Page 454 - Matrix Analysis & Applied Linear Algebra

450 Chapter 5 Norms, Inner Products, and Orthogonality
5.15 ANGLES BETWEEN SUBSPACES

Consider the problem of somehow gauging the separation between a pair of
n
nontrivial but otherwise general subspaces M and N of . Perhaps the first
thing that comes to mind is to measure the angle between them. But defining the
n
“angle” between subspaces in is not as straightforward as the visual geometry
2 3
of or might suggest. There is just too much “wiggle room” in higher
dimensions to make any one definition completely satisfying, and the “correct”
definition usually varies with the specific application under consideration.
Before exploring general angles, recall what has already been said about
some special cases beginning with the angle between a pair of one-dimensional
subspaces. If M and N are spanned by vectors u and v, respectively, and if
u =1= v , then the angle between M and N is defined by the expression
T
cos θ = v u (p. 295). This idea was carried one step further on p. 389 to define
the angle between two complementary subspaces, and an intuitive connection to
norms of projectors was presented. These intuitive ideas are now made rigorous.

Minimal Angle

n
The minimal angle between nonzero subspaces M, N⊆ is defined
to be the number 0 ≤ θ min ≤ π/2 for which
T
cos θ min = max v u. (5.15.1)
u∈M, v∈N
u = v =1
2 2
• If P M and P N are the orthogonal projectors onto M and N,
respectively, then
cos θ min = P N P M . (5.15.2)
2
• If M and N are complementary subspaces, and if P MN is the
oblique projector onto M along N, then
1
sin θ min = . (5.15.3)
P MN
2
• M and N are complementary subspaces if and only if P M − P N
is invertible, and in this case
1
sin θ min = . (5.15.4)
(P M − P N ) −1
2

Proof of (5.15.2). If f : V→ is a function defined on a space V such that
f(αx)= αf(x) for all scalars α ≥ 0, then
max f(x)= max f(x) (see Exercise 5.15.8). (5.15.5)
x =1 x ≤1