Canonical form

In mathematics and computer science, a canonical, normal, or standard form is a special way to show a mathematical object as a mathematical expression. It helps make things simpler and easier to understand. Often, the canonical form is the easiest way to write something and makes sure each object looks the same every time.

For example, the canonical form of a positive integer in decimal representation is just a number without any extra zeros at the beginning. In more advanced math, there are special forms like the Jordan normal form for matrix similarity and the row echelon form for matrices.

In computer algebra, computers can show the same math object in many different ways. A canonical form makes sure each object has just one way to look, which helps computers check if two things are the same. However, choosing this special form can sometimes be tricky and depend on certain rules.

Definition

A canonical form is a special way to show an object so that it looks the same no matter how you start with it. Think of it like having a favorite way to write your name so everyone knows it's you, even if they see it written differently elsewhere.

In math and computer science, this helps us see if two things are really the same by checking if their canonical forms match. For example, in modular arithmetic, we pick the smallest non-negative number to represent a group of numbers that behave the same way. This makes calculations easier and clearer. Sometimes, choosing a canonical form is just a rule, like writing polynomials from highest to lowest power, but other times it comes from important math results.

History

The word "canonical" comes from an Ancient Greek word meaning "regular, according to rule." People have used it to mean "norm," "standard," or "archetype" in many subjects.

Mathematicians started using the term in the 1700s. By the mid-1800s, several writers, including Sylvester, Hesse, Hermite, Borchardt, and Cayley, were using words like "canonical form" or "normal form" to describe the simplest way to write mathematical expressions. One dictionary from 1865 defined canonical form as "a form, usually the simplest or most symmetrical, to which all functions of the same class can be reduced."

Examples

In math and computer science, a canonical form is a special way to write a number or object so it looks the same no matter who is looking at it. This helps people see if two things are really the same even if they look different at first.

Large number notation

Scientists often use a special way to write very big numbers so they are easier to read. This is called scientific notation.

Number theory

There are special ways to write whole numbers and fractions so they are easy to understand.

Linear algebra

Algebra

Geometry

In geometry, there are special ways to write the rules for lines and circles. For example, a line can be written in a certain way, and a circle can be written using a special formula.

Integrable systems

In advanced math, there are special ways to write rules for how things change over time.

Dynamical systems

This area also uses special ways to write rules for how things change.

Three dimensional geometry

In three-dimensional shapes, there are special ways to write down important rules about the shape.

Functional analysis

Classical logic

There are special ways to write logic problems so they are easier to solve.

Set theory

There is a special way to write certain numbers that show order.

Game theory

Games can be written in a special way to study how they work.

Proof theory

In studying how proofs work, there are special ways to write down the steps.

Rewriting systems

Sometimes, math problems can be rewritten into a special form that makes them easier to understand.

Lambda calculus

In computer science, there are special ways to write rules for small pieces of code.

Graph theory

In math, graphs (which look like dots connected by lines) can be written in a special way so it’s easy to see if two graphs are really the same.

Computing

In computers, data is often changed into a special form so it is easier to work with. This is called normalization. It helps make databases and other information organized and safe.

Objects	A is equivalent to B if:	Normal form
Normal matrices over the complex numbers	A = U ∗ B U {\displaystyle A=U^{*}BU} for some unitary matrix U	Diagonal matrices (up to reordering)
Matrices over the complex numbers	A = U B V ∗ {\displaystyle A=UBV^{*}} for some unitary matrices U and V	Diagonal matrices with real non-negative entries (in descending order)
Matrices over an algebraically closed field	A = P − 1 B P {\displaystyle A=P^{-1}BP} for some invertible matrix P	Jordan normal form (up to reordering of blocks)
Matrices over an algebraically closed field	A = P − 1 B P {\displaystyle A=P^{-1}BP} for some invertible matrix P	Weyr canonical form (up to reordering of blocks)
Matrices over a field	A = P − 1 B P {\displaystyle A=P^{-1}BP} for some invertible matrix P	Frobenius normal form
Matrices over a principal ideal domain	A = P − 1 B Q {\displaystyle A=P^{-1}BQ} for some invertible matrices P and Q	Smith normal form
Matrices over the integers	A = U B {\displaystyle A=UB} for some unimodular matrix U	Hermite normal form
Matrices over the integers modulo n		Howell normal form
Finite-dimensional vector spaces over a field K	A and B are isomorphic as vector spaces	K n {\displaystyle K^{n}} , n a non-negative integer

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