• whatever 3 values of n_x(0) you may have started with, if you go far enough into the future (t large) you'll end up approximately with

  n1(t) -> 2a,

  n2(t) -> a,

  n3(t) -> a,

  for some limiting constant a;
• if at the start n₁(0) = 2n₂(0) = 2n₃(0) (for example, the suggested values, 200, 100 and 100), then the populations n_x(t) will remain constant for all t > 0;
• as t gets large, the matrices L^t seem to converge to the matrix

  1/2	2/3	1/3
  1/4	1/3	1/6
  1/4	1/3	1/6

infinity + 1 = infinity

(bear with me here; if you can handle the idea of infinity being a number, then this is simpler than the more rigorous mathematical treatment), we expect

L^% = LL^% = L^%L.

And in fact, multiplying L^% from the left or right by L yields L^% back again.

Continuing on in this vein, recall that

n(t) = L^t n(0),

where n(t) is the column matrix of 3 populations n_x(t). Therefore, let

n(%) = L^% n(0),

which is the column of populations after an infinite number of generations (which won't happen until the year 2089).

n₁(%) = 2n₂(%) = 2n₃(%) = (n₁(0)/2 + 2n₂(0)/3 + n₃(0)/3).

(So n₁(%) + 2n₂(%) + n₃(%) = (n₁(0) + 4n₂(0)/3 + 2n₃(0)/3).) This explains what we should have observed playing with the calculator re the ratios of the 1, 2 and 3 year old populations for large t. Note that

L n(%) = LL^% n(0) = L^%+1 n(0) = L^% n(0) = n(%),.................(eigenvector equation),

and that explains why if n₁(t) = 2n₂(t) = 2n₃(t), then n(t+1) = L n(t) = n(t).

A v = µ v

where A is an nxn matrix, v an nx1 matrix (vector), and µ a number. v is called the eigenvector, and µ its eigenvalue. Note that if v is an eigenvector of some matrix A, then so is cv for any constant c; so eigenvectors determine eigen-"directions".

In the L-eigenvector equation written above, n(%) is the eigenvector, and µ = 1 is the corresponding eigenvalue. (There are two other eigenvalues for L, but they are complex numbers with nonzero imaginary components, and they won't be of interest to us.)

On the next page we'll construct a new set of f_x (but leave the P_x the same), resulting in an eigenvalue bigger than 1, and consequently exponential growth.