CANONICAL FORM

Generally, in mathematics, a 'canonical form' (often called 'normal form') of an object is a standard presentation.
'Canonical form' can also mean a differential form that is defined in a natural (canonical) way; see below.

Contents

Definition

Examples

Classical logic

Game theory

Linear algebra

Proof theory

Lambda calculus

Dynamical systems

Differential forms

Definition

A canonical form is required to have two essential properties. Every object under consideration must have exactly one canonical form, and two objects that have the same canonical form must be the same up to some equivalence. The uniqueness requirement is sometimes relaxed, allowing the forms to be unique up to order 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, Jordan canonical form of a matrix is a deep theorem.
A canonical form solves a classification theorem, and is more data, in that it not only classifies every class, but gives a distinguished (canonical) representative.

Examples

Classical logic

★ Negation normal form

★ Conjunctive normal form

★ Disjunctive normal form

★ Algebraic normal form

★ Canonical form (Boolean algebra)

★ Prenex normal form

Game theory

★ Normal form game

Linear algebra

★ Jordan normal form

★ Frobenius normal form

★ Smith normal form

Proof theory

★ Normal form (natural deduction)

Lambda calculus

★ Beta normal form if no beta reduction is possible

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.