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
1 2 3 1  1  3
2  (1) 4 1

 
 
 

3 4 2 0  3  (2) 4  0  1 4
 Matrix Multiplication
1 23 1 (1)(3)  (2)(2) (1)(1)  (2)(0)  1 1


 
 

3
4
2
0
(3)(3)

(4)(2)
(3)(1)

(4)(0)
1
3


 
 

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
0.7 0.3
P

0.4 0.6
Example: The transition matrix in 2nd and 3rd
year..
0.7 0.3
P

0.4 0.6
.583 .417 
P 

.556 .444 
3
.61 .39 
P 

.52 .48 
2
How many SBC customers will be there 2 years from now?
.61 .39 
P  3 1 
  2.35 1.65  2.35

.52 .48 
2
How many SBC customers will be there 3 years from now?
.583 .417 
P  3 1 
  2.305 1.695  2.305

.556 .444 
3
How many non-SBC customers will be there 2 years from
now?
.61 .39 
P  3 1 
  2.35 1.65  1.65

.52 .48 
2
 How many non SBC customers will be there 3 years
from now?
.583 .417 
P  3 1 
  2.305 1.695  1.695

.556 .444 
3
Thank you!