LA Ch 10 Lab -SGLee

Chapter 10

Jordan Canonical Form

10.1 Finding the Jordan Canonical Form with a Dot Diagram

*10.2 Jordan Canonical Form and Generalized Eigenvectors

10.3 Jordan Canonical Form and CAS

10.4 Exercises

If a matrix is diagonalizable, every thing is much easier. But most of matrices are not diagonalizable. The Jordan canonical form or Jordan normal form is an upper triangular matrix of a particular form called a Jordan matrix (a simple block diagonal matrix) representing an operator with respect to some basis. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs given by its algebraic multiplicity.

Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. Since each matrix has a corresponding Jordan canonical form which is similar to it, all computations can be done with this simple upper triangular matrix. The Jordan normal form is named after Camille Jordan, a French mathematician renowned for his work in various branches of mathematics. In this chapter, we will study how to find a Jordan matrix which is similar to any given matrix and how to find generalized eigenvectors.

 Theorem10.1.2 The number of dots in the first  rows of the dot diagram for  is equal to the dimension of solution space of  (i.e. the nullity of ).

nullity=nullity .

 Theorem10.1.3 For , let  denote the number of dots in the th row of the dot diagram of .  Then, the following are true.      (1)    .    (2)    If , .

By Theorem 10.1.2,
(provided )

Also,  and

(The number of dots in each row,, means the number of blocks of  size at least ).

From Theorem 10.1.3, let’s see how the dot diagram for each  is completely determined by the matrix .

Find the Jordan Canonical Form of .

.

(x - 3) * (x - 2)^3

[3, 2, 2, 2]

The matrix  has characteristic polynomial , so  has two distinct eigenvalues

Here  has algebraic multiplicity 1, and  has algebraic multiplicity 3. Thus, the dot diagram for  has 1 dot

and  has one  Jordan block. That is, .

As well, the dot diagram for  has 3 dots, and

2

1

,
.

Thus, the dot diagram for  is the following.

:        •   (number of Jordan blocks: 2)

:

has one  Jordan block and one  Jordan block. That is,

Hence, the Jordan Canonical form of  is

[3|0 0|0]

[-+--+-]

[0|2 1|0]

[0|0 2|0]

[-+--+-]

[0|0 0|2]                                 ■

Find the Jordan Canonical Form of .

The matrix  has characteristic polynomial ,

so there are two distinct eigenvalues of   and , each with algebraic multiplicity 2.  For ,

.

Therefore, the dot diagram for  is the following.

:  •    (Number of Jordan blocks: )

:  •

So, .

For  is 0(∵The number of dots  is ). Therefore, the dot diagram for   is the following:

:  • •  (Number of Jordan blocks = 2)

So,

Thus, the Jordan Canonical Form of  is

.

[4 1|0|0]

[0 4|0|0]

[--+-+-]

[0 0|2|0]

[--+-+-]

[0 0|0|2]                                 ■

[Remark]

* Jordan Canonical Form Learning Materials

10.2 *Jordan Canonical Form and Generalized Eigenvectors

Reference video: https://youtu.be/YrRnCByzxNM      https://youtu.be/yJ7n0icjtNA

In Section 10.1, we learned the theory and method for finding a matrix ,

called the Jordan Canonical form, such that .

In this section, we will examine a method for finding the matrix  in the above equation.

This method utilizes the concept of generalized eigenvectors.

The following matrix was referred from the wiki

Let .

Consider the matrix . The Jordan normal form is obtained

by some similarity transformation  , i.e. .

Let  have column vectors , then

.

We see that

.

For , we have ,

i.e.  is an eigenvector of  corresponding to the eigenvalue .

For , multiplying both sides by  gives .

But , so . Thus, .

Vectors like   are called generalized eigenvectors of .

Definition [Generalized Eigenvectors]

Definition [Generalized Eigenvectors]

A non zero vector,  is called a generalized eigenvector of a matrix  corresponding to an eigenvalue , if there exists a positive integer  such that . The smallest such is called the order of the generalized eigenvector .

Thus, given an eigenvalue , its corresponding Jordan block gives rise to a Jordan chain.

A Jordan chain of length  corresponding to an eigenvalue  is a sequence of non zero vectors,  such that

.

The generator, or lead vector (say, ) of the chain is a generalized eigenvector such that , where  is the size of the Jordan block.

The vector  is an eigenvector corresponding to . In general,  is the pre-image of  under .

So the lead vector generates the chain via multiplication by .

