Page 326 - Matrix Analysis & Applied Linear Algebra

322 Chapter 5 Norms, Inner Products, and Orthogonality

The geometrical concepts of projection, reflection, and rotation are among
3
2
the most fundamental of all linear transformations in and (see Example
4.7.1 for three simple examples), so pursuing these ideas in higher dimensions
is only natural. The reflector and rotator given in Example 4.7.1 are isometries
(because they preserve length), but the projector is not. We are about to see
that the same is true in more general settings.

Elementary Orthogonal Projectors

n×1
Foravector u ∈C such that u =1, a matrix of the form
Q = I − uu ∗ (5.6.2)

is called an elementary orthogonal projector. More general projec-
tors are discussed on pp. 386 and 429.

To understand the nature of elementary projectors consider the situation in
3
. Suppose that u 3×1 =1, and let u ⊥ denote the space (the plane through
the origin) consisting of all vectors that are perpendicular to u —we call u ⊥ the
orthogonal complement of u (a more general definition appears on p. 403).
The matrix Q = I−uu T is the orthogonal projector onto u ⊥ in the sense that
3×1
Q maps each x ∈ to its orthogonal projection in u ⊥ as shown in Figure
5.6.1.
u u ⊥

T
(I - Q)x = uu x
x

T
Qx = (I - uu )x
0
Figure 5.6.1
To see this, observe that each x can be resolved into two components

x =(I − Q)x + Qx, where (I − Q)x ⊥ Qx.
T
The vector (I − Q)x = u(u x)ison the line determined by u, and Qx is in
T
the plane u ⊥ because u Qx = 0.