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Generally, in mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, 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
Differential form

In the mathematics fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates....
 that is defined in a natural (canonical) way; see below.

ose we have some set S of objects, with an equivalence relation
Equivalence relation

In mathematics, an equivalence relation is, loosely, a binary relation on a Set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets....
. 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.






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Generally, in mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, 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
Differential form

In the mathematics fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates....
 that is defined in a natural (canonical) way; see below.

Definition

Suppose we have some set S of objects, with an equivalence relation
Equivalence relation

In mathematics, an equivalence relation is, loosely, a binary relation on a Set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets....
. 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
Classification theorem

In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class....
 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 recognise 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
Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value — the modulus....
, the canonical form for a residue class is usually taken as the least non-negative integer in it. Operations on classes are carried out by combining these representatives and then reducing the result to its least non-negative 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 x2 + x + 30 than x + 30 + x2, 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 E-equivalent.

Linear algebra

Objects A is equivalent to B if: Normal form Notes
Normal
Normal matrix

A complex number Matrix #Square_matrices_and_related_definitions matrix A is a normal matrix ifwhere A* is the conjugate transpose of A....
 matrices over the complex numbers		 for some unitary
Unitary

Unitary may refer to:* In automotive design, unitary construction is another common term for a unibody or monocoque construction* In Christian doctrine, unitarianism is the belief in a "unitary God" as opposed to the concept of the Trinity....
 U		 Diagonal matrices (up to reordering) This is the Spectral theorem
Spectral theorem

In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrix_....
Matrices over the complex numbers for some unitary
Unitary

Unitary may refer to:* In automotive design, unitary construction is another common term for a unibody or monocoque construction* In Christian doctrine, unitarianism is the belief in a "unitary God" as opposed to the concept of the Trinity....
 U and V		 Diagonal matrices with real positive entries (in descending order) Singular value decomposition
Singular value decomposition

In linear algebra, the singular value decomposition is an important Matrix decomposition of a rectangular real number or complex number matrix , with several applications in signal processing and statistics....
Matrices over an algebraically closed field
Algebraically closed field

In mathematics, a field F is said to be algebraically closed if every polynomial in one variable of degree at least 1, with coefficients in F, has a root in F....
 for some invertible Matrix P Jordan normal form
Jordan normal form

In linear algebra, Jordan normal form shows that a given square matrix M over a field K containing the eigenvalues of M can be transformed into a certain normal form by changing the Basis ....
 (up to reordering of blocks)
Matrices over a field for some invertible Matrix P Frobenius normal form
Frobenius normal form

In linear algebra, the Frobenius normal form, Turner binormal projective form or rational canonical form of a square matrix A is a canonical form for Matrix that reflects the structure of the minimal polynomial of A and provides a means of detecting whether another matrix B is similar matrix to A without field ex...
Matrices over a principal ideal domain
Principal ideal domain

In abstract algebra, a principal ideal domain, or PID is an integral domain in which every ideal is principal ideal, i.e., can be generated by a single element....
 for some invertible Matrices P and Q Smith normal form
Smith normal form

The Smith normal form is a normal form that can be defined for any matrix with entries in a principal ideal domain . The Smith normal form of a matrix is Diagonal matrix, and can be obtained from the original matrix by multiplying on the left and right by invertible square matrices....
 The equivalence is the same as allowing invertible elementary row and column transformations
Finite-dimensional vector spaces over a field K A and B are isomorphic as vector spaces , n a non-negative integer


Classical logic

  • Negation normal form
    Negation normal form

    A logical formula is in negation normal form if negation occurs only immediately above elementary propositions, and are the only allowed Boolean connectives....
  • Conjunctive normal form
    Conjunctive normal form

    In boolean logic, a formula is in conjunctive normal form if it is a logical conjunction of clause , where a clause is a logical disjunction of literal s....
  • Disjunctive normal form
    Disjunctive normal form

