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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!