Page 461 - Matrix Analysis & Applied Linear Algebra

5.15 Angles between Subspaces 457
Exercises for section 5.15

5.15.1. Determine the angles θ min and θ max between the following subspaces
3
of .
(a) M = xy-plane, N = span {(1, 0, 0), (0, 1, 1)} .
(b) M = xy-plane, N = span {(0, 1, 1)} .

3
5.15.2. Determine the principal angles between the following subspaces of .
(a) M = xy-plane, N = span {(1, 0, 0), (0, 1, 1)} .
(b) M = xy-plane, N = span {(0, 1, 1)} .

n
5.15.3. Let θ min be the minimal angle between nonzero subspaces M, N⊆ .
(a) Explain why θ max =0 if and only if M = N.
(b) Explain why θ min =0 if and only if M∩N = 0.
(c) Explain why θ min = π/2if and only if M⊥N.

n
5.15.4. Let θ min be the minimal angle between nonzero subspaces M, N⊂ ,
⊥
and let θ ⊥ denote the minimal angle between M ⊥ and N . Prove
min
n
that if M⊕N = , then θ min = θ ⊥ .
min
˜
n
5.15.5. For nonzero subspaces M, N⊂ , let θ min denote the minimal angle
⊥
between M and N , and let θ max be the maximal angle between M
⊥ n ˜
and N. Prove that if M⊕N = , then cos θ min = sin θ max .
n
5.15.6. For subspaces M, N⊆ , prove that P M −P N is nonsingular if and
only if M and N are complementary.

n
5.15.7. For complementary spaces M, N⊂ , let P = P MN be the oblique
projector onto M along N, and let Q = P M ⊥ N ⊥ be the oblique
⊥ ⊥
projector onto M along N .
(a) Prove that (P M − P N ) −1 = P − Q.
(b) If θ min is the minimal angle between M and N, explain why

1
sin θ min = .
P − Q
2
(c) Explain why P − Q = P .
2 2