MATH 405
Final Exam
W. Adams
December 20, 2002
Show all work. Justify all answers.
This is an open book exam.
1. (20 points each) Let
A
=
"
0
i

i
0
#
.
(a) Show that
A
is unitary.
(b) Find a diagonal matrix
D
and a unitary matrix
U
such that
A
=
UDU
*
.
2. (15 points each) Assume that the minimal polynomial,
m
A
, and characteristic polyno
mial,
P
A
of a matrix
A
are as given in the three cases below. Describe or write down
all possible Jordan canonical (normal) forms for
A
.
(a)
P
A
(
t
) = (
t

3)
4
(
t

2)
3
and
m
A
(
t
) = (
t

3)
3
(
t

2).
(b)
P
A
(
t
) = (
t

2)
5
and
m
A
(
t
) = (
t

2)
3
.
(c)
P
A
(
t
) = (
t

λ
1
)
e
1
(
t

λ
2
)
e
2
· · ·
(
t

λ
m
)
e
m
and
m
A
(
t
) = (
t

λ
1
)
e
1

1
(
t

λ
2
)
e
2

1
· · ·
(
t

λ
m
)
e
m

1
for integers
e
1
≥
2
, . . . , e
m
≥
2.
3. (25 points) Find the Jordan canonical form of the matrix
A
=

4
0
9
0
1
0

4
0
8
.
4. (25 points) Let
V
be a vector space of dimension 3 and let
T
be a linear operator on
V
. Assume that we have a basis
v
1
, v
2
, v
3
of
V
such that the matrix of
T
with respect
to this basis is the matrix
A
in the last problem. Show that 3
v
1
+ 2
v
3
is an eigenvector
for
T
. What is the corresponding eigenvalue?
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 staff
 Math, Linear Algebra, Vector Space, W. Adams

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