Page 299 - A Course in Linear Algebra with Applications

8.2: Systems of Linear Recurrences 283

Let y n and z n be the numbers of people inside and outside
California after n years; then the information given translates
into the system of linear recurrences

Vn+l = .9j/n + -2Zn
z n+i = .ly n + .8z n

Writing
x
-=it)"**=(* :«

we have X n+i = AX n. The matrix A has eigenvalues 1 and
.7, so we could proceed to solve for y n and z n in the usual way.
However this is unnecessary in the present example since it is
only the ultimate behavior of y n and z n that is of interest.
Assuming that the limits exist, we see that the real object
of interest is the vector

X 00= lim X n = (^n^ocVn
n->oo y imn^oo Z n
i
Taking the limit as n —> oo of both sides of the equation
X n+i = AX n, we obtain X = AX; hence X is an eigenvec-
tor of A associated with the eigenvalue 1. An eigenvector is

quickly found to be ( J. Thus Xoo must be a scalar multiple
of this vector. Now the sum of the entries of X^ equals the
total U.S. population, p say, and it follows that

y p
Y —
3 VI

So the (alarming) conclusion is that ultimately two thirds of
the U.S. population will be in California and one third else-
where. This can be confirmed by explicitly calculating y n and
and taking the limit as n —*• oo.
z n