State Matrices and Recurrence Relations

2. Learning Intention and Success
Criteria
 Learning Intention: Students will know what a state
matrix is, and will be able to use transition matrices
and state matrices to model a change in states over
time.
 Success criteria: Students can create a recurrence
relation involving a transition matrix, and use the
recurrence relation and general formula to determine
the state after a particular time period and in the long
term.

3. Example Revisited
 This week, there were 150 people who ate at Mia’s Pizza
and 50 people who ate at Dan’s Dumplings. Research
has shown that each week, 20% of people who ate at
Mia’s will switch to Dan’s, and 30% of people who ate
at Dan’s will switch to Mia’s.
 How many people will eat at Mia’s and at Dan’s next
week?

4. Converting to Matrices
0.8 × 150 + 0.3 × 50 = 135
0.2 × 150 + 0.7 × 50 = 65
 Convert this to a matrix equation
0.8 0.3
0.2 0.7
×
150
50
=
135
65

5. The Matrix Equation
0.8 0.3
0.2 0.7
×
150
50
=
135
65
 In this lesson, we will be focusing on the column
matrices, called the State Matrices
Transition Matrix State Matrices

6. State Matrix
 It is a column matrix
 One row for each possible state
 Tells us the number or percentage of the population in
each state at one moment in time.
 Denoted 𝑆 𝑛, representing the state at time 𝑠.
Initial State Matrix (𝑆0):
 Tells us the number or percentage of the population in
each state at the beginning.
 The initial conditions

7. Calculating State Matrices
 As seen in our example,
𝑆1 = 𝑇 × 𝑆0
135
65
=
0.8 0.3
0.2 0.7
×
150
50
 How many customers will eat at Dan’s and at Mia’s in
the following week?
 Calculate 𝑆2

127.5
72.5
=
0.8 0.3
0.2 0.7
×
135
65

8. Transition Matrices and Recurrence
Relations
 In general, 𝑆 𝑛+1 = 𝑇 × 𝑆 𝑛 or 𝑆 𝑛 = 𝑇 × 𝑆 𝑛−1.
 This is a recurrence relation, like the previous unit
 𝑆 𝑛 = 𝑇 × 𝑆 𝑛−1, 𝑆0 = …
 Each state is found from the previous state, and there
is an initial state matrix.

9. Example
 For the pizza example, we have
𝑆 𝑛+1 =
0.8 0.3
0.2 0.7
× 𝑆 𝑛, 𝑆0 =
150
50
𝑆1 =
0.8 0.3
0.2 0.7
× 𝑆0
=
0.8 0.3
0.2 0.7
×
150
50
=
135
65
𝑆2 =
0.8 0.3
0.2 0.7
× 𝑆1
=
0.8 0.3
0.2 0.7
×
135
65
=
127.5
72.5

10. General Rule
Use the recursive formula to generate a general rule for
𝑆 𝑛.
𝑆1 = 𝑇 × 𝑆0
𝑆2 = 𝑇 × 𝑆1
= 𝑇 × 𝑇 × 𝑆0
= 𝑇2 × 𝑆0
𝑆3 = 𝑇 × 𝑆2
= 𝑇 × 𝑇 × 𝑇 × 𝑆0
= 𝑇3
× 𝑆0
In general, 𝑆 𝑛 = 𝑇 𝑛
× 𝑆0

11. Example
 Calculate the number of people expected at Mia’s and
at Dan’s after:
 4 weeks?
 20 weeks?
 21 weeks?
 Round to the nearest whole number.

12. Example
 Calculate the number of people expected at Mia’s and
at Dan’s after:
 4 weeks?
 Round to the nearest whole number.
𝑆4 =
0.8 0.3
0.2 0.7
4
×
150
50
=
121.875
78.125
122 at Mia’s, 78 at Dan’s

13. Example
 Calculate the number of people expected at Mia’s and
at Dan’s after:
 20 weeks?
 Round to the nearest whole number.
𝑆20 =
0.8 0.3
0.2 0.7
20
×
150
50
=
120.00002861
79.9999713898
120 at Mia’s, 80 at Dan’s

14. Example
 Calculate the number of people expected at Mia’s and
at Dan’s after:
 20 weeks?
 Round to the nearest whole number.
𝑆21 =
0.8 0.3
0.2 0.7
21
×
150
50
=
120.000014305
79.999985649
120 at Mia’s, 80 at Dan’s

15. An Interesting Pattern
 Notice that 𝑆20 = 𝑆21. If we try 𝑆50, 𝑆75, 𝑆100, we get the
same value
120
80
.
 This is called the steady state solution.
 This occurs when, in the long run, the state matrix will
be the same.
 To determine the steady state solution, use a high
value of 𝑛 and calculate 𝑆 𝑛. (Your CAS can handle 𝑆100
easily)

16. Example
 Consider the car buying example from the previous lesson.
In the long term, what percentage of people will buy
Holdens? Initially, there is an equal number of drivers for
all three companies.
𝑇 =
𝑓𝑟𝑜𝑚
𝐻𝑜𝑙𝑑 𝐻𝑜𝑛 𝑉𝑊
0.65 0.08 0.02
0.33 0.85 0.06
0.02 0.07 0.92
𝐻𝑜𝑙𝑑
𝐻𝑜𝑛
𝑉𝑊
𝑡𝑜

17. Example
0.65 0.08 0.02
0.33 0.85 0.06
0.02 0.07 0.92
20
×
100/3
100/3
100/3
=
In the long term, 12.7% of people drive Holdens, 44.9% of
people drive Hondas and 42.4% of people drive Volkswagens.
The long term steady state solution does not depend on the
initial population distribution, just on the total population.