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Unformatted text preview: lution of (I − A)x = 0, so the vector of limiting proportions is p = (1/6, 1/3, 1/3, 1/6)T . Therefore, if many moves are made, then, regardless of where the pea starts, we expect the pea to end up under shell #1 in about 16.7% of the moves, under #2 for about Copyright c 2000 SIAM Buy online from SIAM Buy from 638 Chapter 7 Eigenvalues and Eigenvectors It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] 33.3% of the moves, under #3 for about 33.3% of the moves, and under shell #4 for about 16.7% of the moves. Note: The shell game (and its analysis) is a typical example of a random walk with reflecting barriers, and these problems belong to a broader classification of stochastic processes known as irreducible, periodic Markov chains. (Markov chains are discussed in detail in §8.4 on p. 687.) The shell game is irreducible in the sense of Exercise 4.4.20 (p. 209), and it is periodic because the pea can return to given position only at definite periods, as reflected in the periodicity of the powers of A. More details are given in Example 8.4.3 on p. 694. D E Exercises for section 7.10 7.10.1. Which of the ⎛ −1/2 A=⎝ 1 1 following are convergent, and which are summable? ⎞ ⎛ ⎞ ⎛ ⎞ 3/2 −3/2 010 −1 −2 −3/2 0 −1/2 ⎠. B = ⎝ 0 0 1 ⎠. C = ⎝ 1 2 1 ⎠. −1 1/2 100 1 1 3/2 T H IG R 7.10.2. For the matrices in Exercise 7.10.1, evaluate the limit of each convergent matrix, and evaluate the Ces`ro limit for each summable matrix. a 7.10.3. Verify that the expressions in (7.10.4) are indeed the solutions to the difference equations in (7.10.3). Y P 7.10.4. Determine the limiting vector for the shell game in Example 7.10.8 by first computing the Ces`ro limit G with a full-rank factorization. a O C 7.10.5. Verify that the expressions in (7.10.4) are indeed the solutions to the difference equations in (7.10.3). 7.10.6....
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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