Page 435 - Matrix Analysis & Applied Linear Algebra

5.13 Orthogonal Projection 431
Example 5.13.1

Problem: Let u n×1 = 0, and consider the line L = span {u} . Construct the
orthogonal projector onto L, and then determine the orthogonal projection of
avector x n×1 onto L.
Solution: The vector u by itself is a basis for L, so, according to (5.13.3),

T −1 T uu T
P L = u u u u =
T
u u
is the orthogonal projector onto L. The orthogonal projection of a vector x
onto L is therefore given by
T
uu T u x
P L x = x = u.
T
T
u u u u
T
T
T
Note: If u =1, then P L = uu , so P L x = uu x =(u x)u, and
2
T
T
P L x = |u x| u = |u x|.
2 2
This yields a geometrical interpretation for the magnitude of the standard inner
product. It says that if u is a vector of unit length in L, then, as illustrated
T
in Figure 5.13.2, |u x| is the length of the orthogonal projection of x onto the
line spanned by u.
x
L
P L x
u
0 T x|
|u

Figure 5.13.2
Finally, notice that since P L = uu T is the orthogonal projector onto L, it must
be the case that P L ⊥ = I − P L = I − uu T is the orthogonal projection onto
⊥
L . This was called an elementary orthogonal projector on p. 322—go back
and reexamine Figure 5.6.1.
Example 5.13.2
m
Volume, Gram–Schmidt, and QR. A solid in with parallel opposing
faces whose adjacent sides are defined by vectors from a linearly independent set
{x 1 , x 2 ,..., x n } is called an n-dimensional parallelepiped.As shown in the
shaded portions of Figure 5.13.3, a two-dimensional parallelepiped is a parallel-
ogram, and a three-dimensional parallelepiped is a skewed rectangular box.