Page 328 - Matrix Analysis & Applied Linear Algebra

324 Chapter 5 Norms, Inner Products, and Orthogonality

There is nothing special about the numbers in this example. For every nonzero
n×1 ⊥
vector u ∈C , the orthogonal projectors onto span {u} and u are
uu ∗ uu ∗
P u = and P u ⊥ = I − . (5.6.6)
u u u u
∗
∗
Elementary Reflectors
For u n×1 = 0, the elementary reflector about u ⊥ is defined to be
uu ∗
R = I − 2 (5.6.7)
u u
∗
or, equivalently,
R = I − 2uu ∗ when u =1. (5.6.8)

46
Elementary reflectors are also called Householder transformations, and
they are analogous to the simple reflector given in Example 4.7.1. To understand
3×1 T
why, suppose u ∈ and u =1 so that Q = I − uu is the orthogonal
⊥ 3×1
projector onto the plane u . For each x ∈ , Qx is the orthogonal pro-
T
jection of x onto u ⊥ as shown in Figure 5.6.1. To locate Rx =(I − 2uu )x,
notice that Q(Rx)= Qx. In other words, Qx is simultaneously the orthogo-
nal projection of x onto u ⊥ as well as the orthogonal projection of Rx onto
T
u . This together with x − Qx = |u x| = Qx − Rx implies that Rx
⊥
is the reflection of x about the plane u , exactly as depicted in Figure 5.6.2.
⊥
(Reflections about more general subspaces are examined in Exercise 5.13.21.)
u u ⊥

x
|| x - Qx ||
Qx
0 || Qx - Rx ||
Rx
Figure 5.6.2
46
Alston Scott Householder (1904–1993) was one of the first people to appreciate and promote
the use of elementary reflectors for numerical applications. Although his 1937 Ph.D. disserta-
tion at University of Chicago concerned the calculus of variations, Householder’s passion was
mathematical biology, and this was the thrust of his career until it was derailed by the war
effort in 1944. Householder joined the Mathematics Division of Oak Ridge National Labora-
tory in 1946 and became its director in 1948. He stayed at Oak Ridge for the remainder of his
career, and he became a leading figure in numerical analysis and matrix computations. Like
his counterpart J. Wallace Givens (p. 333) at the Argonne National Laboratory, Householder
was one of the early presidents of SIAM.