Page 462 - Matrix Analysis & Applied Linear Algebra

458 Chapter 5 Norms, Inner Products, and Orthogonality

5.15.8. Prove that if f : V→ is a function defined on a space V such that
f(αx)= αf(x) for scalars α ≥ 0, then

max f(x)= max f(x).
x =1 x ≤1

n
5.15.9. Let M and N be nonzero complementary subspaces of .

†
(a) Explain why P MN = (I − P N )P M , where P M and P N
are the orthogonal projectors onto M and N, respectively,
and P MN is the oblique projector onto M along N.
(b) If θ min is the minimal angle between M and N, explain why

−1

−1

†

†
sin θ min =
(I − P N )P M
=
P M (I − P N )
2 2

−1

−1

†

†
=
(I − P M )P N
=
P N (I − P M )
.
2 2
n
5.15.10. For complementary subspaces M, N⊂ , let θ min be the minimal
¯
angle between M and N, and let θ min denote the minimal angle be-
⊥
tween M and N .
(a) If P MN is the oblique projector onto M along N, prove that

¯

cos θ min =
P †
.

MN
2
¯
(b) Explain why sin θ min ≤ cos θ min .

be the orthogonal matrices
5.15.11. Let U = U 1 | U 2 and V = V 1 | V 2
defined on p. 451.
T
(a) Prove that if U V 2 is nonsingular, then
2
1
T
2

=1 − U V 1
.

T 2 2 2

(U V 2 ) −1
2 2
T
(b) Prove that if U V 1 is nonsingular, then
2

T
2 1

.
U V 2 2 =1 −
2
2
T

(U V 1 ) −1
2 2