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Disadvantages only a dominant eigenpair is

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Unformatted text preview: p://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 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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