[Def]
If
, then v is called the eigenvector
of A associated with the Eigenvalue
.
is a subspace of
is called the Eigen Space
of A associated with
< Characteristic Polynomial :
>
If
is an eigenvalue of A , then
1.
2.
is singular
If
is an eigenvalue
The solution is of the form x(t) = f(t) v
differentiate with respected to t .
because
f' (t) v = A f(t) v
Let
1. Power Method ( Find MAX. eigenvalue ) 2. Inverse Power Method ( Find MIN. eigenvalue) 3. Shift Inverse Power Method ( Find the eigenvalue close to fixed u )
Power Method :
If
are
eigenvalue of A with
and
are eigenvectors and linear indep.
associated with
respectively .
STEP : Let q be any vector in
consider the sequence
because
form a basis of
W.l.o.G. ( Without loss of Generality )
We may assume
=
A2 q =
In general ,
An q =
=
=
+
+
=
An q =
where
when
Let
be a linear functional [ for example :
]
=
=
Example : 4.3.5
A =
,
q0 =
[Sol]
STEP 1. Find a sequence
A q0 =
=
= q1
STEP 2.
=
= 10 =
=
=
STEP 3. repeat STEP 2.
A2 q0
A (A q0) = A q2 =
consider
=
=
= q2
=
=
By Table 4.1 ( P213 )
Eigenvalue : 9.140055
Eigenvector : ( 1 , 0.140055 )
=
An q0 =
, when
=
=
[Thm] If a is nonsingular , if
is an Eigenvalue of A ,
then
isi an Eigenvalue of A-1 .
<Proof> If
is an Eigenvalue of A
Inverse Power Method :
( For finding the smallest Eigenvlaue of A )
1. If
2. and
are L.I. Eigenvectors .
3. A is nonsingular .
apply Power Method to A-1
can be found by regular Power Method on A-1
Shift Inverse Power Method :
[Thm] If
is an Eigenvalue of A , then
is an Eigenvalue of (A - uI) .
<Proof> because
so
is an Eigenvalue of
( A - uI )
1. Given
s.t.
, and
for all
2.
L.I. Eigenvectors
3. A-1 exists
Example :
are Eigenvalue of A
[Sol] Let u = 1.2 , consider
are Eigenvalues of
( A - uI )
1. If
are Eigenvalues of A ,
then
are Eigenvalues of A - uI2. If
are the smallest Eigenvalue of A - uI ,
then Apply the inverse Power Method on (A - uI) ,
we can find
(
can be fonud )
Issue :
1. Power Method : find MAX. Eigenvalue and Eigenvector 2. Inverse Power Method : find MIN. Eigenvalue and Eigenvector 3. Shift Inverse Power Method : find the Eigenvalue colse to fixed u