Page 457 - Matrix Analysis & Applied Linear Algebra

5.15 Angles between Subspaces 453

to be the directed distance from M to N, and notice that δ(M, N) ≤ 1
because (5.2.5) and (5.13.10) can be combined to produce
dist (m, N)= (I − P N )m = P N ⊥m ≤ P N ⊥ m =1.
2 2 2 2
3
Figure 5.15.1 illustrates δ(M, N) for two planes in .

m M
N

δ(M, N)= max dist (m, N)
m∈M
m 2 =1
Figure 5.15.1
This picture is a bit misleading because δ(M, N)= δ(N, M) for this particular
situation. However, δ(M, N) and δ(N, M) need not always agree—that’s why
3
the phrase directed distance is used. For example, if M is the xy-plane in
√
and N = span {(0, 1, 1)} , then δ(N, M)=1/ 2 while δ(M, N)=1. Con-
sequently, using orthogonal distance to gauge the degree of maximal separation
between an arbitrary pair of subspaces requires that both values of δ be taken
into account. Hence we make the following definition.

Gap Between Subspaces
n
The gap between subspaces M, N⊆ is defined to be

gap (M, N)= max δ(M, N),δ(N, M) , (5.15.9)

where δ(M, N)= max dist (m, N).
m∈M
m 2 =1

Evaluating the gap between a given pair of subspaces requires knowing some
properties of directed distance. Observe that (5.15.5) together with the fact that
T
A 2 = A can be used to write
2
δ(M, N)= max dist (m, N)= max (I − P N )m 2
m∈M m∈M
m 2 =1 m 2 =1
= max (I − P N )m = max (I − P N )P M x 2 (5.15.10)
2
m∈M x =1
m 2 ≤1 2
= (I − P N )P M = P M (I − P N ) .
2 2