Page 393 - Matrix Analysis & Applied Linear Algebra

5.9 Complementary Subspaces 389
Example 5.9.2

Angle between Complementary Subspaces. The angle between nonzero
n
vectors u and v in was defined on p. 295 to be the number 0 ≤ θ ≤
T
π/2 such that cos θ = v u/ v u . It’s natural to try to extend this idea
2 2
n
to somehow make sense of angles between subspaces of . Angles between
completely general subspaces are presently out of our reach—they are discussed
in §5.15—but the angle between a pair of complementary subspaces is within
n
our grasp. When = R⊕N with R = 0 = N, the angle (also known as the
minimal angle) between R and N is defined to be the number 0 <θ ≤ π/2
that satisfies
T
v u T
cos θ = max = max v u. (5.9.16)
u∈R v u 2 u∈R, v∈N
2
v∈N u = v =1
2 2
While this is a good definition,it’s not easy to use—especially if one wants to
compute the numerical value of cos θ. The trick in making θ more accessible
2
is to think in terms of projections and sin θ =(1 − cos θ) 1/2 . Let P be the
projector such that R (P)= R and N (P)= N, and recall that the matrix
2-norm (p. 281) of P is
P = max Px . (5.9.17)
2 2
x =1
2
In other words, P is the length of a longest vector in the image of the unit
2
sphere under transformation by P. To understand how sin θ is related to P ,
2
3
consider the situation in . The image of the unit sphere under P is obtained
by projecting the sphere onto R along lines parallel to N. As depicted in
Figure 5.9.2,the result is an ellipse in R.

P
v
= max Px =
x =1
x
θ θ
v

Figure 5.9.2
The norm of a longest vector v on this ellipse equals the norm of P. That is,
v = max x =1 Px = P , and it is apparent from the right triangle in
2 2 2 2