matrix, however, the following argument will be useful. Let us think of an example
after that.
Using (12.140), we have
A À α i E
α 1 À α i
α 2 À α i
Ã
α 2 À α i
Ã
Ã
α 2 À α i
⋱
0 Ã Ã Ã
0 Ã Ã
0 Ã
0
⋱
α s À α i
Ã
α s À α i
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
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
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
Here we are treating the (n, n) matrix A on V
n . Note that a matrix A À α i E is not
nilpotent as a whole. Suppose that the multiplicity of α i is n i ; in (12.140) n i ¼ 4.
Since eigenvalues α 1 , α 2 , Á Á Á, and α s take different values from one another, α 1 À α i ,
α 2 À α i , Á Á Á, and α s À α i 6 ¼ 0. In a triangle matrix diagonal elements give eigenvalues
and, hence, α 1 À α i , α 2 À α i , Á Á Á, and α s À α i are nonzero eigenvalues of A À α i E. We
rewrite (12.140) as
A À α i E $
M
1
ð Þ
M
2
ð Þ
⋱
M
i
ð Þ
⋱
M
s
ð Þ
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
,
ð12:142Þ
where M
1
ð Þ
¼ α 1 À α i
ð
Þ, M
2
ð Þ
¼
α 2 À α i
Ã
Ã
α 2 À α i
Ã
α 2 À α i
0
B
@
1
C
A, Á Á Á,
498
12 Canonical Forms of Matrices