**Unformatted text preview: **igenvector x1 was computed above.
T
Computing a left-hand eigenvector y1 = (1, 2) yields
Tk → G1 = Copyright c 2000 SIAM T
x1 y1
1
=
Tx
3
y1 1 1
1 2
2 . (7.3.13) Buy online from SIAM
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7.3 Functions of Diagonalizable Matrices
http://www.amazon.com/exec/obidos/ASIN/0898714540 531 Example 7.3.5 Population Migration. Suppose that the population migration between two
geographical regions—say, the North and the South—is as follows. Each year,
50% of the population in the North migrates to the South, while only 25% of
the population in the South moves to the North. This situation is depicted by
drawing a transition diagram as shown below in Figure 7.3.1. It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] .5 N .5 S D
E .75 .25 T
H Figure 7.3.1 Problem: If this migration pattern continues, will the population in the North
continually shrink until the entire population is eventually in the South, or will
the population distribution somehow stabilize before the North is completely
deserted? IG
R Solution: Let nk and sk denote the respective proportions of the total population living in the North and South at the end of year k, and assume nk + sk = 1.
The migration pattern dictates that the fractions of the population in each region
at the end of year k + 1 are Y
P nk+1 = nk (.5) + sk (.25)
sk+1 = nk (.5) + sk (.75)
pT
k or, equivalently, pT+1 = pT T,
k
k (7.3.14) where
= ( nk sk ) and pT+1 = ( nk+1 sk+1 ) are the respective population
k
distributions at the end of years k and k + 1, and where O
C N
T=
S N
.5
.25 S
.5
.75 is the associated transition matrix (recall Example 3.6.3). Inducting on
pT = pT T,
1
0 pT = pT T = pT T2 ,
2
1
0 pT = pT T = pT T3 ,
3
2
0 ··· leads to pT = pT Tk , which indicates that the powers of T determine how the
0
k
73
process evolves. Determining the long-run population distribution is therefore
73 The long-run distribution goes by a lot of diﬀerent names. It’s also called the limiting distribution, the steady-state distribution, and the stationary distribution. Copyright c 2000 SIAM Buy online from SIAM
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