Matrices, Digraphs, Markov Chains - PowerPoint PPT Presentation

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Matrices, Digraphs, Markov Chains

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Matrices, Digraphs, Markov Chains & Their Use Introduction to Matrices Matrix arithmetic Introduction to Markov Chains At each time period, every object in the system ... – PowerPoint PPT presentation

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Title: Matrices, Digraphs, Markov Chains


1
Matrices, Digraphs, Markov Chains Their Use
2
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

3
Matrix arithmetic
  • Matrix Addition
  • Matrix Multiplication

4
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.

5
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).

6
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

7
Example The transition matrix in 2nd and 3rd
year..
8
How many SBC customers will be there 2 years from
now?
How many SBC customers will be there 3 years from
now?
9
How many non-SBC customers will be there 2 years
from now?
  • How many non SBC customers will be there 3 years
    from now?

10
  • Thank you!
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