Chapter 4 Eigenvector and Eigenvalue

[`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

=

*A*^{2} *q* =

In general ,

*A*^{n} *q* =
=

=
+
+

=

*A*^{n} *q* =
where
when

Let be a linear functional [ for example : ]

= =

*Example : * 4.3.5

*A* =
,
*q*_{0} =

[*Sol*]

STEP 1. Find a sequence
*A q*_{0} =
=
= *q*_{1}

STEP 2. = = 10 =

= =

STEP 3. repeat STEP 2.

*A*^{2} *q*_{0}
*A* (*A q*_{0}) = *A q*_{2} =

consider
=
=
= *q*_{2}

= =

By Table 4.1 ( P213 )

Eigenvalue : 9.140055

Eigenvector : ( 1 , 0.140055 )

=

*A*^{n} *q*_{0} =

, 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* - *uI*2. 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