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

Matrix arithmetic

- Matrix Addition

- Matrix Multiplication

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.

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).

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

Example The transition matrix in 2nd and 3rd

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?

- Thank you!