Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Operator (mathematics)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
{{Short description|Function acting on function spaces}} {{About|operators in mathematics|other uses|Operator (disambiguation)}} {{distinguish|text=the symbol denoting a [[mathematical operation]] or [[mathematical symbol]]}} In [[mathematics]], an '''operator''' is generally a [[Map (mathematics)|mapping]] or [[function (mathematics)|function]] that acts on elements of a [[space (mathematics)|space]] to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an ''operator'', but the term is often used in place of ''function'' when the [[domain of a function|domain]] is a set of functions or other structured objects. Also, the domain of an operator is often difficult to characterize explicitly (for example in the case of an [[integral operator]]), and may be extended so as to act on related objects (an operator that acts on functions may act also on [[differential equation]]s whose solutions are functions that satisfy the equation). (see [[Operator (physics)]] for other examples) The most basic operators are [[linear map]]s, which act on [[vector space]]s. Linear operators refer to linear maps whose domain and range are the same space, for example from <math>\mathbb{R}^n </math> to <math>\mathbb{R}^n</math>.<ref name=Rudin-1973-Analysis> {{cite book | last=Rudin | first=Walter | year=1976 | chapter=Chapter 9: Functions of several variables | title=Principles of Mathematical Analysis | publisher=McGraw-Hill | edition=3rd | isbn=0-07-054235-X | page=207 | quote=Linear transformations of {{mvar|X}} into {{mvar|X}} are often called '''linear operators''' on {{mvar|X}} . }} </ref><ref name=Roman-2008-LinAlg> {{cite book | last=Roman | first=Steven | year=2008 | chapter=Chapter 2: Linear Transformations | title=Advanced Linear Algebra | edition=3rd | publisher=Springer | isbn=978-0-387-72828-5 | page=59 }} </ref>{{efn|: (1) A linear transformation from {{mvar|V}} to {{mvar|V}} is called a ''linear operator'' on {{mvar|V}}. The set of all linear operators on {{mvar|V}} is denoted {{math|''ℒ''(''V'')}} . A linear operator on a real vector space is called a ''real operator'' and a linear operator on a complex vector space is called a ''complex operator''. ... We should also mention that some authors use the term linear operator for any linear transformation from {{mvar|V}} to {{mvar|W}}. ... : ''Definition:'' The following terms are also employed: : (2) ''endomorphism'' for linear operator ... : (6) ''automorphism'' for bijective linear operator. ::::— Roman (2008)<ref name=Roman-2008-LinAlg/> }} Such operators often preserve properties, such as [[continuous function|continuity]]. For example, [[differentiation (mathematics)|differentiation]] and [[indefinite integration]] are linear operators; operators that are built from them are called [[differential operator]]s, [[integral operator]]s or integro-differential operators. '''Operator''' is also used for denoting the symbol of a [[mathematical operation]]. This is related with the meaning of "operator" in [[computer programming]] (see [[Operator (computer programming)]]). == Linear operators == {{Main|Linear operator}} The most common kind of operators encountered are ''linear operators''. Let {{mvar|U}} and {{mvar|V}} be [[vector space]]s over some [[field (mathematics)|field]] {{mvar|K}}. A [[map (mathematics)|''mapping'']] <math>\operatorname{A} : U \to V </math> is [[linear (mathematics)|''linear'']] if <math display="block">\operatorname{A}\left( \alpha \mathbf{x} + \beta \mathbf{y} \right) = \alpha \operatorname{A} \mathbf{x} + \beta \operatorname{A} \mathbf{y}\ </math> for all {{math|'''x'''}} and {{math|'''y'''}} in {{mvar|U}}, and for all {{math|''α'', ''β''}} in {{mvar|K}}. This means that a linear operator preserves vector space operations, in the sense that it does not matter whether you apply the linear operator before or after the operations of addition and scalar multiplication. In more technical words, linear operators are [[morphism]]s between vector spaces. In the finite-dimensional case linear operators can be represented by [[Matrix (mathematics)|matrices]] in the following way. Let {{mvar|K}} be a field, and <math>U</math> and {{mvar|V}} be finite-dimensional vector spaces over {{mvar|K}}. Let us select a basis <math>\ \mathbf{u}_1, \ldots, \mathbf{u}_n </math> in {{mvar|U}} and <math>\mathbf{v}_1, \ldots, \mathbf{v}_m </math> in {{mvar|V}}. Then let <math>\mathbf{x} = x^i \mathbf{u}_i</math> be an arbitrary vector in <math>U</math> (assuming [[Einstein convention]]), and <math>\operatorname{A}: U \to V </math> be a linear operator. Then<math display="block">\ \operatorname{A}\mathbf{x} = x^i \operatorname{A}\mathbf{u}_i = x^i \left( \operatorname{A}\mathbf{u}_i \right)^j \mathbf{v}_j ~.