I'm curious. If all you need is 2 (b) why did you post the first page?

Anyway, we have a matrix equation:

and an initial population distribution:

where is the set of eigenvectors of A.

Let be the set of eigenvalues corresponding to the (respective) eigenvector.

Then

Iterating this process gives:

Now, if is the dominant eigenvalue (I think this said the first eigenvalue. It was hard to read.) then we may assume

for all . (Obviously only a positive eigenvalue can dominate the population.) This means that for large k there is effectively only one term in the that is significant: the first term. So

for large k.

This means the population is stable in the state represented by eigenvector and has a growth rate represented by .

If there is no dominant eigenvalue then the population will fluctuate in a series of eigenstates with no specific growth rate dominating. However, for large enough k the population should stabilize around a vector that is the sum of eigenvectors with the largest eigenvalues. For example, say that we know that for . Then, given a large enough k we would have:

So the population would settle into a growth state vector with (in this case) a growth of .

-Dan