[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
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 =
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 =
[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) .
so is an Eigenvalue of ( A - uI )
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 )
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