APPLICATIONS OF LINEAR ALGEBRA
ELECTRONIC VERSION OF LECTURE
Dr. Lê Xuân Đại
HoChiMinh City University of Technology
Faculty of Applied Science, Department of Applied Mathematics
Email: ytkadai@hcmut.edu.vn
HCMC — 2019.
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
1 / 25
OUTLINE
1
MARKOV CHAINS
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
2 / 25
OUTLINE
1
MARKOV CHAINS
2
LEONTIEF INPUT-OUTPUT MODELS
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
2 / 25
Markov Chains
Stochastic matrices and Markov chains
Many types of applications
involve
a
n
o
finite set of state S 1, S 2, . . . , S n of a
population.
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
3 / 25
Markov Chains
Stochastic matrices and Markov chains
Many types of applications
involve
a
n
o
finite set of state S 1, S 2, . . . , S n of a
population.
For instance, residents of a city may live
downtown or in the suburbs. Soft drink
consumers may buy Coca-Cola, Pepsi, or
another brand.
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
3 / 25
Markov Chains
Stochastic matrices and Markov chains
Many types of applications
involve
a
n
o
finite set of state S 1, S 2, . . . , S n of a
population.
For instance, residents of a city may live
downtown or in the suburbs. Soft drink
consumers may buy Coca-Cola, Pepsi, or
another brand.
The probability that a member of a
population will change ưfrom the j th
state to the i th state is represented by a
number p i j , where 0 É p i j É 1.
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
3 / 25
Markov Chains
Stochastic matrices and Markov chains
DEFINITION 1.1



The matrix P = 


p 11 p 12 . . . p 1n
p 21 p 22 . . . p 2n 

is called
...
... . . . ... 

p n1 p n2 . . . p nn
the matrix of transition probabilities.
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
4 / 25
Markov Chains
Stochastic matrices and Markov chains
DEFINITION 1.1



The matrix P = 


p 11 p 12 . . . p 1n
p 21 p 22 . . . p 2n 

is called
...
... . . . ... 

p n1 p n2 . . . p nn
the matrix of transition probabilities.
At each transition, each member in a given
state must either stay in that state or change
to another state. It means that
n
P
i =1
Dr. Lê Xuân Đại (HCMUT-OISP)
p i j = 1, ∀ j = 1..n.
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
4 / 25
Markov Chains
Stochastic matrices and Markov chains
DEFINITION 1.2
The n th state matrix of a Markov chain for
which P is the matrix of transition
probabilities and X 0 is the initial state
matrix is
X n = P X n−1 = P 2 X n−2 = . . . = P n X 0
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
(1)
5 / 25
Markov Chains
Stochastic matrices and Markov chains
EXAMPLE 1.1 (A CONSUMER PREFERENCE
MODEL)
Two competing companies offer satellite
television service to a city with 100 000
households. The figure below shows the
changes in satellite subscription each year.
Company A now has 15 000 subscribers and
Company B has 20 000 subscribers. How
many subscribers will each company have in
one year? After 10 year? After 15 year?
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
6 / 25
Markov Chains
Dr. Lê Xuân Đại (HCMUT-OISP)
Stochastic matrices and Markov chains
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
7 / 25
Markov Chains
Stochastic matrices and Markov chains
SOLUTION. The matrix of transition
probabilities is


0.70 0.15 0.15


P =  0.20 0.80 0.15 
0.10 0.05 0.70
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
8 / 25
Markov Chains
Stochastic matrices and Markov chains
SOLUTION. The matrix of transition
probabilities is


0.70 0.15 0.15


P =  0.20 0.80 0.15 
0.10 0.05 0.70
and the initial state matrix representing the
portions of the
in the 3
 total population

15000


states is X 0 =  20000  .
65000
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
8 / 25
Markov Chains
Stochastic matrices and Markov chains
To find the state matrix representing the
portions of the population in the 3 states in
one year, multiply P by X 0 to obtain



0.70 0.15 0.15
15000



X 1 = P X 0 =  0.20 0.80 0.15   20000  =
0.10 0.05 0.70
65000


23250


=  28750 
48000
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
9 / 25
Markov Chains
Stochastic matrices and Markov chains
To find the numbers of subscribers after 10
years, first find X 10


