N
iÀ1 a ρ N
iÀ2 a ρ Á Á ÁNa ρ a ρ
À
Á
N
¼ N
iÀ1 a ρ N
iÀ2 a ρ Á Á ÁNa ρ a ρ
À
Á
0 1
0 1
0
⋮
⋱
⋮
Á Á Á
0 1
0 1
0
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
:
ð12:122Þ
These (i, i) matrices of (12.122) are called i-th order Jordan blocks. Notice that
the number of those Jordan blocks is (r i À r i + 1 ). Let us expressly define this
number as [3, 4]
J i ¼ r i À r iþ1 ,
ð12:123Þ
where J i is the number of the i-th order Jordan blocks. The total number of Jordan
blocks within a whole vector space V
n
¼ W
(ν) is
X ν
i¼1
J i ¼
X ν
i¼1
r i À r iþ1
ð
Þ¼r 1 :
ð12:124Þ
Recalling the dimension theorem mentioned in (11.45), we have
dim V
n
¼ dim Ker N
i
þ dimN
i V
n
ð Þ ¼ dim Ker N
i
þ rank N
i
:
ð12:125Þ
Meanwhile, since W
(i)
¼ Ker N
i
, dim W
(i)
¼ m i ¼ dim Ker N
i . From (12.116),
m 0 0. Then (11.45) is now read as
dim V
n
¼ m i þ rankN
i
:
ð12:126Þ
That is
n ¼ m i þ rankN
i
ð12:127Þ
or
m i ¼ n À rankN
i
:
ð12:128Þ
Meanwhile, from Table 12.1 [3, 4] we have
492
12 Canonical Forms of Matrices