Page 141 - Numerical Methods for Chemical Engineering

130 3 Matrix eigenvalue analysis

w
x

Px
T
Figure 3.6 Effect of P = I − 2ww on an arbitrary vector x.
We next demonstrate how a sequence of such Householder transformations performs the
QR factorization,

 
R 11 R 12 ... R 1N
 R 22 ... R 2N 
T
  QQ = I (3.152)
. . . . 
A = QR = Q 
 . . 
R NN
Let us examine the first column of R that has a nonzero element only at the first component.
N
For any vector x ∈ , we can find a vector v, generating a Householder reflection P, that
zeros all but the first component of x:
b
 
vv 2(v · x)   [1]
T 0
Px = I − 2 x = x − v =  .  = be (3.153)
 
|v| 2 |v| 2  . 
.
0
[1]
−2
[1]
As x − 2(v · x)|v| v = be , v must be a linear combination of x and e ,
v = x + αe [1] (3.154)
As
2
2

|v| = x + αe [1] [1] =|x| + 2αx 1 + α 2
· x + αe
(3.155)
[1] 2
v · x = x + αe · x =|x| + αx 1
the Householder transformation acts on x as
2
2(|x| + αx 1 ) [1]
Px = x − x + αe
2
|x| + 2αx 1 + α 2
2 2
2(|x| + αx 1 ) 2α(|x| + αx 1 ) [1]
= 1 − x − e (3.156)
2
2
|x| + 2αx 1 + α 2 |x| + 2αx 1 + α 2
We obtain Px = be [1] by satisfying
2
α = ε|x|
2 |x| + αx 1
1 = ⇒ −1, x 1 < 0 (3.157)
2
|x| + 2αx 1 + α 2 ε = sign(x 1 ) =
1, x 1 ≥ 0
so that
Px =−ε|x|e [1] (3.158)