**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
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638
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540 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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