Page 394 - Matrix Analysis & Applied Linear Algebra

390 Chapter 5 Norms, Inner Products, and Orthogonality

Figure 5.9.2 that
x 1 1
sin θ = 2 = = . (5.9.18)
v v P
2 2 2
A little reflection on the geometry associated with Figure 5.9.2 should convince
3
you that in anumber θ satisfies (5.9.16) if and only if θ satisfies (5.9.18)—a
n
completely rigorous proof validating this fact in is given in §5.15.
√
Note: Recall from p. 281 that P = λ max , where λ max is the largest
2
T
number λ such that P P − λI is a singular matrix. Consequently,
1 1
sin θ = = √ .
P λ max
2
T
T
Numbers λ such that P P − λI is singular are called eigenvalues of P P
(they are the main topic of discussion in Chapter 7, p. 489), and the numbers
√
λ are the singular values of P discussed on p. 411.
Exercises for section 5.9

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5.9.1. Let X and Y be subspaces of whose respective bases are
     
1 1 1
   
1
2
,
2
B X =     and B Y =   .
   
1 2 3
3
(a) Explain why X and Y are complementary subspaces of .
(b) Determine the projector P onto X along Y as well as the
complementary projector Q onto Y along X.

2
(c) Determine the projection of v = −1 onto Y along X.
1
(d) Verify that P and Q are both idempotent.
(e) Verify that R (P)= X = N (Q) and N (P)= Y = R (Q).
5.9.2. Construct an example of a pair of nontrivial complementary subspaces
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of , and explain why your example is valid.

5.9.3. Construct an example to show that if V = X + Y but X∩ Y = 0, then
avector v ∈V can have two different representations as

and v = x 2 + y 2 ,
v = x 1 + y 1
where x 1 , x 2 ∈X and y 1 , y 2 ∈Y, but x 1 = x 2 and y 1 = y 2 .