# [NEW] Jordan Canonical Form — from Wolfram MathWorld | jordan canonical form – Pickpeup

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# Jordan Canonical Form

The Jordan canonical form, also called the classical canonical form, of a special type of block matrix in which each block consists
of Jordan blocks with possibly differing constants
. In particular, it is a block
matrix
of the form

(1)

(Ayres 1962, p. 206).

A specific example is given by

(2)

which has three Jordan blocks. (Note that the degenerate case of a matrix is considered a Jordan
block
even though it lacks a superdiagonal
to be filled with 1s; cf. Strang 1988, p. 454).

Any complex matrix can be written
in Jordan canonical form by finding a Jordan basis for each Jordan
block
. In fact, any matrix with coefficients in an algebraically closed field
can be put into Jordan canonical form. The dimensions of the blocks corresponding
to the eigenvalue can be recovered
by the sequence

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(3)

The convention that the submatrices have 1s on the subdiagonal instead of the superdiagonal is also used sometimes

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Michael Jordan Shot Block/Steal versus Bulls

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## Example of Jordan Canonical Form: General Properties

Matrix Theory: A real 8×8 matrix A has minimal polynomial m(x) = (x2)^4, and the eigenspace for eigenvalue 2 has dimension 3. Find all possible Jordan Canonical Forms for A.

## R is uncountable

This is also one of my favorite proofs! In this video I not only prove that the rational numbers are countable (that is you can create an infinite list of rational numbers), but also that the real numbers are uncountable, meaning that even if you lived forever, you wouldn’t be able to count all the real number. At the same time I’m presenting a very classical proof method called the cantor diagonal argument, which is used a lot in analysis. Enjoy!

## State Space Representation: Jordan Canonical Form

In this lecture, we discuss development of Jordan canonical form for proper transfer function. Further, an example related to Jordan canonical form has been discussed.

## Advanced Linear Algebra, Lecture 4.7: Jordan canonical form

Advanced Linear Algebra, Lecture 4.7: Jordan canonical form
The spectral theorems says that if A:X→X is a linear map on a finitedimensional vector space over an algebraically closed field, then X has a basis of generalized eigenvectors, and we gave an explicit construction of this in the previous lecture. The matrix form of such a basis is the Jordan canonical form, which is a blockdiagonal matrix of \”Jordan blocks\”. After introducing this, we consider two commuting maps A and B. We show that X always has a basis of common generalized eigenvectors. In the case when A and B are diagonalizable, then A and B are simultaneously diagonalizable. In matrix form, this means that for some matrix P, whose columns are common eigenvectors to A and B, both P^{1}AP and P^{1}BP are diagonal.
Course webpage: http://www.math.clemson.edu/~macaule/math8530online.html

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