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10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES
In Chapter 7 we saw that the eigenvalues of an matrix A are obtained by solving its
characteristic equation
For large values of n, polynomial equations like this one are difficult and time-consuming
to solve. Moreover, numerical techniques for approximating roots of polynomial equations
of high degree are sensitive to rounding errors. In this section we look at an alternative
method for approximating eigenvalues. As presented here, the method can be used only to
find the eigenvalue of A that is largest in absolute value—we call this eigenvalue the
dominant eigenvalue of A. Although this restriction may seem severe, dominant eigenval-
ues are of primary interest in many physical applications.
Not every matrix has a dominant eigenvalue. For instance, the matrix
(with eigenvalues of and ) has no dominant eigenvalue. Similarly, the
matrix
(with eigenvalues of and ) has no dominant eigenvalue.
EXAMPLE 1 Finding a Dominant Eigenvalue
Find the dominant eigenvalue and corresponding eigenvectors of the matrix
Solution From Example 4 of Section 7.1 we know that the characteristic polynomial of A is
Therefore the eigenvalues of A are and
of which the dominant one is From the same example we know that
the dominant eigenvectors of A (those corresponding to ) are of the form
t Þ 0.x 5 t
3
3
1
4
,
l
2
522
l
2
522.
l
2
522,
l
1
521
l
2
1 3
l
1 2 5 (
l
1 1)(
l
1 2).
A 5
3
2
1
212
25
4
.
l
3
5 1
l
1
5 2,
l
2
5 2,
A 5
3
2
0
0
0
2
0
0
0
1
4
l
2
521
l
1
5 1
A 5
3
1
0
0
21
4
l
n
1 c
n21
l
n21
1 c
n22
l
n22
1
. . .
1 c
0
5 0.
n 3 n
550 CHAPTER 10 NUMERICAL METHODS
Definition of Dominant
Eigenvalue and
Dominant Eigenvector
Let and be the eigenvalues of an matrix A. is called the
dominant eigenvalue of A if
The eigenvectors corresponding to are called dominant eigenvectors of A.
l
1
i 5 2, . . . , n.
|
l
1
|
>
|
l
i
|
,
l
1
n 3 n
l
n
l
1
,
l
2
, . . . ,