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
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 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: 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. .
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 non SBC customers will be there 3 years from now?