# canonical

Canonical is an adjective derived from . Canon comes from the Greek word kanon "rule" (perhaps originally from kanna "reed", cognate to cane) is used in various meanings.

basic, canonic, canonical: reduced to the simplest and most significant form possible without loss of generality, e.g. "a basic story line"; "a canonical syllable pattern"

## Religion

This word is used by theologians and canon lawyers to refer to the canons of the Roman Catholic, Eastern Orthodox and Anglican Churches adopted by ecumenical councils. It also refers to later law developed by local churches and dioceses of these churches. The function of this collection of various "canons" is somewhat analogous to the precedents established in common law by case law.

In the 20th century, the Roman Catholic Church revised its canon law in 1917 and then again 1981 into the modern Code of Canon Law. This code is no longer merely a compilation of papal decrees and conciliar legislation, but a more completely developed body of international church law. It is analogous to the English system of Statute law.

Canonical can also mean "part of the canon", i.e., one of the books comprising the biblical canon, as opposed to apocryphal books. Canonization is the process by which a person is recognized as a saint.

## Literature and art

It is used most often when describing bodies of literature or art: those books that all educated people have read make up the "canon", for example the Western canon. (See also canon (fiction)).

## Mathematics

Mathematicians have for perhaps a century or more used the word canonical to refer to concepts that have a kind of uniqueness or naturalness, and are (up to trivial aspects) "independent of coordinates." Examples include the canonical prime factorization of positive integers, the Jordan canonical form of matrices (which is built out of the irreducible factors of the characteristic polynomial of the matrix), and the canonical decomposition of a permutation into a product of disjoint cycles. Various functions in mathematics are also canonical, like the canonical homomorphism of a group onto any of its quotient groups, or the canonical isomorphism between a finite-dimensional vector space and its double dual. Although a finite-dimensional vector space and its dual space are isomorphic, there is no canonical isomorphism. This lack of a canonical isomorphism can be made precise in terms of category theory, but one could say at a simpler level that "any isomorphism you can think of here depends on choosing a basis." As stated by Goguen, "To any canonical construction from one species of structure to another corresponds an adjunction between the corresponding categories." [1]

Being canonical in mathematics is stronger than being a conventional choice. For instance, the vector space Rn has a standard basis which is canonical in the sense that it is not just a choice which makes certain calculations easy; in fact most linear operators on Euclidean space take on a simpler form when written as a matrix relative to some basis other than the standard one (see Jordan form). In contrast, an abstract n-dimensional real vector space V would not have a canonical basis; it is isomorphic to Rn of course, but the choice of isomorphism is not canonical.

The word canonical is also used for a preferred way of writing something, see the main article canonical form.

## Computer science

Some circles in the field of computer science have borrowed this usage from mathematicians. It has come to mean "the usual or standard state or manner of something"; for example, "the canonical way to organize a file system is as a hierarchy, with extensions to make it a directed graph". XML Signature defines canonicalization as the process of converting XML content to a canonical form, to take into account changes that can invalidate a signature over that data (from JWSDP 1.6).

For an illuminating story about the word's use among computer scientists, see the Jargon File's entry for the word[1].

Some people have been known to use the noun canonicality; others use canonicity. In fields other than computer science, canonicity is this word's canonical form.

## Physics

In theoretical physics, the concept of canonical (or conjugate, or canonically conjugate) variables is of major importance. They always occur in complementary pairs, such as spatial location x and linear momentum p, angle φ and angular momentum L, and energy E and time t. They can be defined as any coordinates whose Poisson brackets give a Kronecker delta (or a Dirac delta in the case of continuous variables). The existence of such coordinates is guaranteed under broad circumstances as a consequence of Darboux's theorem. Canonical variables are essential in the Hamiltonian formulation of physics, which is particularly important in quantum mechanics. For instance, the SchrÃ¶dinger equation and the Heisenberg uncertainty relation always incorporate canonical variables. Canonical variables in physics are based on the aforementioned mathematical structure and therefore bear a deeper meaning than being just convenient variables. One facet of this underlying structure is expressed by Noether's theorem, which states that a (continuous) symmetry in a variable implies an invariance of the conjugate variable, and vice versa; for instance symmetry under spatial displacement leads to conservation of momentum, and time-independence implies energy conservation.

