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Eigenvalues and Eigenvectors
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→∞ It is illegal to print, duplicate, or distribute this material
Please report violations to firstname.lastname@example.org = Example 7.3.6 k→∞ n0 + s0
3 k→∞ 2(n0 + s0 )
3 = 1
3 s0 ) 1/3
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
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
R pT = lim pT Tk = pT lim Tk = pT G1 =
k→∞ • k→∞ T
(pT x1 )y1
= T 1 . (7.3.15)
y1 x1 The fact that pT Tk converges to an eigenvector is a special case of the
power method discussed in Example 7.3.7.
Equation (7.3.15) shows why the initial distribution pT always drops away
in the limit. But pT is not completely irrelevant because it always aﬀects
the transient behavior—i.e., the behavior of pT = pT Tk for smaller k ’s.
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