ENSC 483 HW#3 Solution–M. Saif
1
Simon Fraser University
School of Engineering Science
ENSC 483  Modern Control Systems
1. (a) Find the eigenvalues of the following matrices:
A
=
8

8

2
4

3

2
3

4
1
;
B
=
2
1
1
0
3
1
0

1 1
;
C
=
•
α
ω

ω α
‚
;
D
=
•
0
1

ω
2
n
0
‚
(b) Discuss if the above matrices can be diagnonilized.
Solution:
(a) In the case of matrix
A
, the eigenvalues are given by
λ
1
= 1
,λ
2
= 3
,λ
3
= 2
.
For matrix
B
, all the eigenvalues are located at
2
.
For matrix
C
, we have
Δ(
λ
) = (
α

λ
)
2
+
ω
2
= 0 =
⇒
λ
1
,
2
=
α
±
jω
.
For matrix
D
, we have
Δ(
λ
) =
λ
2
+
ω
2
n
= 0 =
⇒
λ
1
,
2
=
±
jω
n
.
(b) Using the deﬁnition (
Av
=
λ
v
), we can ﬁnd three linearly independent eigenvectors corresponding
to these distinct
eigenvalues. One such set is:
v
1
=
1
0
.
75
0
.
5
;
v
2
=
1
0
.
5
0
.
5
;
v
3
=
1
0
.
6667
0
.
3337
Given these linearly independent eigenvectors, we can form a nonsingular transformation that will
diagonalize the matrix
A
.
Since matrix
B
has repeated eigenvalues, we need to investigate whether or not we can ﬁnd three
linearly independent eigenvectors associated with these repeated eigenvalues. To do this check that
ρ
(
B

2
I
) = 1 =
⇒
γ
(
B

2
I
) = 2
. Since the nullity of the matrix
(
B

2
I
)
is two, we conclude
that we can only ﬁnd two linearly independent eigenvectors for this matrix. Therefore, this matrix
is not diagonalizable.
C
is diagonalizable since it has distinct eigenvalues.
D
is diagonilizable since it has distinct eigenvalues.
2. Consider the following
symmetric matrix
A
=
3
0

2
0
2
0

2 0
0
(a) Find the eigenvalues and the eigenvectors of this matrix.
(b) Find the (special) modal matrix
Q
whose columns are eigenvectors normalized to be of unit length.
(c) Form the product
ˆ
A
=
Q
T
AQ
and show that
ˆ
A
is a diagonal matrix.
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2
(d) What do you conclude from this exercise?
Solution:
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 Spring '07
 MehrdadSaif
 Linear Algebra, Matrices, Eigenvalues, Orthogonal matrix, Saif

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