Graphics Programs Reference
In-Depth Information
where
k
⎡
⎣
⎤
⎦
1 0 0 00000
0 1 0 00000
00
c
00
s
00
00 0 1 0000
00 0 01 000
00
k
R
=
(9.10)
s
00
c
00
00 0 0001 0
00 0 00001
−
iscalled the
Jacobi rotation matrix
. Note that
R
is an identitymatrix modifiedbythe
terms
c
=
cos
θ
and
s
=
sin
θ
appearing at the intersections of columns/rows
k
and
is the rotation angle. The rotationmatrix has the useful property of being
orthogonal
,or
unitary
, meaning that
, where
θ
R
−
1
R
T
=
(9.11)
Oneconsequence of orthogonalityisthat the transformationinEq. (9.9)has the
essentialcharacteristicofarotation: it preserves the magnitude of the vector; that is,
|
x
∗
|
x
| = |
.
The similarity transformation corresponding to the plane rotationinEq. (9.9) is
A
∗
=
R
−
1
AR
R
T
AR
=
(9.12)
The matrix
A
∗
not only has the sameeigenvalues as the original matrix
A
, but dueto
orthogonality of
R
it is also symmetric. The transformationinEq. (9.12)changes only
the rows/columns
k
and
of
A
. The formulasfor these changes are
A
kk
=
c
2
A
kk
+
s
2
A
−
2
csA
k
A
∗
=
c
2
A
+
s
2
A
kk
+
2
csA
k
A
k
=
A
∗
k
=
(
c
2
s
2
)
A
k
+
−
cs
(
A
kk
−
A
)
(9.13)
A
ki
A
ik
=
=
cA
ki
−
s A
i
,
i
=
k
,
i
=
A
∗
i
A
i
=
=
cA
i
+
=
=
s A
ki
,
i
k
,
i
Jacobi Diagonalization
in the Jacobi rotationmatrix canbe chosen so that
A
k
=
A
∗
k
=
The angle
0. Thissug-
gests the following idea: whynot diagonalize
A
by looping through all the off-diagonal
terms and eliminate themone by one? This isexactlywhatJacobi diagonalizationdoes.
θ
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