Matrices, Digraphs, Markov Chains

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Matrices, Digraphs, Markov Chains Their Use
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Introduction to Matrices

• A matrix is a rectangular array of numbers
• Matrices are used to solve systems of equations
• Matrices are easy for computers to work with

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Matrix arithmetic

• Matrix Addition

• Matrix Multiplication

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Introduction to Markov Chains

• At each time period, every object in the system
  is in exactly one state, one of 1,,n.
• Objects move according to the transition
  probabilities the probability of going from
  state j to state i is tij
• Transition probabilities do not change over time.

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The transition matrix of a Markov chain

• T tij is an n ? n matrix.
• Each entry tij is the probability of moving from
  state j to state i.
• 0 ? tij ? 1
• Sum of entries in a column must be equal to 1
  (stochastic).

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Example Customers can choose from a major Long
Distance carrier (SBC) or others ores

• Each year 30 of SBC customers switch to other
  carrier, while 40 of other carrier switch to
  SBC.
• Set Up the matrix for this Problem

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Example The transition matrix in 2nd and 3rd
year..
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How many SBC customers will be there 2 years from
now?
How many SBC customers will be there 3 years from
now?
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How many non-SBC customers will be there 2 years
from now?

• How many non SBC customers will be there 3 years
  from now?

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• Thank you!