M
i
ð Þ
¼
0 Ã Ã Ã
0 Ã Ã
0 Ã
0
0
B
B
B
@
1
C
C
C
A
, Á Á Á, M
s
ð Þ
¼
α s À α i
Ã
α s À α i
:
Thus, M
( p) ( p 6 ¼ i) is a non-singular matrix and M
(i) is a nilpotent matrix. Note
that if we can find μ i such that M
i
ð Þ
Â
à μ i À1 6 ¼ 0 and M
i
ð Þ
Â
à μ i ¼ 0 , with a minimal
polynomial φ M
i
ð Þ x
ð Þ for M
(i)
, we have φ M
i
ð Þ x
ð Þ ¼ x
μ i . Consequently, we get
A À α i E
ð
Þ
μ i $
M
1
ð Þ
Â
à μ i
M
2
ð Þ
Â
à μ i
⋱
0
⋱
M
s
ð Þ
Â
à μ i
0
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
A
:
ð12:143Þ
In (12.143) diagonal elements of non-singular triangle matrices M
p
ð Þ
Â
à μ i p 6 ¼ i
ð
Þ
are α p À α i
À
Á μ i 6 ¼ 0
ð
Þ. Thus, we have a “perforated” matrix A À α i E
ð
Þ
μ i where
M
i
ð Þ
Â
à μ i ¼ 0 in (12.143). Putting
Φ A x
ð Þ
Y s
i¼1
x À α i
ð
Þ
μ i ,
we get
Φ A A
ð Þ
Y s
i¼1
A À α i E
ð
Þ
μ i ¼ 0:
A polynomial Φ A (x) gives a minimal polynomial for f A (x).
From the above argument, we can choose μ i for l i in (12.82). Meanwhile, M
(i) in
(12.142) is identical to N i in (12.134). Rewriting ((12.134)), we get
A
i
ð Þ
¼ M
i
ð Þ
þ α i E n i :
ð12:144Þ
Let us think of matrices A
i
ð Þ
À α i E n i
Â
à k and (A À α i E)
k (k μ i ). From (11.45), we
find
dim V
n
¼ n ¼ dim Ker A À α i E
ð
Þ
k þ rank A À α i E
ð
Þ
k ,
ð12:145Þ
and
12.6 Jordan Canonical Form
499