠ and column of⁠B{\displaystyle B}⁠:

cij=∑k=1raikbkj=ai1b1j+ai2b2j+⋯+airbrj{\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

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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. LetU={u1,…,ur}{\displaystyle {\mathcal {U}}=\{u_{1},\ldots ,u_{r}\}} be abasis of U,V={v1,…,vs}{\displaystyle {\mathcal {V}}=\{v_{1},\ldots ,v_{s}\}} be a basis of V andW={w1,…,wt}{\displaystyle {\mathcal {W}}=\{w_{1},\ldots ,w_{t}\}} be a basis of W. In terms of this basis, letA=MVU(f)∈Rs×r{\displaystyle A=M_{\mathcal {V}}^{\mathcal {U}}(f)\in \mathbb {R} ^{s\times r}}be the matrix representing f : U → V andB=MWV(g)∈Rr×t{\displaystyle B=M_{\mathcal {W}}^{\mathcal {V}}(g)\in \mathbb {R} ^{r\times t}} be the matrix representing g : V → W. Then

B⋅A=MWU(g∘f)∈Rs×t{\displaystyle B\cdot A=M_{\mathcal {W}}^{\mathcal {U}}(g\circ f)\in \mathbb {R} ^{s\times t}}

is the matrix representingg∘f:U→W{\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

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Main article:Tensor product

Given two finite dimensional vector spacesV andW, the tensor product of them can be defined as a (2,0)-tensor satisfying:

V⊗W(v,m)=V(v)W(w),∀v∈V∗,∀w∈W∗,{\displaystyle V\otimes W(v,m)=V(v)W(w),\forall v\in V^{*},\forall w\in W^{*},}

whereV* andW* denote thedual spaces ofV andW.[4]

For infinite-dimensional vector spaces, one also has the:

The tensor product,outer product andKronecker 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 itsintrinsic definition. The outer product is simply the Kronecker product, limited to vectors (instead of matrices).

The class of all objects with a tensor product

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In general, whenever one has two mathematicalobjects that can be combined in a way that behaves like a linear algebra tensor product, then this can be most generally understood as theinternal product of amonoidal 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 theclass of all things (of a giventype) that have a tensor product.

Other products in linear algebra

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Other kinds of products in linear algebra include:

Cartesian product

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Inset theory, aCartesian product is amathematical operation which returns aset (orproduct set) from multiple sets. That is, for setsA andB, the Cartesian productA ×B is the set of allordered pairs(a, b)—wherea ∈A andb ∈B.[5]

The class of all things (of a giventype) that have Cartesian products is called aCartesian category. Many of these areCartesian closed categories. Sets are an example of such objects.

Empty product

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Theempty product on numbers and mostalgebraic structures has the value of 1 (theidentity element of multiplication), just like theempty 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 inlogic,set theory,computer programming andcategory theory.

Products over other algebraic structures

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Products over other kinds ofalgebraic structures include:

A few of the above products are examples of the general notion of aninternal product in amonoidal category; the rest are describable by the general notion of aproduct in category theory.

Products in category theory

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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, seeproduct (category theory), which describes how to combine twoobjects of some kind to create an object, possibly of a different kind. But also, in category theory, one has:

Other products

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  • A function'sproduct 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

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Notes

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  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.

References

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  1. ^abWeisstein, Eric W."Product".mathworld.wolfram.com. Retrieved2020-08-16.
  2. ^"Summation and Product Notation".math.illinoisstate.edu. Retrieved2020-08-16.
  3. ^Clarke, Francis (2013).Functional analysis, calculus of variations and optimal control. Dordrecht: Springer. pp. 9–10.ISBN 978-1447148203.
  4. ^Boothby, William M. (1986).An introduction to differentiable manifolds and Riemannian geometry (2nd ed.). Orlando: Academic Press. p. 200.ISBN 0080874398.
  5. ^Moschovakis, Yiannis (2006).Notes on set theory (2nd ed.). New York: Springer. p. 13.ISBN 0387316094.

Bibliography

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