Graphics Programs Reference
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where
u
is avector and
1
2
u
T
u
1
2
|
2
|
H
=
=
u
(9.37)
Note that
uu
T
in Eq. (9.36) is the outerproduct; that is, amatrix with the elements
uu
T
i j
=
u
i
u
j
.Since
Q
isobviously symmetric(
Q
T
=
Q
), wecan write
u
u
T
u
u
T
H
2
I
I
uu
T
H
uu
T
H
2
uu
T
H
Q
T
Q
=
QQ
=
−
−
=
I
−
+
2
uu
T
H
u
(2
H
)
u
T
H
2
=
I
−
+
=
I
which showsthat
Q
is also orthogonal.
Now let
x
be an arbitrary vector and consider the transformation
Qx
.Choosing
u
=
x
+
k
e
1
(9.38)
where
1 00
0
T
k
=±|
x
|
e
1
=
···
we get
I
x
I
x
uu
T
H
k
e
1
)
T
H
u
(
x
+
Qx
=
−
=
−
u
x
T
x
k
e
1
x
u
k
2
kx
1
+
+
=
x
−
=
x
−
H
H
But
k
x
T
e
1
+
e
1
x
+
k
e
1
)
T
(
x
2
k
pt
2
e
1
e
1
2
H
=
(
x
+
+
k
e
1
)
= |
x
|
+
2
k
2
kx
1
k
2
k
2
=
+
2
kx
1
+
=
+
so that
0
T
Qx
=
x
−
u
=−
k
e
1
=
−
k
00
···
(9.39)
Hence the transformation eliminates all elements of
x
except the first one.
Householder Reduction of a Symmetric Matrix
Let us now apply the following transformation to a symmetric
n
×
n
matrix
A
:
1
0
T
0Q
A
11
x
T
xA
A
11
x
T
P
1
A
=
=
(9.40)
QA
Qx
Here
x
is represents the first column of
A
with the first elementomitted, and
A
issimply
A
with its first row and column removed. The matrix
Q
of dimensions
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