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
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Linear Algebra, Algebra, Matrices, linearly independent eigenvectors, Dr. Macauley HW

Click to edit the document details