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Eigenvalues

Chapter 4

Eigenvalues

Let A be an nn matrix. A number (scalar) is called an eigenvalue ofA ifthere exists a nonzero vector called eigenvector v such that

A =v v. (4.1)

Eigenvalues and eigenvectors are widely used in various applications such as to

determine the stability of a finite-difference scheme to solve a partial differential

equation and finding the solution for the system of differential equations.

Rewriting Eq. (4.1) as

,A I=v v

where Iis an identity matrix, yields

( ) 0vA I = . (4.2)

The linear homogeneous system (4.2) has a nontrivial solution 0v if and only ifthe matrix A I is singular that is | | 0A I = . Solving || IA = 0 leads tosolve the characteristics equation which will yields n eigenvalues of matrix A.

Substituting each eigenvalue into Eq. (4.2) will get its corresponding eigenvector.

4.1 Power Method

For an nn matrixA, the most dominant eigenvalue, 1, where|| 1 > || 2 > || n and its corresponding eigenvector, 1, can be obtained by

the power method

)(

1

)1( 1 k

k

k Am

+

+

= , k= 0, 1, 2,

where 1+km is the maximum absolute value of( ).kAv

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Eigenvalues

The iteration is iterated with a given initial eigenvector, )0(v until || 1 kk mm + || 2 > || n

and its corresponding eigenvector, ,n is given by the shifted power method

( 1) ( )

1

1k kshifted

kAm

+

+

= , k= 0, 1, 2,

where 1+km is the maximum absolute value of( ) ,kshiftedA v

1 ,shiftedA A I =

1 is the largest eigenvalue of the matrix .A

The iteration is iterated with a given initial eigenvector, )0(v until || 1 kk mm + || n

and its corresponding eigenvector, ,n is given by the inverse power method

( 1) 1 ( )

1

1k k

k

Am

+

+

= , k= 0, 1, 2,

where 1+km is the maximum absolute value of1 ( )kA v .

The iteration is iterated with a given initial eigenvector, )0(v until || 1 kk mm +