Canonical form

Generally, in mathematics, a canonical form (often called normal form or standard form) of an object is a standard way of presenting that object.
Canonical form can also mean a differential form that is defined in a natural (canonical) way; see below.
Finding a canonical form is called canonization. In some branches of computer science the term canonicalization is adopted.
Contents
Definition
Suppose we have some set S of objects, with an equivalence relation. A canonical form is given by designating some objects of S to be "in canonical form", such that every object under consideration is equivalent to exactly one object in canonical form. In other words, the canonical forms in S represent the equivalence classes, once and only once. To test whether two objects are equivalent, it then suffices to test their canonical forms for equality. A canonical form thus provides a classification theorem and more, in that it not just classifies every class, but gives a distinguished (canonical) representative.
In practical terms, one wants to be able to recognize the canonical forms. There is also a practical, algorithmic question to consider: how to pass from a given object s in S to its canonical form s*? Canonical forms are generally used to make operating with equivalence classes more effective. For example in modular arithmetic, the canonical form for a residue class is usually taken as the least nonnegative integer in it. Operations on classes are carried out by combining these representatives and then reducing the result to its least nonnegative residue. The uniqueness requirement is sometimes relaxed, allowing the forms to be unique up to some finer equivalence relation, like allowing reordering of terms (if there is no natural ordering on terms).
A canonical form may simply be a convention, or a deep theorem.
For example, polynomials are conventionally written with the terms in descending powers: it is more usual to write x^{2} + x + 30 than x + 30 + x^{2}, although the two forms define the same polynomial. By contrast, the existence of Jordan canonical form for a matrix is a deep theorem.
Examples
Note: in this section, "up to" some equivalence relation E means that the canonical form is not unique in general, but that if one object has two different canonical forms, they are Eequivalent.
Linear algebra
Objects A is equivalent to B if: Normal form Notes Normal matrices over the complex numbers A = U ^{*} BU for some unitary matrix U Diagonal matrices (up to reordering) This is the Spectral theorem Matrices over the complex numbers A = UBV ^{*} for some unitary matrices U and V Diagonal matrices with real positive entries (in descending order) Singular value decomposition Matrices over an algebraically closed field A = P ^{− 1}BP for some invertible matrix P Jordan normal form (up to reordering of blocks) Matrices over a field A = P ^{− 1}BP for some invertible matrix P Frobenius normal form Matrices over a principal ideal domain A = P ^{− 1}BQ for some invertible Matrices P and Q Smith normal form The equivalence is the same as allowing invertible elementary row and column transformations Finitedimensional vector spaces over a field K A and B are isomorphic as vector spaces K^{n}, n a nonnegative integer Classical logic
 Negation normal form
 Conjunctive normal form
 Disjunctive normal form
 Algebraic normal form
 Canonical form (Boolean algebra)
 Prenex normal form
 Skolem normal form
Functional analysis
Objects A is equivalent to B if: Normal form Hilbert spaces A and B are isometrically isomorphic as Hilbert spaces sequence spaces (up to exchanging the index set I with another index set of the same cardinality) Commutative C ^{*} algebras with unit A and B are isomorphic as C ^{*} algebras The algebra C(X) of continuous functions on a compact Hausdorff space, up to homeomorphism of the base space. Algebra
Objects A is equivalent to B if: Normal form Finitely generated Rmodules with R a principal ideal domain A and B are isomorphic as Rmodules Primary decomposition (up to reordering) or invariant factor decomposition Geometry
 The equation of a line: Ax + By = C
 C = 0 or C = 1
 The equation of a circle:
By contrast, there are alternative forms for writing equations. For example, the equation of a line may be written as a linear equation in pointslope and slopeintercept form.
Mathematical notation
Standard form is used by many mathematicians and scientists to write extremely large numbers in a more concise and understandable way.
Set theory
 Cantor normal form of an ordinal number
Game theory
 Normal form game
Proof theory
Rewriting systems
 In an abstract rewriting system a normal form is an irreducible object.
Lambda calculus
 Beta normal form if no beta reduction is possible; Lambda calculus is a particular case of an abstract rewriting system.
Dynamical systems
 Normal form of a bifurcation
Graph theory
Main article: Graph canonizationDifferential forms
Canonical differential forms include the canonical oneform and canonical symplectic form, important in the study of Hamiltonian mechanics and symplectic manifolds.
See also
 Canonical class
 Normalization
 Standardization
References
 Shilov, Georgi E. (1977), Silverman, Richard A., ed., Linear Algebra, Dover, ISBN 048663518X.
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Canonical Form — [dt. kanonische Form], kanonisch … UniversalLexikon
Canonical form — canonic ca*non ic (k[.a]*n[o^]n [i^]k), canonical ca*non ic*al (k[.a]*n[o^]n [i^]*kal), a. [L. canonicus, LL. canonicalis, fr. L. canon: cf. F. canonique. See {canon}.] Of or pertaining to a canon; established by, or according to, a canon or… … The Collaborative International Dictionary of English
canonical form — noun : the simplest form of a matrix ; specifically : the form of a square matrix that has zero elements everywhere except along the principal diagonal * * * canonical form UK US noun [countable] [singular canonical form … Useful english dictionary
canonical form — UK / US noun [countable] Word forms canonical form : singular canonical form plural canonical forms linguistics the most basic or standard form of an expression … English dictionary
canonical form — kanoninis pavidalas statusas T sritis fizika atitikmenys: angl. canonical form vok. kanonische Form, f rus. каноническая форма, f pranc. forme canonique, f … Fizikos terminų žodynas
canonical form — kanoninė forma statusas T sritis automatika atitikmenys: angl. canonical form vok. Kanonische Darstellung, f rus. каноническая форма, f pranc. forme canonique, f … Automatikos terminų žodynas
canonical form — kanoninė struktūra statusas T sritis chemija apibrėžtis Struktūra, kurioje yra tik dvicentriai ryšiai. atitikmenys: angl. canonical form rus. каноническая структура … Chemijos terminų aiškinamasis žodynas
canonical form — kanoninė forma statusas T sritis informatika apibrėžtis Patogi naudojimui ir visuotinai priimta teiginio, formulės arba ↑reiškinio užrašymo forma. Dažniausiai naudojama matematikoje ir programavime. Pavyzdžiui, daugianario su vienu kintamuoju x… … Enciklopedinis kompiuterijos žodynas
canonical form — noun a) A standard or normal presentation of a mathematical entity b) Any of a set of representations of the resonance structure of a molecule each of which contributes to the real structure; a contributing structure Syn … Wiktionary
canonical form — noun Date: 1851 the simplest form of something; specifically the form of a square matrix that has zero elements everywhere except along the principal diagonal … New Collegiate Dictionary