Page 343 - Matrix Analysis & Applied Linear Algebra

5.6 Unitary and Orthogonal Matrices 339

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5.6.17. Perform the following sequence of rotations in beginning with
 
1
v 0 =  1  .
−1

1. Rotate v 0 counterclockwise 45 around the x-axis to produce v 1 .
◦
◦
2. Rotate v 1 clockwise 90 around the y-axis to produce v 2 .
3. Rotate v 2 counterclockwise 30 around the z-axis to produce v 3 .
◦
Determine the coordinates of v 3 as well as an orthogonal matrix Q
such that Qv 0 = v 3 .

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5.6.18. Does it matter in what order rotations in are performed? For ex-
3
ample, suppose that a vector v ∈ is first rotated counterclockwise
around the x-axis through an angle θ, and then that vector is rotated
counterclockwise around the y-axis through an angle φ. Is the result
the same as first rotating v counterclockwise around the y-axis through
an angle φ followed by a rotation counterclockwise around the x-axis
through an angle θ?

n
⊥
5.6.19. For each nonzero vector u ∈C , prove that dim u = n − 1.
2
5.6.20. A matrix satisfying A = I is said to be an involution or an involu-
2
tory matrix, and a matrix P satisfying P = P is called a projector
or is said to be an idempotent matrix—properties of such matrices
are developed on p. 386. Show that there is a one-to-one correspondence
n×n
between the set of involutions and the set of projectors in C . Hint:
Consider the relationship between the projectors in (5.6.6) and the re-
flectors (which are involutions) in (5.6.7) on p. 324.

5.6.21. When using a computer to generate and display a three-dimensional
convex polytope such as the one in Example 5.6.4, it is desirable to not
draw those faces that should be hidden from the perspective of a viewer
positioned as shown in Figure 5.6.6. The operation of cross product in
3
(usually introduced in elementary calculus courses) can be used to
decide which faces are visible and which are not. Recall that if
     
u 1 v 1 u 2 v 3 − u 3 v 2
u =  u 2  and v =  v 2  , then u × v =  u 3 v 1 − u 1 v 3  ,
u 3 v 3 u 1 v 2 − u 2 v 1