Graphics Programs Reference
In-Depth Information
Accumulated Transformation Matrix
Since we used similarity transformations, the eigenvalues of the tridiagonal matrix
are the same as those of the original matrix. However,todetermine the eigenvectors
X
of original
A
we must use the transformation
X
=
PX
tridiag
where
P
is the accumulation of the individualtransformations:
P
=
P
1
P
2
···
P
n
−
2
We buildupthe accumulated transformationmatrixbyinitializing
P
to a
n
×
n
identitymatrix and then applying the transformation
P
11
I
i
0
T
0Q
P
11
P
12
P
21
Q
P
←
PP
i
=
=
(b)
P
21
P
22
P
12
P
22
Q
with
i
=
1
,
2
,...,
n
−
2. It can be seen thateach multiplicationaffects only the right-
most
n
i
columnsof
P
(since the first row of
P
12
containsonly zeroes, itcan also be
omittedinthe multiplication). Using the notation
−
P
12
P
22
P
=
wehave
P
12
Q
P
22
Q
P
I
uu
T
H
P
u
H
u
T
P
Q
P
−
P
−
yu
T
=
=
−
=
=
(9.47)
where
P
u
H
y
=
(9.48)
The procedurefor carrying out the matrix multiplicationinEq. (b) is
Retrieve
u
(in our triangularizationprocedure the
u
's are storedinthe columns
of the lower triangular portion of
A
).
2
Compute
H
= |
u
|
/
2.
P
u
Compute
y
=
/
H
.
P
−
Compute the transformation
P
←
yu
T
.
householder
Thisfunctionperforms the Householderreduction on the matrix
A
. Uponreturn,
d
occupies the principal diagonalof
A
and
c
forms the upper subdiagonal; that is,
d = diag(A)
and
c = diag(A,1)
. The portion of
A
below the principal diagonal is
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