A dimension of the invariant subspace corresponding to an eigenvalue 2 is three
(due to the triple root) and, hence, there should be one generalized eigenvector. To
determine it, we examine a following matrix equation:
A À 2E
ð
Þ
2 x ¼ 0:
ð12:160Þ
The matrix representation is
0
0 0
0
0
0 0
0
0
0 0
0
À4 0 À4 4
0
B
B
B
@
1
C
C
C
A
c 1
c 2
c 3
c 4
0
B
B
B
@
1
C
C
C
A
:
That is,
Àc 1 À c 3 þ c 4 ¼ 0:
ð12:161Þ
Furthermore, we have
A À 2E
ð
Þ
3 ¼
0
0 0
0
0
0 0
0
0
0 0
0
À8 0 À8 8
0
B
B
B
@
1
C
C
C
A
:
ð12:162Þ
Moreover, rank (A À 2E)
l
(¼1) remains unchanged for l ! 2 as expected from
(12.149).
It will be convenient to examine a structure of the invariant subspace. For this
purpose, we seek the number of Jordan blocks r
2
ð Þ
1 and their order. Using (12.150),
we have
r
2
ð Þ
1 ¼ 4 À rank A À 2E
ð
Þ¼4 À 2 ¼ 2:
ð12:163Þ
The number of first-order Jordan blocks is
J
2
ð Þ
1 ¼ rank A À 2E
ð
Þ
0 þ rank A À 2E
ð
Þ
2 À 2rank A À 2E
ð
Þ
¼ 4 þ 1 À 2 Â 2 ¼ 1:
ð12:164Þ
In turn, the number of second-order Jordan blocks is
J
2
ð Þ
2 ¼ rank A À 2E
ð
Þþrank A À 2E
ð
Þ
3 À 2rank A À 2E
ð
Þ
2
¼ 2 þ 1 À 2 Â 1 ¼ 1:
ð12:165Þ
504
12 Canonical Forms of Matrices