R
À1 p
j
ð Þ
¼
X n
k¼1
q ik p
j
ð Þ
k ¼ δ
j
ð Þ
i ,
ð12:175Þ
where δ
j
ð Þ
i
denotes a column vector to which only the j-th row is 1, otherwise
0. Thus, a column vector δ
j
ð Þ
i
is an “address” of e
( j ) in reference to
(e
(1) e
(2)
Á Á Áe
( j )
Á Á Áe
(n) ) taken as basis vectors.
In our present case, in fact, we have
R
À1 p
1
ð Þ
¼
À1 0
À1 1
0
0
1
0
1
0
0
0
À1 À1 0
0
0
B
B
B
@
1
C
C
C
A
0
0
0
1
0
B
B
B
@
1
C
C
C
A
¼
1
0
0
0
0
B
B
B
@
1
C
C
C
A
,
R
À1 p
2
ð Þ
¼
À1 0
À1 1
0
0
1
0
1
0
0
0
À1 À1 0
0
0
B
B
B
@
1
C
C
C
A
0
0
1
1
0
B
B
B
@
1
C
C
C
A
¼
0
1
0
0
0
B
B
B
@
1
C
C
C
A
,
R
À1 p
3
ð Þ
¼
À1 0
À1 1
0
0
1
0
1
0
0
0
À1 À1 0
0
0
B
B
B
@
1
C
C
C
A
1
À1
0
1
0
B
B
B
@
1
C
C
C
A
¼
0
0
1
0
0
B
B
B
@
1
C
C
C
A
,
R
À1 p
4
ð Þ
¼
À1 0
À1 1
0
0
1
0
1
0
0
0
À1 À1 0
0
0
B
B
B
@
1
C
C
C
A
0
À1
0
0
0
B
B
B
@
1
C
C
C
A
¼
0
0
0
1
0
B
B
B
@
1
C
C
C
A
:
ð12:176Þ
In (12.176) p
(1) is a column vector representation of e
4
ð Þ
1 ; p
(2) is a column vector
representation of e
2
ð Þ
1 , and so on.
A minimal polynomial Φ A (x) is expressed as
Φ A x
ð Þ ¼ x À 4
ð
Þ x À 2
ð
Þ
2 :
Readers can easily make sure of it.
We remark that a non-singular matrix R pertinent to the similarity transformation
is not uniquely determined, but we have arbitrariness. In (12.169), for example, if we
adopt Àg
2
ð Þ
2 instead of g
2
ð Þ
2 , we should adopt Àe
2
ð Þ
2 instead of e
2
ð Þ
2 accordingly. Thus,
instead of R in (12.169) we may choose
508
12 Canonical Forms of Matrices