Page 339 - Matrix Analysis & Applied Linear Algebra

5.6 Unitary and Orthogonal Matrices 335
Example 5.6.6
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Problem: If x ∈ is a vector such that x =1, explain how to use plane
n
rotations to construct an orthonormal basis for that contains x.
Solution: This is almost the same problem as that posed in Example 5.6.3, and,
as explained there, the goal is to construct an orthogonal matrix Q such that
Q ∗1 = x. But this time we need to use plane rotations rather than an elementary
reflector. Equation (5.6.17) asserts that we can build an orthogonal matrix from
a sequence of plane rotations P = P 1n ··· P 13 P 12 such that Px = e 1 . Thus
T
T
x = P e 1 = P , and hence the columns of Q = P T serve the purpose. For
∗1
example, to extend
 
−1
x = 1  2 


3 0
−2
4
to an orthonormal basis for , sequentially annihilate the second and fourth
components of x by using (5.6.16) to construct the following plane rotations:
√ √ √
−1/ 5 2/ 500 −1 5
     
√ √
 −2/ 5 −1/ 500  1  2  1  0 
0 0 1 0 3 0 3 0
P 12 x =     =   ,
0 0 0 1 −2 −2
√ √
5/300 −2/3 5 1
     
0 1 0 0
  1  0   0 
P 14 P 12 x =     =   .
0 0 1 0 3 0 0
√
2/3 0 0 5/3 −2 0
Therefore, the columns of
√ √
−1/3 −2/ 50 −2/3 5
 
√ √
T
T
Q =(P 14 P 12 ) = P P T  2/3 −1/ 50 4/3 5 
12 14 =  0 0 1 0 
√
−2/3 0 0 5/3
are an orthonormal set containing the specified vector x.
Exercises for section 5.6
5.6.1. Determine which of the following matrices are isometries.
√ √
   
1/ 2 −1/ 2 0 10 1
√ √ √
(a)  1/ 6 1/ 6 −2/ 6  . (b)  10 −1  .
√ √ √
1/ 3 1/ 3 1/ 3 01 0
 
  e iθ 1 0 0
0010 ···
 0 e iθ 2 ··· 0 
(c)  1000  (d)  . . . .  .
 .

0001  . . . . . . . . 
0100 0 0 ··· e iθ n