MARKOV CHAIN

1
MARKOV CHAIN
A, B and C are three towns. Each year 10 of the
residents of A move to B 30 of the residents of
A move to C 20 of the residents of B move to
A 30 of the residents of B move to C 20 of the
residents of C move to A 20 of the residents of
C move to B Everybody else stays put. No one
dies. No one is born.
2
B
A, B, and C are three towns.
A
10 people live in A
20 people live in B
15 people live in C
C
3
B
A
C
4
B
A
.10
.20
C
5
B
A
.10
.20
.20
.30
C
6
B
A
.10
.20
.20
.20
.30
.20
C
7
B
A
.10
.20
.20
.20
.30
.20
C
8
B
A
.10
.20
.20
.20
.30
.20
C
9
B
This year
Next year
Where will everyone be in 10 years?
10
MATRIX ALGEBRA TO THE RESCUE
11
B
A
.10
.20
.20
.20
.30
.20
C
12
B
.10
A
.20
.20
.30
.20
An1 .7An
.20
C
13
B
.10
A
.20
.20
.30
.20
An1 .7An .2Bn .2Cn
.20
C
14
B
.10
A
.20
.20
.30
.20
An1 .7An .2Bn .2Cn
.20
Bn1 .1An .5Bn .2Cn
C
15
B
.10
A
.20
.20
.30
.20
An1 .7An .2Bn .2Cn
.20
Bn1 .1An .5Bn .2Cn
Cn1 .2An .3Bn .6Cn
C
16
B
.10
A
.20
.20
.30
.20
An1 .7An .2Bn .2Cn
.20
Bn1 .1An .5Bn .2Cn
Cn1 .2An .3Bn .6Cn
C
17
An1 .7An .2Bn .2Cn
Bn1 .1An .5Bn .2Cn
Cn1 .2An .3Bn .6Cn
Cn1 .2An .3Bn .6Cn
Cn1 .2An .3Bn .6Cn
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20
Because the sum of the numbers in each column of
M is 1, 1 is an eigenvalue for M, and An
eigenvector belonging to 1 will describe stable
proportions.
The null space of 1I M the null space of
-

21

The null space of
Suppose there are this year 140 people in A, 80
people in B, and 130 people in C. How many will
be in each town next year?
?