Linear Algebra, Theory And Applications, 2012a

Markov Chains And Migration
Processes
11.1 Regular Markov Matrices
The existence of the Jordan form is the basis for the proof of limit theorems for certain
kinds of matrices called Markov matrices.
Definition 11.1.1 An n × n matrix A =(a ij ) , is a Markov matrix if a ij ≥ 0 for all i, j
and
∑
a ij =1.
i
It may also be called a stochastic matrix. A matrix which has nonnegative entries such that
∑
a ij =1
j
will also be called a stochastic matrix. A Markov or stochastic matrix is called regular if
some power of A has all entries strictly positive. A vector, v ∈ R n , is a steady state if
Av = v.
Lemma 11.1.2 The property of being a stochastic matrix is preserved by taking products.
Proof: Suppose the sum over a row equals 1 for A and B. Then letting the entries be
denoted by (a ij )and(b ij ) respectively,
∑ ∑
a ik b kj = ∑ ( ) ∑
a ik b kj = ∑ b kj =1.
i k
k i
k
A similar argument yields the same result in the case where it is the sum over a column
which is equal to 1. It is obvious that when the product is taken, if each a ij ,b ij ≥ 0, then
the same will be true of sums of products of these numbers.
The following theorem is convenient for showing the existence of limits.
Theorem 11.1.3 Let A be a real p × p matrix having the properties
1. a ij ≥ 0
2. Either ∑ p
i=1 a ij =1or ∑ p
j=1 a ij =1.
3. The distinct eigenvalues of A are {1,λ 2 ,...,λ m } where each |λ j | < 1.
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