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SECTION 10.3
❑
SECTION 10.3
0
2
1
4
1
3. A 3
2
5. A 0
0
5
1
2. A 3
1
0
3
1
4
4. A 2
1
2
3
6. A 5
3
5
3
4
5
5
,x
3
2
4
2
1
9. A 2
6
3
10. A 3
1
8. A 1
19. A 3
0
0
7
2
0
0
3
3
3
,x
4
1
2
1
0
13. A 2
2
1
7
1
4
8
12. A 1
1
0
6
14. A 2
3
1
6
17. A 1
2
1
0
1
2
6
7
2
0
0
8
0
0
1
A
2
5
6
2
2
3
1
.
4
2
3
(b) Calculate two iterations of the power method with scaling,
starting with x0 1, 1.
(c) Explain why the method does not seem to converge to a
dominant eigenvector.
A
21
3
0
0
0
1
1
2
0 .
2
1
16. A 0
0
2
7
0
0
1
0
0
18. A 0
2
6
4
1
0
0
1
12
5
has a dominant eigenvalue of 2. Observe that Ax x
implies that
A1x In Exercises 15–18, use the power method with scaling to approximate a dominant eigenvector of the matrix A. Start with
x0 1, 1, 1 and calculate four iterations. Then use x4 to approximate the dominant eigenvalue of A.
3
15. A 1
0
1
20. A 2
6
23. The matrix
In Exercises 11–14, use the power method with scaling to approximate a dominant eigenvector of the matrix A. Start with x0 1, 1
and calculate five iterations. Then use x5 to approximate the dominant eigenvalue of A.
11. A 0
0
2
21. Writing (a) Find the eigenvalues and corresponding eigenvectors of
A
3
3
9 ,x 0
5
1
2
4
2
1
1
0
22. Writing Repeat Exercise 21 using x0 1, 1, 1, for the
matrix
2
1
2 , x 1
3
3
2
5
6
In Exercises 19 and 20, the matrix A does not have a dominant
eigenvalue. Apply the power method with scaling, starting with
x0 1, 1, 1, and observe the results of the first four iterations.
0
3
In Exercises 7–10, use the Rayleigh quotient to compute the eigenvalue of A corresponding to the eigenvector x.
7. A 593
EXERCISES
In Exercises 1–6, use the techniques presented in Chapter 7 to find
the eigenvalues of the matrix A. If A has a dominant eigenvalue, find
a corresponding dominant eigenvector.
1. A EXERCISES
1
x.
Apply five iterations of the power method (with scaling) on
A1 to compute the eigenvalue of A with the smallest magnitude.
24. Repeat Exercise 23 for the matrix
2
A 0
0
3
1
0
1
2 .
3