Page 313 - Advanced engineering mathematics

9.3 Some Special Types of Matrices 293

This transforms the given quadratic form to its standard form
5 9
2
2
2
2
λ 1 y + λ 2 y =− y + y ,
1 2 1 2
2 2
in which there are no cross product y 1 y 2 terms.
SECTION 9.3 PROBLEMS

⎛ ⎞
In each of Problems 1 through 12, find the eigenvalues 5 0 0 0
and associated eigenvectors. Check that the eigenvectors 12. ⎜ 0 0 −1 0 ⎟
⎜
⎟
associated with distinct eigenvalues are orthogonal. Find ⎝ 0 −1 0 0 ⎠
an orthogonal matrix that diagonalizes the matrix. Note 0 0 0 0
Problems 17-22, Section 9.1.
In each of Problems 13 through 21, determine whether
the matrix is unitary, hermitian, skew-hermitian, or none

4 −2 of these. Find the eigenvalues and associated eigenvec-
1.
−2 1 tors. If the matrix is diagonalizable, write a matrix that
diagonalizes it. In Problems 5 and 7, eigenvalues must be

−3 5
2. approximated, so only “approximate eigenvectors" can be
5 4
found. It is instructive to try to diagonalize a matrix using
approximate eigenvectors.
6 1
3.
1 4

0 2i
13.
−13 1 2i 4
4.
1 4

3 4i
⎛ ⎞ 14.
0 1 0 4i −5
5. ⎝ 1 −2 0 ⎠
⎛ ⎞
0 0 3 0 1 0
15. ⎝ −1 0 1 − i ⎠
0 1 1 0 −1 − i 0
⎛ ⎞
6. ⎝ 1 2 0 ⎠ √ √
⎛ ⎞
0 0 3 1/ 2 i/ 2 0
√ √
−1/ 2 i/ 2 0 ⎠
16. ⎝
5 0 2 0 0 1
⎛ ⎞
7. ⎝ 0 0 0 ⎠
2 0 0 ⎛ 3 2 0 ⎞
17. ⎝ 2 0 i ⎠
2 −4 0 0 −i 0
⎛ ⎞
8. ⎝ −4 0 0 ⎠
⎛ ⎞
0 0 0 −1 0 3 − i
18. ⎝ 0 1
⎛ ⎞ 0 ⎠
0 0 0 3 + i 0 0
9. ⎝ 1 1 −2 ⎠
⎛ ⎞
0 −2 0 i 1 0
19. ⎝ −1 0
⎛ ⎞ 2i ⎠
1 3 0 0 2i 0
10. ⎝ 3 0 1 ⎠
⎛ ⎞
0 1 1 3i 0 0
20. ⎝ −1 0
⎛ ⎞ 0 ⎠
0 0 0 0 −i 0 0
⎜ 0 1 −2 0 ⎟
⎜ ⎟
⎛ ⎞
11. ⎜ 0 −2 1 0 ⎟ 8 −1 i
⎜ ⎟
⎝ 0 −3 0 0 ⎠ 21. ⎝ −1 0 0 ⎠
0 0 0 0 −i 0 0
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
October 14, 2010 14:49 THM/NEIL Page-293 27410_09_ch09_p267-294