jordan canonical form: คุณกำลังดูกระทู้

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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

(3)

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

(Faddeeva 1958, p. 50).

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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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