Matrices, Digraphs, Markov Chains & Their Use

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

Matrix arithmetic  Matrix Addition  Matrix Multiplication

 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 t ij  Transition probabilities do not change over time.  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 t ij  Transition probabilities do not change over time. Introduction to Markov Chains

The transition matrix of a Markov chain  T = [t ij ] is an n  n matrix.  Each entry t ij is the probability of moving from state j to state i.  0  t ij  1  Sum of entries in a column must be equal to 1 (stochastic).  T = [t ij ] is an n  n matrix.  Each entry t ij is the probability of moving from state j to state i.  0  t ij  1  Sum of entries in a column must be equal to 1 (stochastic).

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  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

Example: The transition matrix in 2 nd and 3 rd year..

How many SBC customers will be there 2 years from now? How many SBC customers will be there 3 years from now?

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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