704 Chapter 9 / Control Systems Analysis in State SpacebutThus, we have shown that the minimal polynomial and the characteristic polynomial of this matrix
Aare the same.
Next, consider the matrix B. The characteristic polynomial is given byA simple computation reveals that matrix Bhas three eigenvectors, and the Jordan canonical
form of Bis given byThus, the multiple eigenvalues are not linked. To obtain the minimal polynomial, we first compute
:from which it is evident thatHence,As a check, let us compute :For the given matrix B, the degree of the minimal polynomial is lower by 1 than that of the char-
acteristic polynomial. As shown here, if the multiple eigenvalues of an n*nmatrix are not linked
in a Jordan chain, the minimal polynomial is of lower degree than the characteristic polynomial.f(B)=B^2 - 3 B+ 2 I= C
4
0
0
0
4
9
0
0
1
S - 3 C
2
0
0
0
2
3
0
0
1
S + 2 C
1
0
0
0
1
0
0
0
1
S = C
0
0
0
0
0
0
0
0
0
S = 0
f(B)f(l)=∑l I-B∑
d(l)=
(l-2)^2 (l-1)
l- 2=l^2 - 3 l+ 2d(l)=l- 2adj(l I-B)= C
(l- 2 )(l- 1 )
0
00
(l- 2 )(l- 1 )
3 (l- 2 )0
0
(l- 2 )^2S
adj(l I-B)C
2
0
0
0
2
0
0
0
1
S
∑l I-B∑= 3
l- 2
0
00
l- 2- 3
0
0
l- 13 =(l-2)^2 (l-1)
= C
0
0
0
13
0
0
0
0
0
S Z 0
= C
4
0
0
16
4
9
12
0
1
S - 3 C
2
0
0
1
2
3
4
0
1
S + 2 C
1
0
0
0
1
0
0
0
1
S
A^2 - 3 A+ 2 I
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