, , , repeated real eigenvalues ' X A X
:Example 1, 1,5 Solve:
1 2 2
' 2 1 2
2 2 1
X X
3
1
1 for =5
1
K
Sec7.5: Multiple Eigenvalue Solution
Lin. indep eigenvectors
nvv v ,,, 21 One single eigenvector
v
nt
ntt veXveX veX ,,, 2211
, , , repeated real eigenvalues ' X A X
Repeated real EigenvaluesRepeated real Eigenvalues
One single eigenvector 1v
1 0 vIA 21 vvIA 32 vvIA
kk vvIA 1
tevX 11
tevtvX 212
tt evtvvX
322
2
13
tkkk
ktk evtvvX
1)!1(
1
1
Repeated real EigenvaluesRepeated real Eigenvalues
:Example Solve:
3 18'
2 9X X
3, 3
1 0 vIA 21 vvIA 32 vvIA
kk vvIA 1
tevX 11
tevtvX 212
tt evtvvX
322
2
13
tkkk
ktk evtvvX
1)!1(
1
1
Repeated real EigenvaluesRepeated real Eigenvalues
:Example Solve:
2,2,2 2 1 6
' 0 2 5
0 0 2
X X
1 0 vIA 21 vvIA 32 vvIA
kk vvIA 1
tevX 11
tevtvX 212
tt evtvvX
322
2
13
tkkk
ktk evtvvX
1)!1(
1
1
Repeated real EigenvaluesRepeated real Eigenvalues
:Example Solve:
1 0 vIA 21 vvIA 32 vvIA
kk vvIA 1
tevX 11
tevtvX 212
tt evtvvX
322
2
13
tkkk
ktk evtvvX
1)!1(
1
1
XX
010
122
001
' 1 ,1 ,1
Homog Linear System
2X2 system ' X AX
2 complex1,2 = i
2 real distinct
1 2, 2 real repeated ,
2 lin indep eig-vec Chain G-eigvec
21 , vv 21 , vv
3X3 system ' X AX
1 real + 2 complex 3 real distinct 2 real repeated + 1 real
3 lin indep eig-vec 2 lin indep eig-vec
321 , , vvv 121 , , wvv
321 ,, i 3,21,321 ,
3 real repeated
3 li
n i
nd
ep e
ig-v
ec
2 li
n i
nd
ep e
ig-v
ec
3
2
1
,
,
v
v
v
1
1,
w
v
321
1 li
n i
nd
ep e
ig-v
ec
1v
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
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
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 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
, , , repeated real eigenvalues
Another method to compute: Generalized eigenvectorsAnother method to compute: Generalized eigenvectors
1 0 vIA 21 vvIA 32 vvIA
kk vvIA 1
kvvv ,,, 21
Chain of generalized eigenvectors
Compute:
)( IA 2)( IA kIA )(
kk vIA 0)(
Solve:
Compute:
kk vIAv )(1
32 )( vIAv
21 )( vIAv
kvvv ,,, 21
Chain of generalized eigenvectors
:Example Find all generalized eigenvectors:
1 ,1 1,
12 )( kk vIAv
010
122
001
A
Another method to compute: Generalized eigenvectorsAnother method to compute: Generalized eigenvectors
Compute:)( IA 2)( IA kIA )(
kk vIA 0)(
Solve:
Compute:
kk vIAv )(1
32 )( vIAv
21 )( vIAv
kvvv ,,, 21 Chain of generalized eigenvectors
:Example Find all generalized eigenvectors:
1 ,1 1,
12 )( kk vIAv
010
122
001
A
:Solution
0110
0112
0000
0IA
0110
0001
000023 RR
22
1 R
1 lin indepeigenvector
Length of chain =3
321 ,, vvv
110
112
000
)( IA
002
002
000
)( 2IA
000
000
000
)( 3IA
0
0
1
3v
0
2
0
0
0
1
110
112
000
2v
2
2
0
0
2
0
110
112
000
1v