<Text-field style="Heading 2" layout="Heading 2">Maple Knows Eigenvalues!</Text-field>

We are going to study the the calculation and analysis of eigensystems using Maple:

with(LinearAlgebra):

Construct a matrix using shortcut notation and 2D input:

A := < < 1.0 | 2 | 3 > , < 5 | 6 | 7 >, < 9 | 10 | 11 > >;

Evals := Eigenvalues(A);

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Let's test one of these eigenvectors, say the second one.

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Now suppose that you want unitary diagonalization. Try the next two examples.

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Now let's examine Example 5.8 of the text, which models a bird/frog population as a discrete linear dynamical system:

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

Calculate an eigensystem:

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

So we see the bad answer alluded to in the text, page 274, that is, both populations will grow without bound, so it is a bad idea to introduce the bird population into this environment for the purpose of eliminating the frog population.

Let's answer one more question. Are there any kill rates that would ensure that both populations would die out in time? This will require a little bit of programming:

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

Here's an interesting plot:

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Because some numbers in the matrix are negative, you have to be a bit careful about interpreting the model. Now let's do some individual testing of various rates. Try r = 0.35, 0.40 and 0.5.

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

Observe that the population at a given stage is roughly the population at the previous stage times the dominant eigenvalue, if the dominant eigenvalue is a positive number. Thus, a positive dominant eigenvalue \316\273 acts like a growth factor, which means that the growth rate is r = \316\273 - 1.

print();