</math> Then <math>a_i^j \equiv \left( \operatorname{A}\mathbf{u}_i \right)^j </math>, with all <math>a_i^j\in K </math>, is the matrix form of the operator <math> \operatorname{A} </math> in the fixed basis <math>\{ \mathbf{u}_i \}_{i=1}^n</math>. The tensor <math>a_i^j </math> does not depend on the choice of <math>x</math>, and <math>\operatorname{A}\mathbf{x} = \mathbf{y} </math> if <math>a_i^j x^i = y^j</math>. Thus in fixed bases {{mvar|n}}-by-{{mvar|m}} matrices are in [[bijective]] correspondence to linear operators from <math>U </math> to <math>V</math>. The important concepts directly related to operators between finite-dimensional vector spaces are the ones of [[Matrix rank|rank]], [[determinant]], [[inverse operator]], and [[eigenspace]]. Linear operators also play a great role in the infinite-dimensional case. The concepts of rank and determinant cannot be extended to infinite-dimensional matrices. This is why very different techniques are employed when studying linear operators (and operators in general) in the infinite-dimensional case. The study of linear operators in the infinite-dimensional case is known as [[functional analysis]] (so called because various classes of functions form interesting examples of infinite-dimensional vector spaces). The space of [[sequence]]s of real numbers, or more generally sequences of vectors in any vector space, themselves form an infinite-dimensional vector space. The most important cases are sequences of real or complex numbers, and these spaces, together with linear subspaces, are known as [[sequence space]]s. Operators on these spaces are known as [[sequence transformation]]s. Bounded linear operators over a [[Banach space]] form a [[Banach algebra]] in respect to the standard operator norm. The theory of Banach algebras develops a very general concept of [[Spectrum (functional analysis)|spectra]] that elegantly generalizes the theory of eigenspaces. == Bounded operators== {{main|Bounded operator|Operator norm|Banach algebra}} Let {{mvar|U}} and {{mvar|V}} be two vector spaces over the same [[ordered field]] (for example; <math>\mathbb{R} </math>), and they are equipped with [[norm (mathematics)|norm]]s. Then a linear operator from {{mvar|U}} to {{mvar|V}} is called '''bounded''' if there exists {{math|''c'' > 0}} such that <math display="block">\|\operatorname{A}\mathbf{x}\|_V \leq c\ \|\mathbf{x}\|_U </math> for every '''{{math|x}}''' in {{mvar|U}}. Bounded operators form a vector space. On this vector space we can introduce a norm that is compatible with the norms of {{mvar|U}} and {{mvar|V}}: <math display="block">\|\operatorname{A}\| = \inf\{\ c : \|\operatorname{A}\mathbf{x}\|_V \leq c\ \|\mathbf{x}\|_U \}.</math> In case of operators from {{mvar|U}} to itself it can be shown that : <math display="inline">\|\operatorname{A}\operatorname{B}\| \leq \|\operatorname{A}\| \cdot \|\operatorname{B}\|</math>.{{efn| In this expression, the raised dot merely represents multiplication in whatever scalar field is used with {{mvar|V}} . }} Any unital [[normed algebra]] with this property is called a [[Banach algebra]]. It is possible to generalize [[spectral theory]] to such algebras. [[C*-algebra]]s, which are [[Banach algebras]] with some additional structure, play an important role in [[quantum mechanics]]. == Examples == === Analysis (calculus) === {{Main|Differential operator|Integral operator}} From the point of view of [[functional analysis]], [[calculus]] is the study of two linear operators: the [[differential operator]] <math>\frac{\ \mathrm{d}\ }{ \mathrm{d} t }</math>, and the ''[[Volterra operator]]'' <math>\int_0^t</math>. === Fundamental analysis operators on scalar and vector fields === {{Main|Vector calculus|Vector field|Scalar field|Gradient|Divergence|Curl (mathematics)|l6=Curl}} Three operators are key to [[vector calculus]]: * Grad ([[gradient]]), (with operator symbol [[del|<math>\nabla </math>]]) assigns a vector at every point in a scalar field that points in the direction of greatest rate of change of that field and whose norm measures the absolute value of that greatest rate of change. * Div ([[divergence]]), (with operator symbol [[del#Divergence|<math>{\nabla \cdot} </math>]]) is a vector operator that measures a vector field's divergence from or convergence towards a given point. * [[curl (mathematics)|Curl]], (with operator symbol [[del#Curl|<math>\nabla \!