Eigenvalues and Eigenvectors: Finding the Largest Eigenvalue using the Power Method, Slides of Mathematical Methods for Numerical Analysis and Optimization

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Let [

A ] be an

n x n

square matrix.

Suppose, there exists a scalarand a vectorsuch that

1 2 (^

T ) n

X^

x^ x

x

^



[^

](^

)^

(^

)

A^

X^

X 

(^

)^

(^

)

ax^

ax

d^

e^

a e

dx

 2

2

(sin 2

)

(sin

)

d^

ax^

a^

ax

dx^

 

This represents a set of

n

homogeneous equationspossessing non-trivialsolution, provided

0

A^

I  ^



This determinant, onexpansion, gives an n-thdegree polynomial which iscalled characteristicpolynomial of [

A ], which has

n

roots. Corresponding to eachroot, we can solve theseequations in principle, anddetermine a vector calledeigenvector.

Power MethodPower MethodJacobi’s MethodJacobi’s Method

Note!We shall consideronly real and real-symmetric matricesand discuss powerand Jacobi’s methods

To compute the largest eigenvalue and the correspondingeigenvector of the systemwhere [

A ] is a real, symmetric or un-symmetric matrix, thepower method is widely usedin practice.

[^ ](

)^

(^ ) A^ X

X  

ProcedureStep 1: Choose the initial vectorsuch that the largest element isunity.Step 2: The normalized vectoris pre-multiplied by the matrix[ A ].

(0) v docsity.com

At this point, the result looks likeHere,

is the desired largest eigen value and

is the ( corresponding eigenvector. )^

(^ 1)

(^ )

[^ ] k^

k^

k k

u^

A v

q v  ^



q^ k

(^ ) k v

ExampleFind the eigen value of largestmodulus, and the associatedeigenvector of the matrix bypower method

2 3

2 [^ ]^

4 3

5 3 2

9 A

^

 ^

  ^

 ^

 ^



Now we normalize theresultant vector to get

1 2 (1)^

(1) 6

1 7 14

1

u^

q v ^  ^  ^

 ^  ^  ^ 

The second iteration gives,

39 1 2

7

(2)^

(1)^

6

67 7

(^717114) (2) 2

2 3

[^ ]^

u^

A v

^ q v

^ 

^

^

^ 

^ 

^

^

 ^

^ 

^

 ^ ^

^ 

^

^

 ^ ^

^ 

^

^

^

^

^

^