Therorem 1:

ablediagonaliz

matrixnxn A

Under what conditions a given matrix is diagonalizable ???

rseigenvectot independen

linearly n han A

Jordan Block

REMARK:Not all nxn matrices are diagonalizable

20

32A

2)2()( P

0

1A similar to (close to diagonal matrix)

Example

kxk

300

030

003

1) Find charc. Equ.2) Find all eigenvalues3) How many free variables4) How many lin. Indep eigvct

Example

kxk

30000

13000

01000

00300

00130

00013

1) Find charc. Equ.2) Find all eigenvalues3) How many free variables4) How many lin. Indep eigvct

kxk

J

000

100

010

001

Definition:

Jordan block with eigenvalue

Examples

70

17

50

15

200

120

012

9000

1900

0190

0019

Jordan Block

of size k

sJ

J

J

J

00

00

00

2

1Definition:

Where each submatix is a jordan block of the form

iJ

Jordan Normal Form

kxki

i

i

i

iJ

000

100

010

001

Exmples:

90000

19000

00200

00120

00012

1A

1000

1100

0050

0015

2A

200

040

014

3A

100

020

004

4A

1) Find eigenvalues2) multiplicity3) How maany lin. Indep eigenvectors4) How many chain and length

is in Jordan normal form

200

040

024

3B

Note: s = # lin.indep eigvectors

Theorem 1: Any nxn matrix A is similar to a Jordan normal form matrix

Jordan Normal Form

Theorem 1:

Let A be nxn matrix there exits an invertable Q such that:

JQAQ 1

where J is in Jordan normal form

:Example3, 3

1

31v

92

183A

30

13J

:Example

123

031

115

A

300

030

013

J

Find the Jordan form Find the Jordan form

3,3,3

Jordan Normal Form :Example

50000

95000

09500

00043

00004

A

Find the Jordan form

5,5,5,4,4

:Example

50000

05000

09500

00043

00004

A

Find the Jordan form

5,5,5,4,4

Repeated real EigenvaluesRepeated real Eigenvalues

:Example 2,2,2 2 1 6

' 0 2 5

0 0 2

X X

DEF

completenot isit if defective called is 1kty multiplici of eigenvalueAn

rseigenvecto missing ofnumber ofdefect

0

0

1

1v

2 defect

:Example

3 18'

2 9X X

3, 3

1

31v

1 defect

Repeated real EigenvaluesRepeated real Eigenvalues

:Example 2,2,2 2 1 6

' 0 2 5

0 0 2

X X

0

0

1

1v

2 defect

TvIA ]0,0,1[ 02 1

TvvIA ]0,1,0[ 2 21

TvvIA ],,0[ 25

1

5

632

rank 2 generalized eigenvector

rank 3 generalized eigenvector

02

BUT

02

21

22

vIA

vIA

02

BUT

02

22

23

vIA

vIA

DEF: A rank r generalized eigenvctor associated with is a vector v such that

02 BUT 02 1 vIAvIA rr

Repeated real EigenvaluesRepeated real Eigenvalues

:Example 2,2,2 2 1 6

' 0 2 5

0 0 2

X X

0

0

1

1v

2 defect

TvIA ]0,0,1[ 02 1

TvvIA ]0,1,0[ 2 21

TvvIA ],,0[ 25

1

5

632

, , 321 vvv

1r veigenvecto on the based rseigenvecto dgeneralize ofchain

1r veigenvecto on the based rseigenvecto dgeneralize ofchain 3length A

Repeated real EigenvaluesRepeated real Eigenvalues

DEF A length k chain of generalized eigenvectors based on the eigenvector is a set of of k generalized eigenvectors such that

, , , 21 kvvv 1v

1 kk vvIA 21 kk vvIA

12 vvIA

0 kk vIA

Example

kxk

300

030

003

1) Find charc. Equ.2) Find all eigenvalues3) How many free variables4) How many lin. Indep eigvct5) defect

Example

kxk

30000

13000

01000

00300

00130

00013

1) Find charc. Equ.2) Find all eigenvalues3) How many free variables4) How many lin. Indep eigvct5) defect

kxk

J

000

100

010

001

Definition:

Jordan block with eigenvalue

, , , 21 kvvv Chain of generalized eigenvectors Examples

70

17

50

15

200

120

012

9000

1900

0190

0019

Jordan Block

Jordan Normal FormTheorem 1:

Let A be nxn matrix there exits an invertable Q such that:

JQAQ 1

where J is in Jordan normal form

sJ

J

J

00

00

00

2

1

If all generalized eigenvectors are arranged as column vectors in proper order corresponding to the appearance of the Jordan blocks in (*), the results is the matrix Q

30000

03000

01300

00020

00012

J

2,2,3,3,3

:Example

Let A be 5x5 matrix3,3,3

121 ,, wvv

2,2

21,uu],,,,[ 12121 wvvuuQ