Math Biophysics, Fall 2011
Homework 1
(1) Read about finding eigenvalues using Maple. Let
C
=

4

3
3

3
3
2

3
3
3
3

4
3
6
6

6
5
Use Maple to find a nonzero vector
X
such that
CX
= 2
X
.
(2) Use Maple to find the inverses of the matrices
M
=
1
2
3
3
4
5
3
5
6
and
N
=
2
2

i
2 +
i

2
.
(3) Suppose
A
is self adjoint and that
v
1
and
v
2
are eigenvectors corresponding
to distinct eigenvalues. Show that
v
*
1
v
2
= 0. (First read the proof in the
Brief Linear Algebra Review that eigenvalues of selfadjoint matrices are
real. Use a similar method of proof.)
(4) If
A
is a real 2
×
2 matrix such that
A
0
A
=
I
and det
A
= 1, show that for
some
θ
,
A
=
cos
θ

sin
θ
sin
θ
cos
θ
.
(5) Let
R
θ
be the rotation matrix
cos
θ

sin
θ
sin
θ
cos
θ
.
Show that
(
R
θ

I
) (
R

θ

I
) = 2(1

cos
θ
)
I.
(6) Show that if
A
is real and has real eigenvalue
λ
, then there is a real vector
(a vector with real coordinates) which is an eigenvector corresponding to
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 Fall '11
 staff
 Linear Algebra, Cos, Brief Linear Algebra, B1 Rz

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