Canonical form

Formalism (deductive) | Algebra | Mathematical terminology

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 which 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. 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). 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. (Wikipedia).

Example of Rational Canonical Form 3

Matrix Theory: We note two formulations of Rational Canonical Form. A recipe is given for combining and decomposing companion matrices using cyclic vectors.

From playlist Matrix Theory

Example of Rational Canonical Form 2: Several Blocks

Matrix Theory: Let A be a 12x12 real matrix with characteristic polynomial (x^2+1)^6, minimal polynomial (x^2 + 1)^3, and dim(Null(A^2 + I)) = 6. Find all possible rational canonical forms for A.

From playlist Matrix Theory

Example of Rational Canonical Form 1: Single Block

Matrix Theory: Let A be the real matrix [0 -1 1 0 \ 1 0 0 1 \ 0 0 0 -1 \ 0 0 1 0]. Find a matrix P that puts A into rational canonical form over the real numbers. We compare RCF with Jordan canonical form and review companion matrices. (Minor corrections added.)

From playlist Matrix Theory

Overview of Jordan Canonical Form

Matrix Theory: We give an overview of the construction of Jordan canonical form for an nxn matrix A. The main step is the choice of basis that yields JCF. An example is given with two distinct eigenvalues.

From playlist Matrix Theory

F[x]-Module Derivation of Rational and Jordan Canonical Forms

Similar matrices isomorphism proof: https://youtu.be/-ligAAxFM8Y Every module is a direct sum of cyclic modules: https://youtu.be/gWIRI43h0ic Intro to F[x]-modules: https://youtu.be/H44q_Urmts0 The rational canonical form and Jordan normal form of a matrix are very important tools in li

From playlist Ring & Module Theory

Linear Programming, Lecture 9. Artificial Variables;

Sept 20, 2016. Penn State University.

From playlist Math484, Linear Programming, fall 2016

Example of Jordan Canonical Form: Real 4x4 Matrix with Basis 1

Matrix Theory: Find a matrix P that puts the real 4x4 matrix A = [2 0 0 0 \ 0 2 1 0 \ 0 0 2 0 \ 1 0 0 2 ] in Jordan Canonical Form. We show how to find a basis that gives P. The Jordan form has 2 Jordan blocks of size 2.

From playlist Matrix Theory

Example of Jordan Canonical Form: 2x2 Matrix

Matrix Theory: Find the Jordan form for the real 2 x 2 matrix A = [0 -4 \ 1 4]. For this matrix, there is no basis of eigenvectors, so it is not similar to a diagonal matrix. One alternative is to use Jordan canonical form.

From playlist Matrix Theory

Rational Canonical Form vs Smith Normal Form - March 5, 2021 - Rings and Modules

From playlist Course on Rings and Modules (Abstract Algebra 4) [Graduate Course]

8ECM Invited Lecture: Robert Berman

From playlist 8ECM Invited Lectures

R. Berman - Canonical metrics, random point processes and tropicalization

In this talk I will present a survey of the connections between canonical metrics and random point processes on a complex algebraic variety X. When the variety X has positive Kodaira dimension, this leads to a probabilistic construction of the canonical metric on X introduced by Tsuji and

From playlist Complex analytic and differential geometry - a conference in honor of Jean-Pierre Demailly - 6-9 juin 2017

p-adic automorphic forms in the sense of Scholze (Lecture 1) by Debargha Banerjee

PERFECTOID SPACES ORGANIZERS: Debargha Banerjee, Denis Benois, Chitrabhanu Chaudhuri, and Narasimha Kumar Cheraku DATE & TIME: 09 September 2019 to 20 September 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore Scientific committee: Jacques Tilouine (University of Paris, France) Eknath

From playlist Perfectoid Spaces 2019

Elliptic Curves - Lecture 25a - The canonical height (properties) and Z-linear independence

This video is part of a graduate course on elliptic curves that I taught at UConn in Spring 2021. The course is an introduction to the theory of elliptic curves. More information about the course can be found at the course website: https://alozano.clas.uconn.edu/math5020-elliptic-curves/

From playlist An Introduction to the Arithmetic of Elliptic Curves

Transfer Function to State Space

In this video we show how to transform a transfer function to an equivalent state space representation. We will derive various transformations such as controllable canonical form, modal canonical form, and controller canonical form. We will apply this to an example and show how to use Ma

From playlist Control Theory

Schemes 48: The canonical sheaf

This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. In this lecture we define the canonical sheaf, giev a survey of some applications (Riemann-Roch theorem, Serre duality, canonical embeddings, Kodaira dimensio

From playlist Algebraic geometry II: Schemes

Linear Programming, Lecture 5.Canonical form; basic feasible solution; geometric intepretation.

Sept 6, 2016. Penn State University.

From playlist Math484, Linear Programming, fall 2016

Jordan Canonical Form - March 8, 2021 - Rings and Modules

From playlist Course on Rings and Modules (Abstract Algebra 4) [Graduate Course]

Canonical forms for free group automorphisms - Jean Pierre Mutanguha

Arithmetic Groups Topic: Canonical forms for free group automorphisms Speaker: Jean Pierre Mutanguha Affiliation: Member, School of Mathematics Date: March 23, 2022 The Nielsen-Thurston theory of surface homeomorphism can be thought of as a surface analogue to the Jordan Canonical Form.

From playlist Mathematics

Carolina Araujo: - Fano Foliations 1 -Definition, examples and first properties

CIRM VIRTUAL EVENT Recorded during the research school "Geometry and Dynamics of Foliations " the May 11, 2020 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on C

From playlist Virtual Conference

Canonical form

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