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MthSc 853 (Linear Algebra)
Dr. Macauley
HW 7
Due Wednesday, March 9th, 2011
(1) Prove the following properties of the trace function:
(a) tr
AB
= tr
BA
for all
m
×
n
matrices
A
and
n
×
m
matrices
B
.
(b) tr
AA
T
=
∑
a
2
ij
for all
n
×
n
matrices
A
.
(2) Find the eigenvalues and corresponding eigenvectors for the following matrices over
C
.
(
a
)
±
2 4
5 3
²
(
b
)
±
3
2

2 3
²
(
c
)
5

6

6

1
4
2
3

6

4
(
d
)
3 1

1
2 2

1
2 2
0
.
(3) (a) Show that if
A
and
B
are similar, then
A
and
B
have the same eigenvalues.
(b) Is the converse of part (a) true? Prove or disprove.
(4) Let
A
φ
be the 3
×
3 matrix representing a rotation of
R
3
through an angle
φ
about the
y
axis.
(a) Find the eigenvalues for
A
φ
over
C
.
(b) Determine necessary and suﬃcient conditions on
φ
in order for
A
φ
to have 3 linearly
independent eigenvectors in
R
3
. Justify your claim and interpret it geometrically.
(5) Let
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 Spring '08
 Staff
 Linear Algebra, Algebra, Matrices

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