Luke
Johnston
21365614
find eigenvalues
det (
A- 711=0
A- II
[
=
-
find
eigenvectors
A
?
=
(A- SHE
,,
(A
→
,
[
-
-
q
16
-
g)
/ A-It /
=
"
9
-
"
1-
i,
"
9
12
Ri
→
1=5
1%1:)
te R
,
344T
now
"
→
A- SI )=
consider
{
→
Span
the
0
}C
Since
1A
t=3→%4
-
→
Ker B
SIK
ker( A-
=/ I ]
is
{ [ ;] }
increasing
A
5,
is
C Ker
,
2×2
.
=
eigenvector
.
.
.
this sequence stabilizes
( I f)
SI )
Bi
an
of subspaces
order
IR2
o
101+25=0
form
3%-4+-0
Ker (
-
(7-5)<=0
I :/ ¥:p :| :/ "
=
-17×+77+5+144=0
119
I
É
=
1/41=1;)
matrix
"
V, =t
XI)
-
↳
12
augmented
-
(7+7) (17-7)+16 (a)
-
=
IT
=
I
7-
at
BY
is
only
eigenvalue
This
,
any
→
vector
nor
in
Karl A- SI)
,
but
Span { to ] / I ] } form Jordan
a
p=
/ ;)
'
o
→
P "AP=
in
R'
Basis
5=8 :]
is in
her ( A-SIT
A-
+I
=
-
0
I
,
I
,
2-7
0
/
der ( A- ¥ )
=
=
=
=
(2-7)/1-+1 (-2-11-(-1×1))
(2-7×7112+7)
( A -2112=0
→
→
chain
of
lengths
-
a-
-
(t
by
-
2)
(✗
+
1) 2=0
7=-1
or
=/I ;iÉf
+
,
Cayley hamilton theorem
-
ATI has chain
Jordan normal
of
high 2
form of A
2-7
-73+3×+2
7=2
(1--21)=1 %)
1-
is
| /
}? ?
O
O
-
I
?⃝
eigenvutostt-2-i-t.fi?i-i fr=?fai :|
O
O
R,
O
→
-
R
° °
"
of
[
f. B) ;)
"
0
°
.
,
t.tt/R7Vytl3Vz-Vz
"
-7
→
-
O
.
o
o
=
0
→
7
Vy=¥t
→
F-
"
"
9
" ""
Vx 1-44+104=0
→
Vx
=
-
lot
"
+
¥t
-1¥
=
*i÷
V×=t
4-
=D
se.IR/t,s-,
2-
c-
÷÷
R
I
→
v→
=
,
Vz=O
to
eigenspace.lu?k=(A-iI)vsi-f: :lH=H-JodanBasiscreate
a
generalized
Span
{E://t.IE//
let Ii
A
is
be
diagonal
.
of
eigenvector
an
2- cable
→
→
it
has
h
distinct
AÑ=tÑ
'
A- AT
Ji
-
I.
is
A-
'
'
,
A
distinct
i
c-
,
diagonali.zea66.mn
eigenvectors
G. ]
n
Hi
.
Hi
'
scalar
→
IT
is
scalar
a
of A is
4. of A
has
an
a
eigenvector of
A
A
has
'
A- is
n
distinct
eigenvalues
diagonal
-
able
2-
"
corresponding eigenvalue ¥
.
→
A
eigenvalues
"
Since
matrix
tÉvi= A- Ii
eigenvalue
every
Square
a
every eigenvector É
→
for
a
=
A-
=
the
,
A
"
also
does
A
assume
has
A
→
A'
→
=
form
J
I
=
]
=
PIP
,
f- f. vi. the
"
A
=
=
the
"
-
A-
'
=
✗
generalized
jordan
eigenspaa
basis
-
'
A
=
JI p
if
draws in
A
PJ P
'
matrix
the basis of
2
P J P
-
normal
i. e.
PJP
→
.
A
( P5F)
→
diagonal zeable
not
jordan
PJÉ
→
is
'
Ap
=
J
J1J
A
E :] :|: :|
comms
-
ftp..fi/.E:fearhiiB
we
[ E)
get
1: :\ : :|
→
this
requires
and
2>
I
=L
=
⇐
I ⇐>
+
=
£
7=1
}
Contradiction
→
A
must
be
diagonals
#
able
A" (2-7)
✗
algebraic
=
0
multiplicity
geometric multiplicity
C( A -0-1-1=2
dim /A- II)
"
>
( A- It}
dim
7--2,7--0
at
4
=
for
for
3
=
→
>
7=0
7--0
N( A
-
OI
dire / A
-
)
by
=3
)
II
,
rank
nullity
theorem
sufficiently large K
Such that
diu(A-XI )
did
/A
→
-
for
✗
It? dim (A- ✗ I)
=
dim ( A- II)
off geometric multiplicity algebraic
=
4
→
multiplicity
.
.
→
.
z
>
did A- II )
dim ( A
-
OI
}
>
3
dim ( A- OI ) ! dial AT
=
.
.
.
4
"
multiplicity
1=0
dim ( A- II )
algebraic
""
geometric multiplicity
A and
e-
det(C-
B
both
are
II
)
( 2- xp
=
2
jordan
Jordan Normal
of
→
normal
form
and AT
,
=
B
matrices
A B
,
7--2
→
µ{§ )
=
→
if
jordan
1¥:)
f- 2--1)
→
in
Kerk -21=1
,
blocks
form
=
C
are
are
similar
not
,
I
spmtfjf.fm
of length
Iµ
they have
similar
=
2
.
f- A
the same
to C
I of
,
B
IN F
.
length I
A
is
similar
to
B