Graphics Programs Reference
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where
A
∗
=
P
−
1
AP
.Because
was untouchedbythe transformation, the eigenval-
ues of
A
are also the eigenvalues of
A
∗
. Matrices thathave the sameeigenvalues are
deemed to be
similar
, and the transformationbetween themiscalleda
similarity
transformation
.
Similarity transformations arefrequentlyused to change an eigenvalue problem
to a form that iseasier to solve. Suppose that we managedbysome meanstofind a
P
that diagonalizes
A
∗
,sothat Eqs. (9.6) are
λ
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
A
11
−
λ
x
1
x
2
.
x
n
0
···
0
0
0
.
0
A
22
−
λ
0
···
0
=
.
.
.
.
.
.
A
nn
−
λ
0
0
···
The solution of these equations is
A
11
A
22
A
nn
λ
1
=
λ
2
=
···
λ
n
=
(9.7)
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
1
0
.
0
0
1
.
0
0
0
.
1
x
1
=
x
2
=
x
n
=
···
or
x
1
x
n
X
∗
=
x
2
···
=
I
According to Eq. (9.5) the eigenvectormatrix of
A
is
PX
∗
=
X
=
PI
=
P
(9.8)
Hence the transformationmatrix
P
is the eigenvectormatrix of
A
and the eigenvalues
of
A
are the diagonaltermsof
A
∗
.
Jacobi Rotation
A specialtransformationis the plane rotation
Rx
∗
x
=
(9.9)
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