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532
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540
accomplished by analyzing limk→∞ Tk . The results of Example 7.3.4 together
with n0 + s0 = 1 yield the long-run (or limiting) population distribution as
pT = lim pT = lim pT Tk = pT lim Tk = ( n0
∞
k
0
0
k→∞ It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] = Example 7.3.6 k→∞ n0 + s0
3 k→∞ 2(n0 + s0 )
3 = 1
3 2
3 s0 ) 1/3
1/3 2/3
2/3 . So if the migration pattern continues to hold, then the population distribution
will eventually stabilize with 1/3 of the population being in the North and 2/3 of
the population in the South. And this is independent of the initial distribution! D
E Observations: This is an example of a broader class of evolutionary processes
known as Markov chains (p. 687), and the following observations are typical.
• It’s clear from (7.3.12) or (7.3.13) that the rate at which the population
distribution stabilizes is governed by how fast (1/4)k → 0. In other words,
the magnitude of the largest subdominant eigenvalue of T determines the
rate of evolution.
• For the dominant eigenvalue λ1 = 1, the column, x1 , of 1’s is a righthand eigenvector (because T has unit row sums). This forces the limiting
distribution pT to be a particular left-hand eigenvector associated with
∞
T
λ1 = 1 because for an arbitrary left-hand eigenvector y1 associated with
λ1 = 1, equation (7.3.13) in Example 7.3.4 insures that T
H Y
P IG
R pT = lim pT Tk = pT lim Tk = pT G1 =
∞
0
0
0
k→∞ • k→∞ T
(pT x1 )y1
yT
0
= T 1 . (7.3.15)
Tx
y1 1
y1 x1 The fact that pT Tk converges to an eigenvector is a special case of the
0
power method discussed in Example 7.3.7.
Equation (7.3.15) shows why the initial distribution pT always drops away
0
in the limit. But pT is not completely irrelevant because it always aﬀects
0
the transient behavior—i.e., the behavior of pT = pT Tk for smaller k ’s.
0
k O
C Cayley–Hamilton...

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