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Interpreting the Eigenvalues in a Symmetric
Stochastic Matrix
Kevin D. Salyer
April 11, 2003
Consider the following n-state Markov process for the random variable, x
t
:
x
t
=
?
?
?
?
?
?
?
x
1
x
2
.
.
.
x
n
(1)
The one-period transition probability matrix, with the entry in the ith row
and jth column denoting the conditional probability of going from state i to
state j,is:
? =
?
?
?
?
?
?
?
?
1??
n?1
1??
n?1
···
1??
n?1
1??
n?1
? ··· ···
1??
n?1
1??
n?1
···
.
.
.
...
1??
n?1
1??
n?1
··· ··· ?
1??
n?1
1??
n?1
··· ··· ··· ?
?
?
?
?
?
?
?
(2)
There will be n eigenvalues associated with the stochastic matrix but since
the columns (and rows) sum to one, we know that one of the eigenvalues will
be equal to 1. (The proof is easy: ?·1 = 1). The unconditional probabilities,
p
0
=(p
1,
p
2
,...,p
n
) aregivenbythesolutionto?
T
p = p.Sinceamatrixand
its transpose have the same eigenvalues, we see that the eigenvector associated
w...

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