Math 235
Assignment 9
Due: Wednesday, Dec 1st
1.
For any
θ
∈
R
, let
R
θ
=
±
cos
θ

sin
θ
sin
θ
cos
θ
²
.
a) Diagonalize
R
θ
over
C
.
b) Verify your answer in a) is correct, by showing the matrix
P
and diagonal matrix
D
from part a) satisfy
P

1
R
θ
P
=
D
for
θ
= 0 and
θ
=
π
4
.
2.
Let
A
=
0
2 1

2 3 0
1
0 2
.
a) Given that
λ
= 2 +
i
is an eigenvalue of
A
, determine the other eigenvalues of
A
.
b) Determine a real canonical form of
A
and give a change of basis matrix
P
that brings
the matrix into this form.
3.
Suppose that
A
is an
n
×
n
matrix with real entries, and that
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 Fall '08
 CELMIN
 Math, Linear Algebra, Matrices, Singular value decomposition, A. Let, Diagonalize Rθ

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