Matrices Digraphs Markov Chains Their Use Introduction to

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Matrices, Digraphs, Markov Chains & Their Use

Matrices, Digraphs, Markov Chains & Their Use

Introduction to Matrices s A matrix is a rectangular array of numbers s Matrices

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

Matrix arithmetic s Matrix Addition s Matrix Multiplication

Matrix arithmetic s Matrix Addition s Matrix Multiplication

Introduction to Markov Chains s At each time period, every object in the system

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

The transition matrix of a Markov chain s T = [tij] is an n

The transition matrix of a Markov chain s T = [tij] is an n n matrix. s Each entry tij is the probability of moving from state j to state i. s 0 tij 1 s 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:

Example: Customers can choose from a major Long Distance carrier (SBC) or others ores: s Each year 30% of SBC customers switch to other carrier, while 40% of other carrier switch to SBC. s Set Up the matrix for this Problem

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

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

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? s How many

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

Thank you!

Thank you!