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Spectrum of A
σ(A)=the spectrum of A
=the set of all eigenvalues of A
PropositionablediagonalizunitarilyisMA n
Aofrseigenvectoofconsisting
basislorthonormaanhasCn
),(,
)()(
,)()(
A
AINAIN
andAINCA
n
Similarity matrix
If ,then we say that
A is transformed to B under
similarity via similarity matrix P
BAPP 1
Exercise 1.2.4
If are similar over C,
then A and B are similar over R.
)(, RMBA n
組合矩陣理論 第一章 Exercise.doc
Proof of Exercise 1.2.4
.
)()(
,sin)(
)()(
)2()1(
)(,sin,)2(
)1(
.
1
1
RoversimilarareBandATherefore
BSSASS
gularnonisandRMSSSince
BSSSSA
BSSBSAAS
RMBAceBSSA
SBAS
SBAS
BASS
tsMS
CoversimilarareBandA
n
n
n
Schur’s unitary triangularilation Theorem
nMAanyFor
n
UA
0
*2
1
~
unitarily similar
can be in any prescribed
order
Normal matrix
normalisMA n
AAAA **
e.g Hermitian matrix, real symmetric matrix, unitary matrix, real orthogonal matrix, skew-Hermitian matrix, skew-symmetric matrix.
強調與 complex symmetric matric 作區別
Remark about normal matrix
Normal matices can not form a
subspace .
Fact (*) for Normal matrix
normalisMA n
CnormalisIA ,
Proof in next page
.
)()(
)()(
)()(
))((
))((
*
*
*
**
**
*
*
**
normalisIAHence
IAIA
IAIA
IAIIAA
IIAIAAA
IAAAA
IAIA
IAIA
AAAA
normalisA
Spectrum Thm for normal matix.normalisMA n
tsUmatrixunitaryan .
Cwhere
UUA
n
n
,,1
*1
0
0
注意
Appling Schur’s unitary triangulariation Theorem to prove.
Real Version of Spectrum Thm for normal matix
).()( AAAAnormalisRMA TTn
tsQmatrixorthogonalreala .
n
T
A
A
AQQ
0
01
formtheofmatrix
orisAwhere j
22
11
R
,It is normal.The proof is
in next page
.
(*),
,0
0
0
0
:2
0
0
0
0
:1
2
22
22
22
22
normalis
Factbyandnormalisit
symmetricskewrealisSince
I
Method
Method
Proposition for eigenvaluenMAanyFor
thentatataatpLet kk ,)( 2
210 k
kAaAaAaIaAp 2210)(
AofseigenvalueareIf n ,,, 21
)(
)(,),(),( 21
Apofseigenvalue
arepppthen n
Proof of privious Proposition
n
U
A
0
*2
1
~
n
U
pAp
0
*2
1
~)(
)(
)(
)(
~)(
0
*2
1
n
U
p
p
p
Ap
1.3
Jordan Form and Minimal Polynomial
Elementary Jordan Block
kk
kJ
0
1
0
10
1
)(
0
0
main diagonal
elementary jordan block
super diagonal
sub diagonal
kk
kk IJ
0
1
10
)(
0
0
kk
kNLet
0
1
10
0
0
It is Nilpotent matrix.(see ne
xt page)
kk
kN
0
0
1
00
100
0
0
2
kk
kkN
0
0
0
000
0000
0
1
1
0 kkN
Jordan Matrix
nnkn
n
n
kJ
J
J
J
)(
)(
)(
0
02
1
2
1
jordan matrix
distinctnecessarynotare
andnnnnwhere
k
k
,,, 21
21
Jordan Canonical Form Theorem
nMAanyFor
unique up to the ordering of elementary Jordan blocks along the block diagonal.
