Page 456 - Matrix Analysis & Applied Linear Algebra

452 Chapter 5 Norms, Inner Products, and Orthogonality

T
T
)
where U V 1 =(C −1 T is nonsingular. To see that U V 2 is also nonsingular,
2
1
T
suppose dim M = r so that dim N = n − r and U V 2 is n − r × n − r. Use
2
the formula for the rank of a product (4.5.1) to write
T T T
rank U V 2 = rank U 2 −dim N U 2 ∩R (V 2 )= n−r−dim M∩N = n−r.
2
It now follows from (5.15.7) that P M − P N is nonsingular, and
T −1
(U V 1 ) 0
T
−1
V (P M − P N ) U = 1 T −1 .
0 −(U V 2 )
2
n
(Showing that P M − P N is nonsingular implies M⊕N = is Exercise
5.15.6.) Formula (5.2.12) on p. 283 for the 2-norm of a block-diagonal matrix
can now be applied to yield
5 6

T

T

(P M − P N ) −1
= max
(U V 1 ) −1
, (U V 2 ) −1
. (5.15.8)

2 1 2 2 2

T

T
But
(U V 1 ) −1
2 = (U V 2 ) −1
2 because we can again use (5.2.6) to write

2
1
1
T
2 T T T

= min
U V 1 x
= min x V U 1 U V 1 x

T 2 1 2 1 1
2

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

T
2
T
2
=1 − max
U V 1 x
=1 − U V 1
.

2
2
x =1 2 2
2

2

2
T
T
By a similar argument, 1/ (U V 2 ) −1
2 =1− U V 1
2 (Exercise 5.15.11(a)).

2
2
Therefore,

T

(P M − P N ) −1
= (U V 1 ) −1
= C T
= C = P MN .

2 1 2 2 2 2
While the minimal angle works fine for complementary spaces, it may not
convey much information about the separation between noncomplementary sub-
spaces. For example, θ min =0 whenever M and N have a nontrivial inter-
section, but there nevertheless might be a nontrivial “gap” between M and
N —look at Figure 5.15.1. Rather than thinking about angles to measure such a
gap, consider orthogonal distances as discussed in (5.13.13). Define
δ(M, N)= max dist (m, N)= max (I − P N )m 2
m∈M m∈M
m 2 =1 m 2 =1