Page 333 - Matrix Analysis & Applied Linear Algebra

5.6 Unitary and Orthogonal Matrices 329
z z

v
π/4
π/2
y y
x View (a) v x View (b)
z
z

π/3

y
v y
x x
v
View (c) View (d)
Figure 5.6.5
Problem: If the coordinates of each vertex in View (a) are specified, what are
the coordinates of each vertex in View (d)?
Solution: If P x is the rotator that maps points in View (a) to corresponding
points in View (b), and if P y and P z are the respective rotators carrying View
(b) to View (c) and View (c) to View (d), then
√
     
10 0 1 1 √ 0 1 √ 1/2 − 3/2 0
P x =  00 −1 , P y = √  0 20 , P z =  3/2 1/2 0 ,



2
01 0 −1 0 1 0 0 1
so
 √ 
1 1 6
1 √ √ √
P = P z P y P x = √  3 3 − 2  (5.6.15)
2 2
−2 2 0
is the orthogonal matrix that maps points in View (a) to their corresponding
images in View (d). For example, focus on the vertex labeled v in View (a), and
let v a , v b , v c , and v d denote its respective coordinates in Views (a), (b), (c),
T T
and (d). If v a =( 110 ) , then v b = P x v a =( 101 ) ,
 √
 √  
2 2/2
√
v c = P y v b = P y P x v a =  0  , and v d = P z v c = P z P y P x v a =  6/2 .

0 0