A is similar to a jordan matrix
If A is real with only real eigenvalues, then the similarity matrix can be taken to be real
By Exercise
1.2.4
Observation 1 for Jordan matrix
ablediagonalizisA
.matrixdiagonalaisJ A
the jordan matrix of
A
Observation 2 for Jordan matrix
ofmultiplegeometric
toingcorrespondJinblocksof A#
the proof in next page
)(
)(
)(
)(
.
11
1
kn
mn
n
n
A
A
A
k
m
m
J
J
J
J
J
formtheintakebecanJThen
toingcorrespond
JinblocksofnumberthebemLet
kminJrank
andminJrankSince
J
J
J
J
IJ
iin
in
kn
mn
n
n
A
i
i
k
m
m
,,1)(
,,,11)0(
)(
)(
)0(
)0(
11
1
.
)(
)(
)(
)(dim
)1()(
1
11
toingcorrespond
Jinblocksofnumberthe
mmnn
JIrankn
AIrankn
AIN
ofmultiplegeometric
mnmn
nnJIrank
A
A
k
ii
k
mii
m
iiA
Observation 3 for Jordan matrix
ofmultipleebraica lg
toingcorrespondJin
blocksofsizestheofsumthe
A
Observation 4 for Jordan matrix
ofmultiplegeometric
ofmultipleebraica
lg
blocksofareto
ingcorrespondJinblocksall A
11
Observation 5 for Jordan matrix
Given counter example in next
page
The algebraic and geometric multiple of λ
can not determine completely the
Jordan structure corresponding to λ
Assume that 1 is an eigenvalue of A and
geometric multiple of 1 is 3
algebraic multiple of 1 is 5
then 3 blocks in AJ corresponding to λ
the sum of sizes of these blocks is 5
Therefore (see next page)
AJinareJJJ )1(),1(),1( 311
AJinareJJJ )1(),1(),1( 221
or
Annihilating polynomial for A
In next page we show that A has an annihilating
polynomial.
Let p(t) be a polynomial.
If p(A)=0, then we say p(t) annihilates A and
p(t) is an annihilating polynomial for A
.)(..
0)(
,)(
0
..,,,
,,,,
,dim
,,,,
2
2
2
2
2
2
2
10
10
10
2
2
2
Aforpolynomialngannihilatianistpei
Ap
thentataatpLet
AaAaIa
tszeroallnotCaaa
dependentlinearlyareAAAI
nMSince
MAAAI
nn
nn
n
n
n
nn
Minimal polynomial of A
)(tmA
The minimal polynomial of A is
monic polynomial of least degree that
annihilates A and is denoted by
)()(0)( tptmAp A
the proof in next page
)()(
0)(
0)(
)()()()(
)(deg)(deg0)(
)()()()(
lg
tptm
tr
Ar
ArAmAqAp
tmtrortrwhere
trtmtqtp
orithmadivisionBy
A
A
A
A
Caley-Hamilton Theorem
0)( AcA
)()( tctm AA
This Theorem implies that
Minimal Polynomial when A~B
)()(~ tmtmBA BA
the proof in next page
)()(
)()(,
)()(
0)()(
)(.)(~)(
)(.
)()()(
.
~
11
1
tmtmTherefore
tmtmSimilarly
tmtm
BmAm
tppolyallforApBp
tppolyallfor
QApQAQQpBp
AQQB
tsQ
BA
BA
BA
AB
AA
Mimimal poly. of Jordan matrix AofseigenvaluedistinctbeLet k ,,, 21
.
