Page 298 - A Course in Linear Algebra with Applications

282 Chapter Eight: Eigenvalues and Eigenvectors

system of first order recurrences

Vn+l =Vn + Z n
Zn+l = Vn

with initial conditions y 0 = 0 and z 0 = 1. The coefficient
matrix A = I J has eigenvalues (1 + /5)/2 and
v
/
(1 — y 5)/2, so it is diagonalizable. Diagonalizing A as in
Example 8.1.5, we find that

1 (1 + V5)/2
D-S- AS-( ° "i

where
r
S = ( ( 1 + V5)/2 ( l - V 5 ) / 2 y

n 1 n 1
Then Y n = A Y 0 = (SBS' )^ = SD S' Y 0. This yields
the rather unexpected formula

for the (n + l)th Fibonacci number.

Markov processes
In order to motivate the concept of a Markov process, we
consider a problem about population movement.

Example 8.2.4
Each year 10% of the population of California leave the state
for some other part of the United States, while 20% of the
U.S. population outside California enter the state. Assum-
ing a constant total population of the country, what will the
ultimate population distribution be?