Canonical Form

he technique that is used to represent the mathematical entities or matrix in its standard form (or mathematical expression) is termed as canonical form. The term canonicalization is also known as standardization or normalization with respect to the equivalence relation.
Define an equivalence relation R on a set U with a map  such that for all , then the following conditions are hold:



The standard representation of canonical form is used in many fields like Linear algebra, graph theory, Operations Research, Computer science, and so on. The triangular form, Jordan canonical form and row echelon form are some major canonical forms in Linear Algebra.

Triangular canonical form :

Let  be a linear operator. Then, U has a basis B such that

This matrix representation is known as a triangular canonical form.

The characteristic polynomial of triangular canonical form is

Jordan Canonical Form:

Any matrix  can be represented as a Jordan canonical form by using a similarity transformation. That is,