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

=

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