Page 304 - Advanced engineering mathematics

284 CHAPTER 9 Eigenvalues, Diagonalization, and Special Matrices

⎛ ⎞
1 0 0 0 In each of Problems 12 through 15, use the idea of
0 4 1 0
⎜ ⎟ Problem 11 to compute the indicated power of the matrix.
9. ⎜ ⎟
⎝ 0 0 −3 1 ⎠
0 0 1 −2 −3 −3
12. A = ;A 16
⎛ ⎞ −2 4
−2 0 0 0
−4 −2 0 0
⎜ ⎟
10. ⎜ ⎟ −1 0 18
⎝ 0 0 −2 0 ⎠ 13. A = 1 −5 ;A
0 0 0 −2

11. Let A have eigenvalues λ 1 ,··· ,λ n , and suppose that P 14. A = −2 3 ;A 31
diagonalizes A. Show that, for any positive integer k, 3 −4
⎛ k ⎞
λ 0 ··· 0 0 2
1 15. A = ;A 43
⎜ 0 λ k ··· 0 ⎟ 1 0
−1
k
A = P⎜ . . 2 . . ⎟P .
⎜
⎟
⎝ . . . . . . . . ⎠ 16. Suppose A is diagonalizable. Prove that A is diago-
2
0 0 ··· λ n k nalizable.
9.3 Some Special Types of Matrices
In this section, we will discuss several types of matrices having special properties.
9.3.1 Orthogonal Matrices

An n × n matrix is orthogonal if its transpose is its inverse:
−1
t
A = A .
In this event,
t
t
AA = A A = I n .

For example, it is routine to check that
√
⎛ √ ⎞
01/ 5 2/ 5
A = 1 0 0 ⎠
⎝
√ √
0 2 5 −1 5
is orthogonal. Just multiply this matrix by its transpose to obtain I 3 .
t t
Because (A ) = A, a matrix is orthogonal exactly when its transpose is orthogonal.
It is also easy to verify that an orthogonal matrix must have determinant 1 or −1.

THEOREM 9.7
If A is orthogonal, then |A|=±1.

Proof Because a matrix and its transpose have the same determinant,
|I n |= 1 =|AA |=|AA |=|A||A |=|A| .
2
t
t
−1
The name “orthogonal matrix” derives from the following property.

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October 14, 2010 14:49 THM/NEIL Page-284 27410_09_ch09_p267-294