In statistical mechanics, the canonical ensemble, the grand canonical ensemble, and the microcanonical ensemble are archetypal probability distributions for the (unknown) microscopic state of a thermal system, applying respectively in the physical cases of:- a closed system at fixed temperature (able to exchange energy with its environment); an open system at fixed temperature (able to exchange both energy and particles); and a closed thermally isolated system (able to exchange neither). These probability distributions can be applied directly to practical problems in thermodynamics.

## References

1. ^ Goguen J. "A categorical manifesto". Math. Struct. Comp. Sci., 1(1):49--67, 1991
In grammar, an adjective is a word whose main syntactic role is to modify a noun or pronoun (called the adjective's subject), giving more information about what the noun or pronoun refers to.

A cane is a long, straight wooden stick, generally of bamboo, Malacca (rattan) or some similar plant, mainly used as a support, such as a walking stick, or as an instrument of punishment.
Christianity

Foundations
Jesus Christ
Church Theology
New Covenant Supersessionism
Dispensationalism
Apostles Kingdom Gospel
History of Christianity Timeline
Bible
Old Testament New Testament
Books Canon Apocrypha
Christianity

Foundations
Jesus Christ
Church Theology
New Covenant Supersessionism
Dispensationalism
Apostles Kingdom Gospel
History of Christianity Timeline
Bible
Old Testament New Testament
Books Canon Apocrypha
Christianity

Foundations
Jesus Christ
Church Theology
New Covenant Supersessionism
Dispensationalism
Apostles Kingdom Gospel
History of Christianity Timeline
Bible
Old Testament New Testament
Books Canon Apocrypha
In common law legal systems, the law is created and/or refined by judges: a decision in the case currently pending depends on decisions in previous cases and affects the law to be applied in future cases.
Case law (also known as decisional law) is that body of reported judicial opinions in countries that have common law legal systems that are published and thereby become precedent, i.e. the basis for future decisions).
Canon Law, the ecclesiastical law of the Catholic Church, is a fully developed legal system, with all the necessary elements: courts, lawyers, judges, a fully articulated legal code and principles of legal interpretation. The academic degrees in canon law are the J.C.B.
A statute is a formal, written law of a country or state, written and enacted by its legislative authority, perhaps to then be ratified by the highest executive in the government, and finally published. Typically, statutes command, prohibit, or declare policy.
A biblical canon is a list of Biblical books which establishes the set of books which are considered to be authoritative as scripture by a particular Jewish or Christian community.
Apocrypha (from the Greek word ἀπόκρυφα, meaning "those having been hidden away"[1]) are texts of uncertain authenticity or writings where the authorship is questioned.
saint is one who is sanctified (cf. 2 Chron. 6:41). The early Christians were all called saints. (Heb. 13:24; Jud. 1:3; Phile. 1:5, 7) Over time, the traditional usage of the term saint
Western canon is a term used to denote a of books, and, more widely, music and art, that has been the most influential in shaping Western culture. It asserts a compendium of the greatest Work of art of artistic merit.
Canon, in the context of a fictional universe, comprises those novels, stories, films, etc., that are considered to be genuine or officially sanctioned, and those events, characters, settings, etc., that are considered to have existence within the fictional universe.
In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e.
In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid in about 300 BC.
factorization (British English: also factorisation) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original.
The integers (from the Latin integer, which means with untouched integrity, whole, entire) are the set of numbers including the whole numbers (0, 1, 2, 3, …) and their negatives (0, −1, −2, −3, …).
In linear algebra, Jordan normal form (often called Jordan canonical form)[1] 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.
matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied.
characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace.

## Motivation

Given a square matrix , we want to find a polynomial whose roots are precisely the eigenvalues of .
Permutation is the rearrangement of objects or symbols into distinguishable sequences. Each unique ordering is called a permutation.
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces).
group is a set with a binary operation that satisfies certain axioms, detailed below. For example, the set of integers with addition is a group. The branch of mathematics which studies groups is called group theory.
In mathematics, an isomorphism (Greek: isos "equal", and morphe "shape") is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings.
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are