\times </math>]]) is a vector operator that measures a vector field's curling (winding around, rotating around) trend about a given point. As an extension of vector calculus operators to physics, engineering and tensor spaces, grad, div and curl operators also are often associated with [[tensor calculus]] as well as vector calculus.<ref name=Schey-2005>{{cite book |first=H.M. |last=Schey |year=2005 |title=Div, Grad, Curl, and All That |location=New York, NY |publisher=W.W. Norton |isbn=0-393-92516-1}}</ref> === Geometry === {{Main|General linear group|Isometry}} In [[geometry]], additional structures on [[vector space]]s are sometimes studied. Operators that map such vector spaces to themselves [[bijective]]ly are very useful in these studies, they naturally form [[group (mathematics)|group]]s by composition. For example, bijective operators preserving the structure of a vector space are precisely the [[invertible function|invertible]] [[linear operator]]s. They form the [[general linear group]] under composition. However, they ''do not'' form a vector space under operator addition; since, for example, both the identity and −identity are [[invertible]] (bijective), but their sum, 0, is not. Operators preserving the [[Euclidean metric]] on such a space form the [[isometry group]], and those that fix the origin form a subgroup known as the [[orthogonal group]]. Operators in the orthogonal group that also preserve the orientation of vector tuples form the [[special orthogonal group]], or the group of rotations. === Probability theory === {{Main|Probability theory}} Operators are also involved in probability theory, such as [[expected value|expectation]], [[variance]], and [[covariance]], which are used to name both number statistics and the operators which produce them. Indeed, every covariance is basically a [[dot product]]: Every variance is a dot product of a vector with itself, and thus is a [[quadratic norm]]; every standard deviation is a norm (square root of the quadratic norm); the corresponding cosine to this dot product is the [[Pearson correlation coefficient]]; expected value is basically an integral operator (used to measure weighted shapes in the space). ==== Fourier series and Fourier transform ==== {{Main|Fourier series|Fourier transform}} The Fourier transform is useful in applied mathematics, particularly physics and signal processing. It is another integral operator; it is useful mainly because it converts a function on one (temporal) domain to a function on another (frequency) domain, in a way effectively [[invertible function|invertible]]. No information is lost, as there is an inverse transform operator. In the simple case of [[periodic function]]s, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of [[sine wave]]s and cosine waves:<math display="block">f(t)=\frac{\ a_0\ }{2}+\sum_{n=1}^{\infty}\ a_n\cos(\omega\ n\ t) + b_n\sin(\omega\ n\ t) </math> The tuple {{math|( ''a''{{sub|0}}, ''a''{{sub|1}}, ''b''{{sub|1}}, ''a''{{sub|2}}, ''b''{{sub|2}}, ... )}} is in fact an element of an infinite-dimensional vector space [[Sequence space|{{math|''ℓ''{{i sup|2}} }}]], and thus Fourier series is a linear operator. When dealing with general function <math>\mathbb{R} \to \mathbb{C}</math>, the transform takes on an [[integral]] form: :<math display="block">f(t) = {1\over\sqrt{2\pi}}\int_{-\infty}^{+\infty}{g(\omega)\ e^{i\ \omega\ t}\ \mathrm{d}\ \omega} </math> ==== Laplace transform ==== {{Main|Laplace transform}} The ''Laplace transform'' is another integral operator and is involved in simplifying the process of solving differential equations. Given {{nobr|{{math|''f'' {{=}} ''f''(''s'')}}}}, it is defined by:<math display="block"> F(s)=\operatorname\mathcal{L}\{f\}(s)=\int_0^\infty e^{-s\ t}\ f(t)\ \mathrm{d}\ t </math> ==Footnotes== {{notelist}} == See also == * [[Function (mathematics)|Function]] * [[Operator algebra]] * [[List of operators]] == References == {{reflist|25em}} [[Category:Algebra]] [[Category:Functional analysis]] [[Category:Mathematical notation]]
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)
Pages transcluded onto the current version of this page
(
help
)
:
Template:About
(
edit
)
Template:Cite book
(
edit
)
Template:Distinguish
(
edit
)
Template:Efn
(
edit
)
Template:Main
(
edit
)
Template:Math
(
edit
)
Template:Mvar
(
edit
)
Template:Nobr
(
edit
)
Template:Notelist
(
edit
)
Template:Reflist
(
edit
)
Template:Short description
(
edit
)