What is "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 canons. ``The oath of canonical obedience.'' --Hallam. 2. Appearing in a Biblical canon; as, a canonical book of the Christian New Testament. 3. Accepted as authoritative; recognized. 4. (Math.) In its standard form, usually also the simplest form; -- of an equation or coordinate. 5. (Linguistics) Reduced to the simplest and most significant form possible without loss of generality; as, a canonical syllable pattern. Opposite of nonstandard. Syn: standard. [WordNet

1. 5]

   6. Pertaining to or resembling a musical canon.

   Canonical books, or Canonical Scriptures, those books which are declared by the canons of the church to be of divine inspiration; -- called collectively the canon. The Roman Catholic Church holds as canonical several books which Protestants reject as apocryphal.

   Canonical epistles, an appellation given to the epistles called also general or catholic. See Catholic epistles, under Canholic.

   Canonical form (Math.), the simples or most symmetrical form to which all functions of the same class can be reduced without lose of generality.

   Canonical hours, certain stated times of the day, fixed by ecclesiastical laws, and appropriated to the offices of prayer and devotion; also, certain portions of the Breviary, to be used at stated hours of the day. In England, this name is also given to the hours from 8 a. m. to 3 p. m. (formerly 8 a. m. to 12 m.) before and after which marriage can not be legally performed in any parish church.

   Canonical letters, letters of several kinds, formerly given by a bishop to traveling clergymen or laymen, to show that they were entitled to receive the communion, and to distinguish them from heretics.

   Canonical life, the method or rule of living prescribed by the ancient clergy who lived in community; a course of living prescribed for the clergy, less rigid than the monastic, and more restrained that the secular.

   Canonical obedience, submission to the canons of a church, especially the submission of the inferior clergy to their bishops, and of other religious orders to their superiors.

   Canonical punishments, such as the church may inflict, as excommunication, degradation, penance, etc.

   Canonical sins (Anc. Church.), those for which capital punishment or public penance decreed by the canon was inflicted, as idolatry, murder, adultery, heresy.

n. 1 (context mathematics computing English) A standard or normal presentation of a mathematical entity 2 (context chemistry English) Any of a set of representations of the resonance structure of a molecule each of which contributes to the real structure; a contributing structure 3 (context linguistics rare English) dictionary form. a basic form of a word used as a dictionary entry.

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. The distinction between "canonical" and "normal" forms varies by 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, and 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. Thus, the equality of two objects can easily be tested by testing the equality of their canonical forms. However canonical forms frequently depend on arbitrary choices (like ordering the variables), and this introduces 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.

In computer science, data that has more than one possible representation can often be canonicalized into a completely unique representation called its canonical form. Putting something into canonical form is canonicalization.