    In boolean logic, a disjunctive normal form is a standardization of a logical formula which is a disjunction of conjunctive clause . As a normal form, it is useful in automated theorem proving....
  • Algebraic normal form
    Algebraic normal form

    In Boolean logic, the algebraic normal form is a method of standardizing and normalizing logical formulas. As a normal form, it can be used in automated theorem proving , but is more commonly used in the design of cryptography random number generators, specifically linear feedback shift registers ....
  • Canonical form (Boolean algebra)
    Canonical form (Boolean algebra)

    In Boolean algebra , any Boolean function can be expressed in a canonical form using the dual concepts of minterms and maxterms. Minterms are called products because they are the AND of a set of variables, and maxterms are called sums because they are the OR of a set of variables ....
  • Prenex normal form
    Prenex normal form

    A formula of the predicate calculus is in prenex normal form if it is written as a string of quantifiers followed by a quantifier-free part ....
  • Skolem normal form
    Skolem normal form

    A formula of first-order logic is in Skolem normal form if it is in conjunctive prenex normal form with only Universal quantification. Every first-order formula can be converted into Skolem normal form while not changing its satisfiability via a process called Skolemization....


Functional analysis

Objects A is equivalent to B if: Normal form
Hilbert spaces A and B are isometrically isomorphic as Hilbert spaces sequence spaces
Hilbert space

The mathematics concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces....
 (up to exchanging the index set I with another index set of the same cardinality
Cardinality

In mathematics, the cardinality of a set is a measure of the "number of Element of the set". For example, the set A = contains 3 elements, and therefore A has a cardinality of 3....
)
Commutative -algebras with unit A and B are isomorphic as -algebras The algebra of continuous functions on a compact Hausdorff space
Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhood ....
, up to homeomorphism
Homeomorphism

In the mathematics field of topology, a homeomorphism or topological isomorphism is a continuous function between two topological spaces....
 of the base space.


Algebra

Objects A is equivalent to B if: Normal form
Finitely generated R-modules with R a principal ideal domain
Principal ideal domain

In abstract algebra, a principal ideal domain, or PID is an integral domain in which every ideal is principal ideal, i.e., can be generated by a single element....
 A and B are isomorphic as R-modules Primary decomposition (up to reordering) or invariant factor decomposition
Structure theorem for finitely generated modules over a principal ideal domain

In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules can be uniquely decomposed in much the same way that integers have a prime f...


Geometry

  • The equation of a line:
    • 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
Linear equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.Linear equations can have one or more variables....
 in point-slope and slope-intercept form.

Standard form is used by many mathematicians and scientists to write extremely large numbers
Large numbers

Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions....
 in a more concise and understandable way.

Set theory

  • Cantor normal form of an ordinal number
    Ordinal number

    In set theory, an ordinal number, or just ordinal, is the order type of a well-order. They are usually identified with hereditarily transitive sets....


Game theory

  • Normal form game
    Normal form game

    In game theory, normal form is a way of describing a game. Unlike extensive-form game, normal-form representations are not graphical per se, but rather represent the game by way of a matrix ....


Proof theory

  • Normal form (natural deduction)
    Normal form (natural deduction)

    An inference of natural deduction is a normal form, according to Dag Prawitz, if no formula occurrence is both the principal premise of an elimination rule and the conclusion of an introduction rule....


Lambda calculus

  • Beta normal form
    Beta normal form

    In the lambda calculus, a term is in beta normal form if no lambda calculus#?-reduction is possible. A term is in beta-eta normal form if neither a beta reduction nor an lambda calculus#?-conversion is possible....
     if no beta reduction is possible


Dynamical systems

  • Normal form of a bifurcation


Differential forms

Canonical differential form
Differential form

In the mathematics fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates....
s include the canonical one-form and canonical symplectic form, important in the study of Hamiltonian mechanics
Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without recourse to Lagrangian mechanics using sym...
 and symplectic manifold
Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a Closed and exact differential forms, nondegenerate form, differential form, ?, called the symplectic form....
s.

See also

  • Canonical class