A x
ð Þ ¼ e 1 e 2 e 3 e 4
ð
Þ RR
À1 ARR
À1
x 1
x 2
x 3
x 4
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
¼ e
4
ð Þ
1 e
2
ð Þ
1 e
2
ð Þ
2 g
2
ð Þ
2
4 0 0 0
0 2 0 0
0 0 2 1
0 0 0 2
0
B
B
B
@
1
C
C
C
A
x
0
1
x
0
2
x
0
3
x
0
4
0
B
B
B
@
1
C
C
C
A
,
ð12:172Þ
where we have
e
4
ð Þ
1 e
2
ð Þ
1 e
2
ð Þ
2 g
2
ð Þ
2
¼ e 1 e 2 e 3 e 4
ð
Þ R,
x
0
1
x
0
2
x
0
3
x
0
4
0
B
B
B
B
B
@
1
C
C
C
C
C
A
¼ R
À1
x 1
x 2
x 3
x 4
0
B
B
B
B
@
1
C
C
C
C
A
:
ð12:173Þ
After (11.84), we have
x = e 1 e 2 e 3 e 4
ð
Þ
x 1
x 2
x 3
x 4
0
B
B
B
B
@
1
C
C
C
C
A
¼ e 1 e 2 e 3 e 4
ð
Þ RR
À1
x 1
x 2
x 3
x 4
0
B
B
B
B
@
1
C
C
C
C
A
¼ e
4
ð Þ
1 e
2
ð Þ
1 e
2
ð Þ
2 g
2
ð Þ
2
x
0
1
x
0
2
x
0
3
x
0
4
0
B
B
B
B
B
@
1
C
C
C
C
C
A
:
ð12:174Þ
As for V
n in general, let us put R ¼ ( p) ij and R
À1
¼ (q) ij . Also represent j-th
(generalized) eigenvectors by a column vector and denote them by p
( j ) . There we
display individual (generalized) eigenvectors in the order of (e
(1) e
(2)
Á Á Áe
( j )
Á Á Áe
(n)
),
where e
( j ) (1
j
n) denotes either a proper eigenvector or a generalized
eigenvector according to (12.174). Each p
( j ) is represented in reference to original
basis vectors (e 1 Á Á Áe n ). Then we have
12.6 Jordan Canonical Form
507