e
2
ð Þ
1 ¼
2
1
:
Another eigenvector is a generalized eigenvector g
2
ð Þ
1
of rank 2. This can be
decided such that
A À 2E
ð
Þg
2
ð Þ
1 ¼
À2 4
À1 2
g
2
ð Þ
1 ¼ e
2
ð Þ
1 :
As an option, we get
g
2
ð Þ
1 ¼
1
1
:
Thus, we can choose R for a diagonalizing matrix together with an inverse matrix
R
À1 such that
R ¼
2 1
1 1
, and R
À1
¼
1 À1
À1 2
:
Therefore, with a Jordan canonical form we have
R
À1 AR ¼
2 1
0 2
:
ð12:179Þ
As before, putting
S ¼
2 0
0 2
and N ¼
0 1
0 0
,
we have
R
À1 AR ¼ S þ N, i:e:, A ¼ R S þ N
ð
ÞR
À1
¼ RSR
À1
þ RNR
À1
:
Putting e S ¼ RSR
À1 and e
N ¼ RNR
À1 , we get
A ¼ e S þ e
N
with
e S ¼
2 0
0 2
and e
N ¼
À2 4
À1 2
:
ð12:180Þ
12.6 Jordan Canonical Form
511