R
À1 AR ¼
À1 0
À1 1
0
0
1
0
1
0
0
0
À1 À1 0
0
0
B
B
B
B
B
@
1
C
C
C
C
C
A
1
À1 0
0
1
3
0
0
0
0
2
0
À3 À1 À2 4
0
B
B
B
B
B
@
1
C
C
C
C
C
A
0 0 1
0
0 0 À1 À1
0 1 0
0
1 1 1
0
0
B
B
B
B
B
@
1
C
C
C
C
C
A
¼
4 0 0 0
0 2 0 0
0 0 2 1
0 0 0 2
0
B
B
B
B
B
@
1
C
C
C
C
C
A
:
ð12:170Þ
The structure of Jordan blocks is shown in Fig. 12.2. Notice here that the trace of
A remains unchanged before and after similarity transformation.
Next, we consider column vector representations. According to (11.37), let us
view the matrix A as a linear transformation over V
4 . Then A is given by
A x
ð Þ ¼ e 1 e 2 e 3 e 4
ð
Þ A
x 1
x 2
x 3
x 4
0
B
B
B
@
1
C
C
C
A
¼ e 1 e 2 e 3 e 4
ð
Þ
1
À1 0
0
1
3
0
0
0
0
2
0
À3 À1 À2 4
0
B
B
B
@
1
C
C
C
A
x 1
x 2
x 3
x 4
0
B
B
B
@
1
C
C
C
A
,
ð12:171Þ
where e 1 , e 2 , e 3 , and e 4 are basis vectors and x 1 , x 2 , x 3 , and x 4 are corresponding
coordinates of a vector x =
P n
i¼1 x i e i 2 V
4
À
Á
. We rewrite (12.171) as
( )
(LJHQYDOXH
(LJHQYDOXH
( )
( )
( )
-RUGDQEORFNV
Fig. 12.2 Structure of
Jordan blocks of a matrix
shown in (12.170)
506
12 Canonical Forms of Matrices