arg
iA
i
toingcorrespondJinblock
Jordanestlofsizethem
i
A
mi
k
iJ ttm )()(
1
Given example to
explain in next page
1)()(
..int
h
iAh
iA IJrankIJrank
tshegerenonnegativleastthe
13
12
12
1
3
2
2
1
1
1
1
1
0
0
10
010
1
1
01
IJ
JLet
A
A
313
312
212
312
31
213
212
122
12
21
)(
)(
)(3)(
0
0
00
000
)(
)(
)(
)(2)(
0
0
00
100
)(
IJ
IJ
A
A
Similarly,
0
)(
*)(
)(
)(
*)(
**)(
)(
)(
0
00
)(
)(
*)(
**)(
)(
32
32
31
31
31
31
3
213
221
221
221
221
22
IJ
IJ
A
A
)()()()(
,0)()()(
6)()(
5)()(
,6)(
3)()(
,4)(
,5)(
32
23
1
32
23
1
223
32
22
2
41
31
21
1
ttttm
andIJIJIJ
IJrankIJrank
IJrankIJrank
IJrank
IJrankIJrank
IJrank
IJrank
AJ
AAA
AA
AA
A
AA
A
A
Mimimal poly. of Jordan matrixAofseigenvaluedistinctareLet n ,,, 21
.
arg
iA
i
toingcorrespondJinblock
Jordanestlofsizetheismwhere
i
A
mi
k
iJ ttm )()(
1
Proof in next page
1)()(
..int h
iAh
iA
i
IJrankIJrank
tshegerenonnegativleasttheism
,,1)(0)(
)()()(
,,,,
)(,
,
)()(
.
arg
11
11
1
1
iJpJp
JpJpJp
where
JJandkthen
Jinblocksofnumberthebe
letandxtpLet
toingcorrespondJin
blockestltheofsizethebemLet
inA
ini
ini
A
k
ini
A
A
mi
k
i
iA
i
i
ii
i
i
)(0
))(()0(
))(())((
))(())((
0)(,
0))((,
,1
,1
1
rj
mnijn
k
rii
mn
rjrj
mnijn
k
rii
mnrjn
mnijn
k
ijn
A
jn
mn
IJJ
rsomeformnandwhere
IJIJ
IJJp
JphaveweprovedisthisOnce
JpjfixedeachforthatshowTo
i
jj
r
j
i
jj
r
jj
i
jjj
j
jsjs
jj
ii
ii
k
iiA
AA
mnandtssCertainly
mtsjSuppose
kimClaim
ThmHamiltonCayleythegotalreadywe
mwherettm
tpdividestmsoandJforpolynomial
ngannihilatianistpthatprovedhaveWe
i
.
.
,,1:
)(
0)()(
)()(
)(
1
,0)(
0
))(()0(
))(())((
))((
))((
))((
))(())(()(
,1
,1
1
11
AA
ijn
k
jiim
nijn
k
jiimjjm
mijm
k
i
jmA
snA
inA
l
iin
l
iAAA
Jm
JJ
IJIJ
IJ
Jm
Jm
JmJmJm
i
j
j
j
i
jj
j
jj
i
jj
j
s
ii
index of eigenvalue p.1
AofseigenvaluedistinctbeLet n ,,, 21
)()(,~ tmtmJASinceAJAA
k
i
miA
ittm1
)()( See next page
index of eigenvalue p.2
.
arg
iA
i
toingcorrespondJinblock
Jordanestlofsizethem
1)()(
..int
h
ih
i IArankIArank
tshegerenonnegativleastthe
imi
k
iA ttm )()(
1
ii ofindexthecalledism
index of eigenvalue p.3
nIArankIArank 10 )()(
0 ofindexthe
thenAIf ),(
)(AvbydenotedisofindexThe
Observation 6 for Jordan matrix p.1
)(ALet
Akk JinsJfnumberthebebLet )(0
andnonegativearebbbThen 321 ,,
bycompletelyederare mindet
32 )(,)(),( IArankIArankIArank
….
