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=
=
29
λ
9
9 0
−
26
λ
3
3 3
−
=
=
(1.34)
−
λ
−
λ
=
λ
2
=
λ
2
−
79
λ
+ 1369 =
−
79
λ
+ 1369 =
=0
,
=0
,
I
:= tr[JJ
∗
] = tr[J
∗
J] = 79
,
II
:= det[JJ
∗
]=det[J
∗
J] = 1369
,
(1.35)
1
=
λ
1
=7
.
302 692
,
λ
1
=53
.
329 317
,
(1.36)
2
=
λ
2
=5
.
066 624
.
λ
2
=25
.
670 683
,
The left eigenspace is spanned by the left eigencolumns (
u
1
,
u
2
), the right eigenspace by the
right eigencolumns (
v
1
,
v
2
), namely
(JJ
∗
−
(J
∗
J
λ
1
I
2
)
u
1
=
0
,
−
λ
1
I
2
)
v
1
=
0
,
(1.37)
(JJ
∗
− λ
2
I
2
)
u
2
=
0
,
(J
∗
J
− λ
2
I
2
)
v
2
=
0
,
or
−
u
11
u
21
=
0
,
−
v
11
v
21
=
0
,
24
.
329 317
9
27
.
329 317
3
9
−
3
.
329317
3
−
0
.
329 317
(1.38)
3
.
329 317 9
9
.
329 317
u
12
u
22
=
0
,
0
.
329 317 3
3
.
329 317
v
12
v
22
=
0
.
Note that the matrices JJ
∗
−
λ
I
2
and J
∗
J
λ
I
2
have only rank one. Accordingly, in order to solve
the homogenous linear equations uniquely, we need an additional constraint. Conventionally, this
problem is solved by postulating
normalized eigencolumns
,namely
−
u
11
+
u
21
=1
,u
12
+
u
22
=1
,
11
+
v
21
=1
,v
12
+
v
22
=1
,
(1.39)
u
1
=
u
2
=1
,
v
1
=
v
2
=1
.
The left eigencolumns, which are here denoted as (
u
1
,
u
2
), are constructed from the following
system of equations:
−
24
.
329 317
u
11
+9
u
21
=0
,
+3
.
329 317
u
12
+9
u
22
=0
,
(1.40)
u
11
+
u
21
=1
,
12
+
u
22
=1
.
This system of equations leads to two solutions. In the frame of the example to bc considered
here, we have chosen the following result:
u
11
=+0
.
346 946
, u
12
=+0
.
937 885
,
(1.41)
u
21
=+0
.
937 885
, u
22
=
−
0
.
346 946
.
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