Matrix normal forms | Matrix decompositions | Linear algebra | Matrix theory
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to some basis. Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. Let V be a vector space over a field K. Then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is algebraically closed (for instance, if it is the field of complex numbers). The diagonal entries of the normal form are the eigenvalues (of the operator), and the number of times each eigenvalue occurs is called the algebraic multiplicity of the eigenvalue. If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size. The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form. The Jordan normal form is named after Camille Jordan, who first stated the Jordan decomposition theorem in 1870. (Wikipedia).
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From playlist Lineare Algebra
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
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
The Normal Distribution (1 of 3: Introductory definition)
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From playlist The Normal Distribution
Jordan Normal Form - Part 1 - Overview
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From playlist Linear algebra (English)
Example of Jordan Canonical Form: General Properties
Matrix Theory: A real 8x8 matrix A has minimal polynomial m(x) = (x-2)^4, and the eigenspace for eigenvalue 2 has dimension 3. Find all possible Jordan Canonical Forms for A.
From playlist Matrix Theory
Jordan Normal Form - Part 2 - An Example
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From playlist Linear algebra (English)
Systems of Differential Equations: Diagonalization and Jordan Canonical Form
It is only possible to perfectly diagonalize certain systems of linear differential equations. For the more general cases, it is possible to "block-diagonalize" the system into what is known as Jordan Canonical Form. This video explores these various options and derives the fully general
From playlist Engineering Math: Differential Equations and Dynamical Systems
A nice basis for a nilpotent operator. Jordan basis. Jordan form for an operator on a finite-dimensional complex vector space.
From playlist Linear Algebra Done Right
Pre-recorded lecture 7: Single Jordan block with non-constant eigenvalue and Complex normal forms
***Apologies, but the original files to some of these lectures are broken, and thus freeze part way through, however the lecture slides can be found here: https://mathematical-research-institute.sydney.edu.au/wp-content/uploads/2022/02/Lecture7_Nijenhuis.pdf*** MATRIX-SMRI Symposium: Nije
From playlist MATRIX-SMRI Symposium: Nijenhuis Geometry companion lectures (Sino-Russian Mathematical Centre)
Lecture 13 | Introduction to Linear Dynamical Systems
Professor Stephen Boyd, of the Electrical Engineering department at Stanford University, lectures on generalized eigenvectors, diagonalization, and Jordan canonical form for the course, Introduction to Linear Dynamical Systems (EE263). Introduction to applied linear algebra and linear d
From playlist Lecture Collection | Linear Dynamical Systems
Pre-recorded lecture 6: Constant normal forms, nilpotent Nijenhuis operators and Thompson theorem
MATRIX-SMRI Symposium: Nijenhuis Geometry and integrable systems Pre-recorded lecture: These lectures were recorded as part of a cooperation between the Chinese-Russian Mathematical Center (Beijing) and the Moscow Center of Fundamental and Applied Mathematics (Moscow). Nijenhuis Geomet
From playlist MATRIX-SMRI Symposium: Nijenhuis Geometry companion lectures (Sino-Russian Mathematical Centre)
Determinantal varieties and asymptotic expansion of Bergman kernels by Harald Upmeier
DISCUSSION MEETING ANALYTIC AND ALGEBRAIC GEOMETRY DATE:19 March 2018 to 24 March 2018 VENUE:Madhava Lecture Hall, ICTS, Bangalore. Complex analytic geometry is a very broad area of mathematics straddling differential geometry, algebraic geometry and analysis. Much of the interactions be
From playlist Analytic and Algebraic Geometry-2018
Hörsaalübung 6 - Lineare Algebra - PLR-Zerlegung, QR-Zerlegung, Schurzerlegung, Jordan-Normalform
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From playlist Lineare Algebra II - SoSe 2020
Pre-recorded lecture 15: gl-regular Nijenhuis operators (part 4)
MATRIX-SMRI Symposium: Nijenhuis Geometry and integrable systems Pre-recorded lecture: These lectures were recorded as part of a cooperation between the Chinese-Russian Mathematical Center (Beijing) and the Moscow Center of Fundamental and Applied Mathematics (Moscow). Nijenhuis Geomet
From playlist MATRIX-SMRI Symposium: Nijenhuis Geometry companion lectures (Sino-Russian Mathematical Centre)
Nijenhuis Geometry Chair's Talk 4 (Alexey Bolsinov)
SMRI -MATRIX Symposium: Nijenhuis Geometry and Integrable Systems Chair's Talk 4 (Alexey Bolsinov) 10 February 2022 ---------------------------------------------------------------------------------------------------------------------- SMRI-MATRIX Joint Symposium, 7 – 18 February 2022 Wee
From playlist MATRIX-SMRI Symposium: Nijenhuis Geometry and integrable systems
Example of Jordan Canonical Form: Real 4x4 Matrix with Basis 2
Matrix Theory: Find a matrix P that puts the following real 4x4 matrix A = [2 0 0 0 \ 0 2 0 0 \ 0 0 2 1 \ 1 0 0 2] into Jordan Canonical Form. Here the JCF has blocks of size 3 and 1. We focus on finding a vector that generates the 3x3 block.
From playlist Matrix Theory
Pre-recorded lecture 8: Differentially non-degenerate singular points and global theorems
MATRIX-SMRI Symposium: Nijenhuis Geometry and integrable systems Pre-recorded lecture: These lectures were recorded as part of a cooperation between the Chinese-Russian Mathematical Center (Beijing) and the Moscow Center of Fundamental and Applied Mathematics (Moscow). Nijenhuis Geomet
From playlist MATRIX-SMRI Symposium: Nijenhuis Geometry companion lectures (Sino-Russian Mathematical Centre)