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.

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 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 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 − 1BP for some invertible matrix P Jordan normal form (up to reordering of blocks)
Matrices over a field A = P − 1BP for some invertible matrix P Frobenius normal form
Matrices over a principal ideal domain A = P − 1BQ for some invertible Matrices P and Q Smith normal form 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 Kn, n a non-negative integer

Functional analysis

Objects A is equivalent to B if: Normal form
Hilbert spaces A and B are isometrically isomorphic as Hilbert spaces $\ell^2(I)$ 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 R-modules with R a principal ideal domain A and B are isomorphic as R-modules 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: $(x - h)^2 + (y - k)^2 = r^2\,$

By contrast, there are alternative forms for writing equations. For example, the equation of a line may be written as a linear equation in point-slope and slope-intercept form.

Mathematical notation

Standard form is used by many mathematicians and scientists to write extremely large numbers in a more concise and understandable way.

Game theory

• Normal form game

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

Differential forms

Canonical differential forms include the canonical one-form and canonical symplectic form, important in the study of Hamiltonian mechanics and symplectic manifolds.

References

• Shilov, Georgi E. (1977), Silverman, Richard A., ed., Linear Algebra, Dover, ISBN 0-486-63518-X .

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• Canonical Form —   [dt. kanonische Form], kanonisch …   Universal-Lexikon

• 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

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• 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