Page 459 - Matrix Analysis & Applied Linear Algebra

5.15 Angles between Subspaces 455

⊥ ⊥ ⊥
When dim M∩N = dim M ∩N > 0, there are vectors x ∈M ∩N
⊥
and y ∈M ∩ N such that x =1= y . Hence, (I − P M )P N x =
2 2 2
x =1, and (I − P N )P M y = y =1, so
2 2 2
δ(N, M)= (I − P M )P N =1= (I − P N )P M = δ(M, N).
2 2
T
⊥ ⊥
Proof of (5.15.15). If dim M∩N = dim M ∩N =0, then U V 1 is
2
nonsingular because it is r × r and has rank r—apply the formula (4.5.1) for
the rank of a product. From (5.15.11) we have
2
T
2
T
2
T
2

1
1
2
δ (M, N)= U V 1
2 = U 1 U V 1
2 = (I − U 2 U )V 1
2

2
T
T
T
T
= max x V (I − U 2 U )V 1 x = max 1 − U V 1 x

1
2
2
x =1 x =1 2
2 2

T
2 1
=1 − min
U V 1 x
=1 −

< 1 (recall (5.2.6)).
2
2
T
x =1 2
(U V 1 ) −1
2
2 2

2

2
2 T T −1

A similar argument shows δ (N, M)= U V 2
2 =1 − 1/ (U V 1 ) 2 (Ex-
2
2
ercise 5.15.11(b)), so δ(N, M)= δ(M, N) < 1.
Because 0 ≤ gap (M, N) ≤ 1, the gap measure defines another angle be-
tween M and N.
Maximal Angle
n
The maximal angle between subspaces M, N⊆ is defined to be
the number 0 ≤ θ max ≤ π/2 for which
sin θ max = gap (M, N)= P M − P N . (5.15.16)
2
For applications requiring knowledge of the degree of separation between
a pair of nontrivial complementary subspaces, the minimal angle does the job.
Similarly, the maximal angle adequately handles the task for subspaces of equal
dimension. However, neither the minimal nor maximal angle may be of much
help for more general subspaces. For example, if M and N are subspaces
of unequal dimension that have a nontrivial intersection, then θ min =0 and
θ max = π/2, but neither of these numbers might convey the desired information.
Consequently, it seems natural to try to formulate definitions of “intermediate”
angles between θ min and θ max . There are a host of such angles known as the
principal or canonical angles, and they are derived as follows.