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4.2 Applications of Eigenvalues/Eigenvectors – Markov Chains
A Markov chain
is a discrete-time stochastic process used to model certain
systems in physics and statistics, for example.
With a Markov Chain, a transition
matrix is used to predict future states of the system.
Eigenvalues and
eigenvectors are used to calculate the steady state of the system after a long
period of time.
We will need two things to get started in a Markov Chain problem:
the initial
state vector, x
0
, and the transition matrix, A.
Example:
Suppose we gather data about a population of birds that inhabit three
islands, named (simply) X, Y, and Z.
We find that birds fly back and forth among
the three islands (so the population of birds is constant) and suppose we find that
the daily process happens as such:
Birds on Island X:
20% stay at X; 50% fly to Y; 30% fly to Z
Birds on Island Y: 20% stay on Y; 40% fly to Z; 40% fly to X
Birds on Island Z:
20% stay on Z; 30% fly to X; 50% fly to Y
This information is summarized below.
From X
From Y
From Z
To X
.2
.4
.3
To Y
.5
.2
.5
To Z
.3
.4
.2
Because each column represents probabilities and the population of birds is
constant, each column sums to 1 (100%).
This forms our transition matrix, A =
.2
.4
.3
.5
.2
.5
.3
.4
.2
.
Suppose at the time that we begin collecting data, there are 400 birds on Island
X, 300 on Island Y and 800 on Island Z.
This gives us an initial state vector,
x
0
=
4 0 0
3 0 0
8 0 0
.
Note that this population, 1500, will be a constant throughout this process--we

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