Simon Fraser University
School of Engineering Science
ENSC 483  Modern Control Systems
Spring 2007–Assignment 4
1. Find the modal matrix and the Jordan Canonical Form of the following matrices:
A
=

1
0

1
1
1
3
0
0
1
0
0
0
0
0
2
1
2

1

1

6
0

2
0

1
2
1
3
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0

1

1
0
1
2
4
1
B
=

1

2
3
0

2
3
0
2
3
0
0
2

4
0
1

2
1

1

4

1
0
0

1
0
0
0
0
0
0
0
0
5

7

1
2

4
1

2

7

1
0

1
1
0

1
0
0
0
1
0
0
3

4
0
1

3
1

1

4

1
0
1

1
0
1

1

1

1

1
0
0

1
1
0
0
0
0

1
1
0
0
0
0
0
0
0
0
0

1
0
0
4

5
0
2

3
1

2

5

2
Hint:
A
has all of its eigenvalues located at
λ
= 1, whereas eigenvalues of
B
are all at
λ
=

1.
2. Let
z
be an arbitrary
n
dimensional vector such that
P
≡
[
z
Az
· · ·
A
n

1
z
] is
nonsingular. Use the CayleyHamilton Theorem to show that the solution of the linear
equation problem
P x
=
A
n
z
is
x
= [

a
0

a
1
· · ·

a
n

1
]
T
where
a
0
, a
1
,
· · ·
, a
n

1
are the coefficients of the
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 Spring '07
 MehrdadSaif
 Linear Algebra, Characteristic polynomial, following matrices, previous problem, Modern Control Systems

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