x = a 1 a 2 Á Á Áa n
ð
Þ
x 1
x 2
⋮
x n
0
B
B
B
@
1
C
C
C
A
:
Operating A on x, we get
A x
ð Þ ¼ a 1 a 2 Á Á Áa n
ð
Þ A
x 1
x 2
⋮
x n
0
B
B
B
B
B
@
1
C
C
C
C
C
A
¼ a 1 A a 2 AÁ Á Áa n A
ð
Þ
x 1
x 2
⋮
x n
0
B
B
B
B
B
@
1
C
C
C
C
C
A
¼ α 1 a 1 α 2 a 2 Á Á Áα n a n
ð
Þ
x 1
x 2
⋮
x n
0
B
B
B
B
B
@
1
C
C
C
C
C
A
¼ a 1 a 2 Á Á Áa n
ð
Þ
α 1
α 2
⋱
α n
0
B
B
B
B
B
@
1
C
C
C
C
C
A
x 1
x 2
⋮
x n
0
B
B
B
B
B
@
1
C
C
C
C
C
A
ð12:192Þ
where with the second equality we used the notation (11.40); with the third equality
some of α i (1 i n) may be identical; a i is an eigenvector that corresponds to an
eigenvalue α i . Suppose that a 1 , a 2 , Á Á Á, and a n are obtained by transforming an
“original” basis set e 1 , e 2 , Á Á Á, and e n by R. Then, we have
A x
ð Þ ¼ e 1 e 2 Á Á Áe n
ð
ÞR
α 1
α 2
⋱
α n
0
B
B
B
B
@
1
C
C
C
C
A
R
À1
x
0
ð Þ
1
x
0
ð Þ
2
⋮
x
0
ð Þ
n
0
B
B
B
B
@
1
C
C
C
C
A
:
We denote the transformation A with respect to a basis set e 1 , e 2 , Á Á Á, and e n by
A 0 ; see (11.82) with the notation. Then, we have
516
12 Canonical Forms of Matrices