Therefore, the statement that every square matrix  can be put in Jordan normal form is equivalent to the claim that

there exists a basis consisting only of eigenvectors and generalized eigenvectors of .

PS: More details about the Jordan Canonical Form can be found at

10.3 Jordan Canonical Form and CAS

Reference video: http://youtu.be/LxY6RcNTEE0

In practice, in order to find the Jordan Canonical Form of a 10  10 matrix, you need to find the roots of a characteristic polynomial of degree 10.

The factorization and rigorous calculation of these roots is impossible.

Moreover, a 10 10 matrix requires us to calculate many exponents and coefficients.

In order to calculate these coefficients, the Gaussian elimination and related computations can be performed by various computer programs.

e.g. HLINPRAC, MATHEMATICA,  MATLAB, and the recently developed open-source program, Sage.

The use of software for computationally complex mathematics is necessary in an increasingly technological society.

The following links provide more information about the Jordan Canonical Form and tools that allow you to explicitly find the Jordan Canonical Form for a given matrix without arduous calculations by hand.

1. Theory and tools : http://matrix.skku.ac.kr/JCF/

2. Jordan Canonical Form; an algorithmic approach:

3. Jordan Canonical Form (step by step) tool:

4. CAS Tool : http://matrix.skku.ac.kr/2014-Album/MC-2.html

Solved by 신영준
Finalized and Final OK by SGLee

Problem 1

Let  be a 5×5 matrix with the only one eigenvalue with algebraic multiplicity of 5.

Find all possible types of  Jordan Canonical forms of when the number of linearly independent eigenvectors corresponding  is 2.

Sol)

Type 1

the dot diagram for  is the following.

=2 : ⦁⦁

=1 : ⦁

=1 : ⦁

=1 : ⦁

So,

Type 2

the dot diagram for  is the following.

=2 : ⦁⦁

=2 : ⦁⦁

=2 : ⦁

So,

Note:  5= 1+4, 2+3 (there are only 2 cases in the given decreasing order.)

Problem 2

For the given Jordan Canonical form , calculate the following:

(1)

Sol)

(2)

Sol)

(3)

Sol)

(4)

Sol)    ■

Chapter 10,  Problem 3

Find the Jordan Canonical Form of the given matrix.

Sol)

The matrix  has its characteristic polynomial  = .

So  has two distinct eigenvalues .

Here =25 has algebraic multiplicity 1, =0 has algebraic multiplicity 4.

Thus the dot diagram for  has 1 dot.

=1 : ⦁    (number of Jordan block: 1)

Here  has one Jordan block of 11.    That is, =[25].

As well, the dot diagram for  has 4 dot, and

= 5-rank(-) = 5-1 = 4

= rank() - rank[] = 1-1 =0

= rank[] - rank[] = 1-1 =0

= rank[] - rank[] = 1-1 =0

Thus, the dot diagram for  is the following.

=4 :  ⦁⦁⦁⦁   (number of Jordan block: 4)

has 4 Jordan blocks of 11 .     That is, =.

Hence, the Jordan Canonical form of  is

∴ ==.

Doble checked by Sage

Chapter 10,  Problem 4

Find the Jordan Canonical Form of the given matrix.

Sol)

The matrix  has characteristic polynomial  = .

So A has three distinct eigenvalues . Here =3 has algebraic multiplicity 1, =2 has algebraic multiplicity 1 and =1 has algebraic multiplicity 3.

Thus the dot diagram for  has 1 dot.

=1 ; ⦁(number of Jordan block: 1)

Here  has one 11 Jordan block. That is, =[3].

And also the dot diagram for  has 1 dot.

=1 : ⦁(number of Jordan block: 1)

Here  has one  11 Jordan block. That is, =[2].

As well, the dot diagram for  has 3 dot, and

= 5-rank(-) = 5-4 = 1

= rank() - rank[] = 4-3 =1

= rank[] - rank[] = 3-2 =1

Thus, the dot diagram for  is the following.

=1 : ⦁ (number of Jordan block: 1)

=1 : ⦁

=1 : ⦁

has one 33 Jordan block. That is, =.

Hence, the Jordan Canonical form of  is

∴ ==

Doble checked by Sage

Chapter 10,  problem 5

Find the Jordan Canonical Form of the given matrix.

Sol)

The matrix  has characteristic polynomial  = .

So A has three distinct eigenvalues .

Here =5 has algebraic multiplicity 1, =1 has algebraic multiplicity 1 and =-3 has algebraic multiplicity 1.

Thus the dot diagram for  has 1 dot.

=1 ; ⦁ (number of Jordan block: 1)

Here  has one 11 Jordan block. That is, =[5].

