Section 9–2 / State-Space Representations of Transfer-Function Systems 653
(9–12)
the transformation x=Pz, where
P=
l 1 ,l 2 ,p,ln=ndistinct eigenvalues of A
will transform P–1APinto the diagonal matrix, or
If the matrix Adefined by Equation (9–12) involves multiple eigenvalues, then
diagonalization is impossible. For example, if the 3*3matrixA, where
has the eigenvalues l 1 ,l 1 ,l 3 ,then the transformation x=Sz, where
will yield
This is in the Jordan canonical form.
S-^1 AS=C
l 1
0
0
1
l 1
0
0
0
l 3
S
S= C
1
l 1
l 12
0
1
2 l 1
1
l 3
l 32
S
A=C
0
0
- a 3
1
0
- a 2
0
1
- a 1
S
P-^1 AP= F
l 1
0
l 2
0
ln
V
G
1
l 1
l 12
l 1 n-^1
1
l 2
l 22
l 2 n-^1
p
p
p
p
1
ln
ln^2
lnn-^1
W
A= G
0 0 0
- an
1 0 0
- an- 1
0 1 0
- an- 2
p
p
p
p
0 0 1
- a 1
W