Canonical form

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.^[1]

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).^[2] 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.

^ In some occasions, the term "canonical" and "normal" can also be used interchangeably, as in Jordan canonical form and Jordan normal form (see Jordan normal form on MathWorks).
^ The term 'canonization' is sometimes incorrectly used for this.

[1] In some occasions, the term "canonical" and "normal" can also be used interchangeably, as in Jordan canonical form and Jordan normal form (see Jordan normal form on MathWorks).

[2] The term 'canonization' is sometimes incorrectly used for this.

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