A
κ
ð Þ
À α i E κ i
h
i κ i À1
a σ A
κ
ð Þ
À α i E κ i Š
κ i À2 a σ Á Á Áa σ
h
A
κ
ð Þ
À α i E κ i
h
i
¼ A
κ
ð Þ
À α i E κ i
h
i κ i À1
a σ A
κ
ð Þ
À α i E κ i
h
i κ i À2
a σ Á Á Áa σ
Â
0 1
0 1
0 ⋱
⋮
⋱ ⋱ ⋮
0
1
Á Á Á
0 1
0
0
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
A
:
ð12:136Þ
A vector a σ stands for a vector associated with the κ-th Jordan block of A
(i) .
From (12.136) we obtain
A
κ
ð Þ
À α i E κ i
h
i
A
κ
ð Þ
À α i E κ i
h
i κ i À1
a σ
&
'
= 0:
ð12:137Þ
Namely,
A
κ
ð Þ
A
κ
ð Þ
À α i E κ i
h
i κ i À1
a σ
&
'
= α i A
κ
ð Þ
À α i E κ i
h
i κ i À1
a σ
&
'
:
ð12:138Þ
This shows that A
κ
ð Þ
À α i E κ i
Â
à κ i À1 a σ is a proper eigenvector of A
(κ) that corresponds to an eigenvalue α i . On the other hand, a σ is a generalized eigenvector of rank
κ i . There are another (κ i À 2) generalized eigenvectors of
A
κ
ð Þ
À α i E κ i
Â
à μ a σ 1 μ κ i À 2
ð
Þ . In total, there are κ i eigenvectors [a proper
eigenvector and (κ i À 1) generalized eigenvectors]. Also we see that the sole proper
eigenvector can be found for each Jordan block.
In reference to these κ i eigenvectors as the basis vectors, the (κ i ,κ i )-matrix A
(κ)
(i.e., a Jordan block) is expressed as
( )
=
⎜
⎜
⎜
⎜
⎛
⎟
⎟
⎟
⎟
⎞
.
1
1
⋮
⋱
⋮
1
⋱
⋱
ð12:139Þ
12.6 Jordan Canonical Form
495