Biology Reference
In-Depth Information
For example, suppose that a certain matrix
A
is defined as:
⎛
⎞
100
0
⎝
⎠
.
A
=
40
00
−
−
4
Then
⎛
⎞
λ
−
1
0
0
⎝
⎠
,
A
−
λ
I
=
0
λ
+
4
0
0
0
λ
+
4
and
2
det
(
A
−
λ
I
)
=
(λ
−
1
)(λ
+
4
)
.
Therefore, the eigenvalues of
A
are
4.
Each eigenvalue has at least one eigenvector. The total possible number of linearly
independent (that is, different no matter how they are scaled) eigenvectors for an
eigenvalue is the
algebraic dimension
of the eigenvalue. In our example above, the
eigenvalue 1 has only one linearly independent eigenvector, and the eigenvalue
λ
=
1 and
λ
=−
−
4
has a maximum of two linearly independent eigenvectors. However, there could be
fewer such eigenvectors. The actual number of eigenvectors for each eigenvalue can
be determined only by more calculations (see [
11
]). The
geometric dimension
of the
eigenvalue is the number of linearly independent eigenvectors of an eigenvalue that
actually exist. Thus, the geometric dimension is less than or equal to the algebraic
dimension. The set of linear combinations of the eigenvectors for an eigenvalue is its
eigenspace
. Fortunately for us, if the matrix we are using to describe the population
meets certain characteristics, we can isolate a single eigenvalue with one associated
eigenvector (and therefore an eigenspace of 1) that describes the growth and stable
distribution of our population. The Perron-Frobenius theorem makes this possible.
7.8.2
The Perron-Frobenius Theorem
In calculating the stable distribution that is generated by a particular projectionmatrix,
we will take advantage of a theorem in matrix algebra; the
Perron-Frobenius theorem
,
named after the German mathematicians who proved it in the early 20
th
century. The
theorem describes the properties of a matrix that satisfies a certain set of conditions
(see [
12
] for a proof). If the projection matrices generated from biological data meet
these conditions, the theorem allows us to use eigenvalues and eigenvectors to predict
population growth and the stable distribution of individuals across stages. First, we
will examine the theorem and its assumptions, and then we will explain how to apply
it to calculating the stable distribution.
The Perron-Frobenius theorem states:
If
A
is a positive
n
×
n
matrix, that is if
(
A
)
i
,
j
>
0 for all
i
and
j
, then
a.
There is an eigenvalue
r
of
A
for which
r
is positive and
r
>
|
λ
|
for any other
eigenvalue
λ
of
A
. (Note that
A
may have complex eigenvalues.)
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