Matrices, Digraphs, Markov Chains & Their Use

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 2ndand 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!