Page 329 - Matrix Analysis & Applied Linear Algebra

5.6 Unitary and Orthogonal Matrices 325

Properties of Elementary Reflectors

• All elementary reflectors R are unitary, hermitian, and involutory
2
( R = I ). That is,
R = R = R −1 . (5.6.9)
∗
• If x n×1 is a vector whose first entry is x 1 =0, and if

1 if x 1 is real,
u = x ± µ x e 1 , where µ = (5.6.10)
x 1 /|x 1 | if x 1 is not real,
is used to build the elementary reflector R in (5.6.7), then

Rx = ∓µ x e 1 . (5.6.11)

In other words, this R “reflects” x onto the first coordinate axis.
Computational Note: To avoid cancellation when using floating-
point arithmetic for real matrices, set u = x + sign(x 1 ) x e 1 .

∗
Proof of (5.6.9). It is clear that R = R , and the fact that R = R −1 is
2
established simply by verifying that R = I.
Proof of (5.6.10). Observe that R = I − 2ˆ uˆ u , where ˆ u = u/ u .
∗
Proof of (5.6.11). Write Rx = x − 2uu x/u u = x − (2u x/u u)u and verify
∗
∗
∗
∗
that 2u x = u u to conclude Rx = x − u = ∓µ x e 1 .
∗
∗
Example 5.6.3
n×1
Problem: Given x ∈C such that x =1, construct an orthonormal basis
n
for C that contains x.
Solution: An efficient solution is to build a unitary matrix that contains x as
∗
its first column. Set u = x±µe 1 in R = I−2(uu /u u) and notice that (5.6.11)
∗
guarantees Rx = ∓µe 1 , so multiplication on the left by R (remembering that
2
R = I) produces x = ∓µRe 1 =[∓µR] . Since |∓ µ| =1, U = ∓µR
∗1
is a unitary matrix with U ∗1 = x, so the columns of U provide the desired
4
orthonormal basis. For example, to construct an orthonormal basis for that
T
includes x =(1/3) ( −120 − 2) , set
−4 −120 −2
   
uu T 1  220 1 
1  2 
u = x − e 1 =   and compute R = I − 2 =   .
T
3 0 u u 3 003 0
−2 −210 2
The columns of R do the job.