Householder
QR
Algorithmically, this method is very close to G.E. with no pivoting. There, a Gauss
transform
M
k
=
I
+
m
k
e
t
k
was used to introduce zeros below the (
k, k
) element of
A
(
k

1)
, giving
A
(
k
)
G
=
M
k
A
(
k

1)
G
. Here, a Householder reflector
H
k
=
I

βu
k
u
t
k
replaces the Gauss transform as the operator that introduces zeros below the (
k, k
)
element:
A
(
k
)
H
=
H
k
A
(
k

1)
H
.
Let
A
∈
R
m
×
n
, with
m
≥
n
, and let
p
= min (
n, m

1). The Householder
QR
factorization of
A
can be coarsely described as
A
(0) =
A
For k = 1:p
Compute
u
so that (
I

βuu
t
)
A
(
k

1)
has zeros below its (
k, k
) entry
Compute
A
(
k
)
=
H
k
A
(
k

1)
End
There are some important details to consider yet, but it is essentially this simple.
We know that if
u
is a Householder vector for
x
, then it is a multiple of
x
±
x
2
e
1
,
and that
Hx
= (
I

βuu
t
)
x
=
±
x
2
e
1
, with
β
= 2
/
(
u
t
u
). As with G.E. we can
view the
k
th
step as
A
(
k
)
=
R
(
k
)
X
(
k
)
0
˜
A
(
k
)
=
I
0
0
˜
H
k
R
(
k

1)
X
(
k

1)
0
˜
A
(
k

1)
=
H
k
A
(
k

1)
,
where
R
(
k
)
is
k
×
k
upper triangular, and
˜
A
(
k
)
is (
m

k
)
×
(
n

k
). Here
u
t
k
= (0
t
,
˜
u
t
k
), where ˜
u
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 Fall '10
 MARK
 Determinant, UK, Triangular matrix, Householder QR

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