Observation 6 for Jordan matrix p.2
11 )()(2)( kkk
k
IArankIArankIArank
b
the proof in next page
11
11
1
1
211
211
)()(2)(
.)(~)(,~
)()(2)(
)()(
)()()1()2(
)()()2(
)()()1(
kkkk
AA
kA
kA
kAk
kk
Ak
A
kA
kA
kkk
Ak
A
kkkk
Ak
A
IArankIArankIArankb
ppolyanyforJpApJASince
IJrankIJrankIJrankb
bIJrankIJrank
IJrankIJrank
bbIJrankIJrank
bbbIJrankIJrank
Observation 7 for Jordan matrix p.2
kk IArankIArank )()( 1
the proof in next page
The number of blocks in
AJ
of size k ≧is
toingcorrespond
ksizeofJinblocksofnumberthethen
IArankIArankIArank
andmmkb
mk
IArankIArankIArankb
ThenJinsJ
ofnumberthebebandtoingcorrespond
JinblockestltheofsizethebemLet
A
mmm
k
kkkk
Ak
k
A
21
11
)()()(
,,2,10
,,1
)()(2)(
.)(
arg
kk
mm
kk
m
ki
ii
m
ki
ii
m
ki
ii
m
ki
ii
m
ki
iii
m
kii
IArankIArank
IArankIArank
IArankIArank
IArankIArank
IArankIArank
IArankIArank
IArankIArank
IArankIArankIArank
b
)()(
)()(
)()(
)()(
)()(
)()(
)()(
)()(2)(
1
1
1
1
1
1
1
1
1
11
thenRMALet n ),(
)()( AA
ofmultipleebraicathe
ofmultipleebraicathe
lg
lg
ofmultiplegeometricthe
ofmultiplegeometricthe
See next page
組合矩陣理論 第一章 Exercise.doc
Jordan structures for ,
11 )()(2)( kkk IArankIArankIArank
)(# kJ
The Jordan structure of A
corresponding to and that corresponding
to are the same. Because
)(# kJ11 )()(2)( kkk IArankIArankIArank
)(RM n
0
0)(DLet
bia
11
iiS
),()( 1 baCab
baSSD
denote
The proof is in next page.
ab
ba
aibi
biai
i
baibaibiabia
biabiabaibai
i
baibia
baibiaii
i
i
i
bia
biaii
iSSD
i
i
iiiiS
iiS
T
22
22
2
1
2
1
112
1
1
1
0
0
112
1)(
1
1
2
111
2
1
11
1
1
000
100
000
001
)(0
0)(
2
2
J
J
)(0
)( 2~
D
IDP
Permutation similarity
The proof is in next page
)(0
)(
000
000
100
010
~
000
000
100
001
~
00
100
000
001
)(0
0)(
2
2
2
D
ID
J
J
..
1000
0010
0100
0001
tsPHence
)(0
)(
)(
)( 2
2
21
D
IDP
J
JP
),(0
),(
0
0
)(0
)(
0
0 21
12
baC
IbaC
S
S
D
ID
S
S
),(0
),(~
)(0
0)( 2
2
2
baC
IbaC
J
J
similarly
),(
),(
0),(
~)(0
0)(
02
2
3
3
baC
IbaC
IbaC
J
J
Proved in next page
),(0
),(
)(0
)(
)(0
)(
0
0
0
0
)(0
)(
0
0
2
12
1
1
11
1
12
baC
IbaC
SSD
ISSD
SD
SSD
S
S
S
S
D
ID
S
S
similarly
kkk
k
J
J
22)(0
0)(
kkbaC
I
baC
IbaC
22
2
2
),(
),(
),(
~
0
0
Theorem 1.3.4thenRMALet n ),(
)(
)(
),(
),(
~1
11
1
1
rm
m
ppn
n
r
p
J
J
baC
baC
A
pkforAofeigenvaluerealnon
isibawhere kkk
,,1
Aofseigenvaluerealarer ,,1
)()(~),( knknkkn kkkJJbaC
kk nnkk
kk
kk
baC
I
baC
IbaC
22
2
2
),(
),(
),(
)()( RMJRMA nAn
A
n
JAPP
tsMP
1
..
By Exercise 1.2.4 組合矩陣理論 第一章 Exercise.doc
A
n
JAQQ
tsRMQ
1
..)(
