Page 848 - Advanced_Engineering_Mathematics o'neil

828 Answers to Selected Problems

13. Not hermitian, not skew-hermitian, not unitary; 2 is a repeated eigenvalue and all eigenvectors are scalar multiples of

i
1 .
This matrix is not diagonalizable.
15. Skew-hermitian;
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1
2 √ √ 1 √ √
3i − 3i
0, ⎝ 0 ⎠ ; 3i, ⎝ ⎠ ;− 3i, ⎝ ⎠
1 + i −1 − i −1 − i
The matrix is diagonalized by
⎛ ⎞
2 1 1
√ √
0 3i − 3i
P = ⎝ ⎠
1 + i −1 − i −1 − i
17. Hermitian; approximate eigenvalues and eigenvectors are
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 1 1
4.051374, ⎝ 0.525687 ⎠ ;0.482696, ⎝ −1.258652 ⎠ ;−1.53407, ⎝ −2.267035 ⎠
−0.129755i 2.607546i −1.477791i
Use these eigenvectors as columns of p.
19. skew-hermitian; approximate eigenvalues and eigenvectors are
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
−i i i
−2.164248i, ⎝ −3.164248 ⎠ ;0.772866i, ⎝ 0.227134 ⎠ ; 2.391382i, ⎝ −1.391382 ⎠
2.924109 0.587771 −1.163664
Use these eigenvectors as columns of P.
21. Hermitian,
⎛ ⎞ ⎛ √ ⎞ ⎛ √ ⎞
0 √ 4 + 3 2 √ 4 − 3 2
−1 ⎠ ;4 − 3 2, ⎝ −1 ⎠ ,
0, ⎝ i ⎠ ;4 + 3 2, ⎝
1 −i −i
use these eigenvectors as columns of P.
23.

4 −6
A =
−6 1
and the standard form is
√ √

5 + 153 2 5 − 153 2
y + y
2 1 2 2
25.

4 −2
A =
−2 1
and the standard form is
√ √ 2

3 + 17 2 3 − 17 2
y + y
2 1 2 2
27.

5 2
A =
2 2
2
and the standard form is y + 6y 2
1 2

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October 14, 2010 17:50 THM/NEIL Page-828 27410_25_Ans_p801-866