This preview shows page 1. Sign up to view the full content.
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
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 09/21/2009 for the course MATH 113 taught by Professor Staff during the Spring '08 term at Maryland.
 Spring '08
 staff
 Math

Click to edit the document details