Matrices, Digraphs, Markov Chains & Their Use

Matrices, Digraphs, Markov Chains & Their Use

Introduction to MatricesIntroduction 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 arithmeticMatrix arithmetic Matrix Addition Matrix Addition

1 2

3 4

3 1

2 0

1 3 2 ( 1)

3 ( 2) 4 0

4 1

1 4

1 2

3 4

3 1

2 0

(1)(3) (2)( 2) (1)( 1) (2)(0)

(3)(3) (4)( 2) (3)( 1) (4)(0)

1 1

1 3

Matrix Multiplication 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 tij

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 tij

Transition probabilities do not change over time.

Introduction to Markov Chains

Introduction to Markov Chains

The transition matrix of a Markov chainThe 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).

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:

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

6.04.0

3.07.0P

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

6.04.0

3.07.0P 2 .61 .39

.52 .48 P

3 .583 .417

.556 .444P

How many SBC customers will be there 2 years

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

from now?

2 .61 .39

.52 .48 3 1 2.35 1.65 2.35P

3 .583 .417

.556 .4443 1 2.305 1.695 2.305P

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 2 years

from now?

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

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

2 .61 .39

.52 .48 3 1 2.35 1.65 1.65P

3 .583 .417

.556 .4443 1 2.305 1.695 1.695P

Thank you!Thank you!