Page 847 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 827

1 0
18
13. A = 18 18
(1 − 5 )/4 5

0 2 22
43
15. A = 21
2 0
Section 9.3 Some Special Types of Matrices
In Problems 1 through 12, Q is an orthogonal matrix that diagonalizes the given matrix.
√
√
1 −2 1/ 5 −2/ 5
1. 0, ;5, ;Q = √ √
2 1 2/ 5 1/ 5
√ √
√ 1 + 2 √ 1 − 2
3. 5 + 2, ;5 − 2,
1 1
√ √ √ √
(1 + 2)/( 4 + 2 2) (1 − 2)/( 4 − 2 2)
Q = √ √
1/ 4 + 2 2 1/ 4 − 2 2
⎛ ⎞ ⎛ √ ⎞ ⎛ √ ⎞
0 1 + 2 1 − 2
√ √
5. 3, ⎝ 0 ⎠ ;−1 + 2, ⎝ 1 ⎠ ;−1 − 2, ⎝ 1 ⎠
1 0 0
√ √ √
⎛ √ ⎞
0 (1 + 2)/ 4 + 2 2 (1 − 2)/ 4 − 2 2
√ √

Q = ⎝ 0 1/ 4 + 2 2 1/ 4 − 2 2 ⎠
1 0 0
√ √
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
0 5 + 41 5 − 41
√ √
7. 0, ⎝ 1 ⎠ ;(5 + 41)/2, ⎝ 0 ⎠ ;(5 − 41)/2, ⎝ 0 ⎠
0 4 4
√ √ √ √
⎛ ⎞
0 (5 + 41)/ 82 + 10 41 (5 − 41)/ 82 − 10 41
Q = ⎝ 1 0 0 ⎠
√ √

0 4/ 82 + 10 41 4/ 82 − 10 41
⎛ ⎞ ⎛ ⎞
⎛ ⎞
1 0 0
√ √ √ √
9. 0, ⎝ 0 ⎠ ;(1 + 17)/2, ⎝ −1 − 17 ;(1 − 17)/2, ⎝ −1 − 17 ⎠
⎠
0 4 4
1 0 0
⎛ ⎞
√ √ √ √
Q = ⎝ 0 (−1 − 17)/ 34 + 2 17 (−1 + 17)/ 34 + 2 17 ⎠
√ √

0 4/ 34 + 2 17 4/ 34 + 2 17
11. 0 is an eigenvalue of multiplicity 2, with independent eigenvectors
1 0
⎛ ⎞ ⎛ ⎞
⎜0⎟ ⎜0⎟
0 0
⎝ ⎠ , ⎝ ⎠
0 1
0 0
⎛ ⎞ ⎛ ⎞
⎜1⎟ ⎜−1⎟
1 1 ⎠
−1, ⎝ ⎠ ;3, ⎝
0 0
⎛ ⎞
1 0 0 0
√ √
⎜ 0 0 1/ 2 −1/ 2 ⎟
Q = ⎜ √ √ ⎟
⎝ 0 0 1/ 2 1/ 2 ⎠
0 1 0 0
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October 14, 2010 17:50 THM/NEIL Page-827 27410_25_Ans_p801-866