And also the dot diagram for  has 1 dot.

=1 : ⦁ (number of Jordan block: 1)

Here  has one 11 Jordan block. That is, =[1].

As well, the dot diagram for  has 1 dot, and.

=1 : ⦁ (number of Jordan block: 1)

has one 11 Jordan block. That is, =.

Hence, the Jordan Canonical form of  is

∴ ==

Doble checked by Sage

Problem 6 Find the Jordan Canonical Form of the given matrix.

Sol) The matrix   has characteristic polynomial  = .

So A has two distinct eigenvalues of =2 and =0, each with algebraic multiplicity 2.

For =2,

= 4 - rank() = 4-2 = 2

= rank() - rank[] = 2-2 = 0

Therefore, the dot diagram for  is the following.

=1 : ⦁  ( Number of Jordan blocks : 1)

=1 : ⦁

So, =

For =0

= 4 - rank = 4 - 3 = 1

= rank - rank[] = 3 - 2 = 1

=1 : ⦁  ( Number of Jordan blocks : 1)

=1 : ⦁

So, =

Hence, the Jordan Canonical form of  is

∴ == .

Doble checked by Sage

Problem7.

Find the Jordan Canonical Form of the given matrix.

Sol.

The matrix  has characteristic polynomial  = . So A has three distinct eigenvalues .

Here =4 has algebraic multiplicity 1, =2 has algebraic multiplicity 1 and =1 has algebraic multiplicity 3.

Thus the dot diagram for  has 1 dot.

⦁

Here  has 11 Jordan block. That is, =[4].

And also the dot diagram for  has 1 dot.

⦁

Here  has 11 Jordan block. That is, =[2].

Now the dot diagram for  has 3 dot, and

= 5-rank(-) = 5-4 = 1

= rank() - rank[] = 4-3 =1

= rank[] - rank[] = 3-2 =1

Thus, the dot diagram for  is the following. (number of Jordan block: 1)

=1 : ⦁

=1 : ⦁

=1 : ⦁

has ons 33 Jordan block. That is,    =. Hence, the Jordan Canonical form of  is

∴ ==

Doble checked by Sage

Problem8.

Find the Jordan Canonical Form of the given matrix.

Sol.

The matrix  has characteristic polynomial  = . So A has two distinct eigenvalues of =4 and =2, each with alegebraic multiplicity 2.

For  = 4,

= 4 - rank() = 4-3 = 1

= rank() - rank[] = 3-2 = 1

Therefore, the dot diagram for  is the following.  ( Number of Jordan blocks : 1)

=1 : ⦁  (Number of Jordan blocks = 1 )

=2    :  ⦁

So, =.

For =2,

= 4 - rank() = 4-2 = 2

= rank() - rank[] = 2-2 = 0

Therefore, the dot diagram for  is the following.

=2 : ⦁⦁ (Number of Jordan blocks = 2 )

So, =.

Thus, the Jordan Canonical Form of  is =.

Doble checked by Sage

Ch 10  Jordan Canonical Forms (Reference)

http://matrix.skku.ac.kr/JCF/

http://matrix.skku.ac.kr/JCF/JordanCanonicalForm-SKKU.html

MT Chapter 10 Jordan Canonical Forms

[ 1. Jordan Canonical Form

- Definition - Link

[ 2. Generalized  Eigenvectors

- Definition - Link

- Finding a Jordan Canonical Form using Generalized Eigenvectors  - Link

- Finding Generalized Eigenvectors using a Jordan Canonical Form - Link

[ 3. Characteristic Polynomial and Minimal Polynomial

- Characteristic Polynomial and Minimal Polynomial of a Jordan form  - Link

- Diagonalization and Minimal Polynomial  - Link

- Cayley-Hamilton theorem- Link

[ 4. Applications  ]

- Computing J^k where J is a Jordan block  - Link

- Computing A^k using Jordan Canonical Form - Link

- Computation of e^A using Jordan Canonical Form- Link

- Solving a Linear Ordinal Differential Equation - Link

Reference :

1) The algebraic sum of currents in a network of conductors meeting at a point is zero.

2) The directed sum of the electrical potential differences (voltage) around any closed network is zero.

http://www.math.washington.edu/~king/coursedir/m308a01/Projects/m308a01-pdf/kirkham.pdf

6) Image Compression Using Singular Value Decomposition, Ian Cooper and Craig Lorence, December 15, 2006

“Low Complexity Power Allocation Scheme for MIMO Multiple Relay System With Weighted Diagonalization”