Computing Eigenvalues & Eigenvectors: Power & Jacobi Methods, 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

T ) n X^ x^ x

x   [^ ](^

)^ (^

) A^ X^

X  docsity.com

Power MethodPower MethodJacobi’s MethodJacobi’s Method

To compute the largest eigenvalue and the correspondingeigenvector of the system^ [^ ](^ where [ A ] is a real, symmetricor 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

We calculate

2 1 1 1 2 2 1

1 r^

r r^ r^

n n n

A v^ c v^

c^ v^

 c v ^ 



^

^ ^

^ 

^

^ ^

^   ^ 

^

^ ^

^ 

^

1 1

2 1 1 1 2 2 1

1

(^ )

r^

r r^ r^

n n n

A^ v^ c v

c^ v^

 c v  

^  

^  

^ ^

^ 

^

^ ^

^   ^ 

^

^ ^

^ 

^

  docsity.com

Now, the eigen valuecan be computed as the limit ofthe ratio of the correspondingcomponents of

and

That is, Here, the index

p^ stands for the p-th component in the corresponding vector

^1 r^1 r  A v. A^ v 1

1 (^ )^111

,^ 1, 2,^

rrA vp Lt pr rr ( ) p

n A v

    

^  docsity.com

The inverse matrix has aset of eigen values whichare the reciprocals of theeigen values of [

A ]. 1

1 [^ ](^

)^ (^

) A^ X^

X  ^ 

Thus, for finding theeigen value of theleast magnitude ofthe matrix [

A ], we have to apply powermethod to the inverseof [ A ].

DefinitionAn^ n x n^ matrix [

A ] is said to be orthogonal if

1 [^ ] [^ ]

[ ], i.e.[^ ]^ T T [^ ] A^ A^

I   A A^ 

If [ A ] is an

n x n^ real symmetric matrix, its eigenvalues are real, and thereexists an orthogonal matrix[ S ] such that the diagonalmatrix D is^1 [^

][^ ][^

] S^ A ^ S

Let^ A^ be an

n x n^ real symmetric matrix. Supposebe numerically the largestelement amongst the off-diagonal elements of

A****. We construct an orthogonal matrix S defined as^1

a^ ij

sin^ ,^ sin

s^ s^ ^ ij^ ji cos^ ,^ cos s^ sii^ jj

 

^ 

i-th 1 column^ -th^ column 1 0 0 0

0 0 cos^

sin^0 i-th

row 0 0 sin^

cos^0

-th^ row 0 0 0

j 0 1 S

j ^    

^

^

^

^

^

 ^

^

^ 

^

^

^

^

^

^ ^

^ ^

^ ^ ^

^    

^ ^ ^

^    

^ ^ ^

^    