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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Canonical form | 1/2 | https://en.wikipedia.org/wiki/Canonical_form | reference | science, encyclopedia | 2026-05-05T07:23:25.219824+00:00 | kb-cron |
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an object and allows it to be identified in a unique way. The distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a unique representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness. The canonical form of a positive integer in decimal representation is a finite sequence of digits that does not begin with zero. More generally, for a class of objects on which an equivalence relation is defined, a canonical form consists in the choice of a specific object in each class. For example:
Jordan normal form is a canonical form for matrix similarity. The row echelon form is a canonical form, when one considers as equivalent a matrix and its left product by an invertible matrix. In computer science, and more specifically in computer algebra, when representing mathematical objects in a computer, there are usually many different ways to represent the same object. In this context, a canonical form is a representation such that every object has a unique representation (with canonicalization being the process through which a representation is put into its canonical form). Thus, the equality of two objects can easily be tested by testing the equality of their canonical forms. Despite this advantage, canonical forms frequently depend on arbitrary choices (like ordering the variables), which introduce difficulties for testing the equality of two objects resulting on independent computations. Therefore, in computer algebra, normal form is a weaker notion: A normal form is a representation such that zero is uniquely represented. This allows testing for equality by putting the difference of two objects in normal form. Canonical form can also mean a differential form that is defined in a natural (canonical) way.
== Definition == Given a set S of objects with an equivalence relation R on S, 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 equality on their canonical forms. A canonical form thus provides a classification theorem and more, in that it not only classifies every class, but also gives a distinguished (canonical) representative for each object in the class. Formally, a canonicalization with respect to an equivalence relation R on a set S is a mapping c:S→S such that for all s, s1, s2 ∈ S:
c(s) = c(c(s)) (idempotence), s1 R s2 if and only if c(s1) = c(s2) (decisiveness), and s R c(s) (representativeness). Property 3 is redundant; it follows by applying 2 to 1. In practical terms, it is often advantageous 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, such as allowing for 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.
== History == According to OED and LSJ, the term canonical stems from the Ancient Greek word kanonikós (κανονικός, "regular, according to rule") from kanṓn (κᾰνών, "rod, rule"). The sense of norm, standard, or archetype has been used in many disciplines. Mathematical usage is attested in a 1738 letter from Logan. The German term kanonische Form is attested in a 1846 paper by Eisenstein, later the same year Richelot uses the term Normalform in a paper, and in 1851 Sylvester writes:
"I now proceed to [...] the mode of reducing Algebraical Functions to their simplest and most symmetrical, or as my admirable friend M. Hermite well proposes to call them, their Canonical forms." In the same period, usage is attested by Hesse ("Normalform"), Hermite ("forme canonique"), Borchardt ("forme canonique"), and Cayley ("canonical form"). In 1865, the Dictionary of Science, Literature and Art defines canonical form as:
"In Mathematics, denotes a form, usually the simplest or most symmetrical, to which, without loss of generality, all functions of the same class can be reduced."
== 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.
=== Large number notation === Standard form is used by many mathematicians and scientists to write extremely large numbers in a more concise and understandable way, the most prominent of which being the scientific notation.
=== Number theory === Canonical representation of a positive integer The canonical form of a continued fraction for representing a number is the simple continued fraction
=== Linear algebra ===
=== Algebra ===
=== Geometry === In analytic geometry: