Approximate the three real eigenvalues of A to the
nearest integer. One of the three eigenvalues of A is
negative. Find a good approximation for this eigen
value and a corresponding eigenvector by using the
procedure outlined in part (a). You are not asked to
do the same for the two other eigenvalues.
35.
Demonstrate the formula
tr A =
k
j -f-
k.2
+ • • • +
k
n,
where the A/ are the complex eigenvalues of the
matrix A, counted with their algebraic multiplicities.
1
2
3
4
5
6
7
8
10

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356
C H A PTER 7
Eigenvalues and Eigenvectors
36.
Hint:
Consider the coefficient of
k
n~ 1
in
(A.)
=
(k\ —
k)(k
2
—
A
.) • • •
(kn —k)y
and compare the result
with Fact 7.2.5.
In 1990, the population of the African country Benin
was about 4.6 million people. Its composition by age
was as follows:
Age Bracket 0-15 15-30 30-45 45-60 60-75 75-90
Percent of
46.6
25.714.78.43.80.8
Population
We represent these data in a state vector whose compo
nents are the populations in the various age brackets, in
millions:
i(0) = 4.6
0.466
0.257
0.147
0.084
0.038
0.008
2.14
1.18
0.68
0.39
0.17
0.04
1.1
1.6
0.6
0
0
0
0.82
0
0
0
0
0
0
0.89
0
0
0
0
0
0
0.81
0
0
0
0
0
0
0.53
0
0
0
0
0
0
0.29
0
We measure time in increments of 15 years, with
t =
0
in 1990. For example, jc(3) gives the age composition in
the year 2035 (1990 + 3
■
15). If current age-dependent
birth and death rates are extrapolated, we have the fol
lowing model:
i(/ + l) =
=
Ax(t).
a.
Explain the significance of all the entries in the ma
trix
A
in terms of population dynamics.
b.
Find the eigenvalue of
A
with largest modulus and an
associated eigenvector. (Use technology.) What is the
significance of these quantities in terms of popula
tion dynamics? (For a summary on matrix techniques
used in the study of age-structured populations, see
Dmitrii O. Logofet,
Matrices and Graphs: Stability
Problems in Mathematical Ecology,
Chapters 2 and
3, CRC Press, 1993.)
37. Consider the set
IHI
of all complex
2 x 2
matrices of the
form
A
=
- z
w
where
w
and
z
are arbitrary complex numbers.
a.
Show that
IHI
is closed under addition and multiplica
tion. (That is, show that the sum and the product of
two matrices in
IHI
are again in
IHI.)
b.
Which matrices in H are invertible?
C 4 =
c.
If a matrix in
HI
is invertible, is the inverse in
IHI
as
well?
d.
Find two matrices
A
and
B
in
M
such that
AB
^
BA.
IHI
is an example of a
skew field:
It satisfies all ax
ioms for a field, except for the commutativity of mul
tiplication. [The skew field
IHI
was introduced by the
Irish mathematician Sir William Hamilton
(1805-
1865);
its elements are called the
quaternions.
An
other way to define the quaternions is discussed in
Exercise
5.3.64.]