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.

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. A canonical form provides a classification theorem, and is more data, in that it not only classifies every class, but gives a distinguished (canonical) representative.

In practical terms, one wants to be able to recognise the canonical forms. There is also a practical, algorithmic question to consider: how to pass from a given 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 ${\displaystyle A=U^{*}BU}$ for some unitary U Diagonal matrices (up to reordering) This is the Spectral theorem
Matrices over the complex numbers ${\displaystyle A=U^{*}BV}$ for some unitary U and V Diagonal matrices with real positive entries (in descending order) Singular value decomposition
Matrices over an algebraically closed field ${\displaystyle A=P^{-1}BP}$ for some invertible Matrix P Jordan normal form (up to reordering of blocks)
Matrices over a field ${\displaystyle A=P^{-1}BP}$ for some invertible Matrix P Frobenius normal form
Matrices over a principal ideal domain ${\displaystyle A=P^{-1}BQ}$ 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 ${\displaystyle K^{n}}$, 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 ${\displaystyle \ell ^{2}(I)}$ sequence spaces (up to exchanging the index set I with another index set of the same cardinality)
Commutative ${\displaystyle C^{*}}$-algebras with unit A and B are isomorphic as ${\displaystyle C^{*}}$-algebras The algebra ${\displaystyle C(X)}$ 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

Differential forms

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