Upload
farah-sofhia-mohd-zin
View
227
Download
0
Embed Size (px)
Citation preview
8/3/2019 Chap4 25oct11 Edited
1/20
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
65
8/3/2019 Chap4 25oct11 Edited
2/20
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 +