33290


X 10 = P 10 X 0 ≈  47150 
19570
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
10 / 25
Markov Chains
Stochastic matrices and Markov chains
To find the numbers of subscribers after 10
years, first find X 10


33290


X 10 = P 10 X 0 ≈  47150 
19570
To find the numbers of subscribers after 15
years, first find X 15


33330


X 15 = P 15 X 0 ≈  47560 
19110
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
10 / 25
Markov Chains
Steady State matrix of a Markov chain
Note. There is little difference between the
numbers of subscribers after 10 years and
after 15 years. If we continue this process,
then the state matrix X n eventually reaches
a steady state.
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
11 / 25
Markov Chains
Steady State matrix of a Markov chain
Note. There is little difference between the
numbers of subscribers after 10 years and
after 15 years. If we continue this process,
then the state matrix X n eventually reaches
a steady state.
DEFINITION 1.3
A stochastic matrix P is said to be regular if
P or some positive power of P has all
positive entries, and a Markov chain whose
transition matrix is regular is said to be a
regular Markov chain.
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
11 / 25
Markov Chains
Steady State matrix of a Markov chain
THEOREM 1.1
If P is the transition matrix for a regular
Markov chain, then
There is a unique probability vector X ,
which is called the steady-state vector of
the Markov chain, with positive entries
such that P X = X .
1
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
12 / 25
Markov Chains
Steady State matrix of a Markov chain
THEOREM 1.1
If P is the transition matrix for a regular
Markov chain, then
There is a unique probability vector X ,
which is called the steady-state vector of
the Markov chain, with positive entries
such that P X = X .
For any initial probability vector X 0, the
sequence of state vectors
X 0 , P X 0 , . . . , P n X 0 , . . . converges to X
1
2
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
12 / 25
Markov Chains
Steady State matrix of a Markov chain
EXAMPLE 1.2
Find the steady state matrix X of the Markov
chain whose matrix of transition
probabilities is the regular matrix


0.70 0.15 0.15


P =  0.20 0.80 0.15 
0.10 0.05 0.70
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
13 / 25
Markov Chains

Steady State matrix of a Markov chain

x1


SOLUTION. Let X =  x 2  . Then use the
x3
matrix equation P X = X to obtain


 

0.70 0.15 0.15
x1
x1


 

 0.20 0.80 0.15   x 2  =  x 2 
0.10 0.05 0.70
x3
x3
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
14 / 25
Markov Chains
Steady State matrix of a Markov chain
Use these equations and the fact that
x 1 + x 2 + x 3 = 100000,
 we can
 find
 the steady


state matrix X = 
Dr. Lê Xuân Đại (HCMUT-OISP)
100000
3
1000000
21
400000
21
33333
 

 ≈  47619 
19048
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
15 / 25
Leontief input-output models
LEONTIEF INPUT-OUTPUT MODELS
Consider an economic system that has n
different industries I 1, I 2, . . . , I n , each having
input needs (raw materials, utilities, etc.)
and an output (finished product). In
producing each unit of output, an industry
may use the outputs of other industries,
including itself. For example, an electric
utility uses outputs from other industries,
such as coal and water, and also uses its
own electricity.
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
16 / 25
Leontief input-output models
Let di j be the amount of output the j th
industry needs from the i th industry to
produce one unit of output per year. The
matrix of these coefficients is the
input-output matrix

d 11 d 12
d
 21 p 22
D =  ..
...
 .
d n1 d n2
(0 É d i j É 1, ∀i , j = 1..n;
Dr. Lê Xuân Đại (HCMUT-OISP)

. . . d 1n
. . . d 2n 


.
. . . .. 
. . . d nn
n
P
d i j É 1, ∀ j = 1..n).
i =1
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
17 / 25
Leontief input-output models
FORMING AN INPUT-OUTPUT MATRIX
EXAMPLE 2.1
Consider a simple economic system consisting of 3
industries: electricity, water, and coal. Production, or
output, of one unit of electricity requires 0.5 unit of
itself, 0.25 unit of water, and 0.25 unit of coal.
Production of one unit of water requires 0.1 unit of
electricity, 0.6 unit of itself, and 0 units of coal.
Production of one unit of coal requires 0.2 unit of
electricity, 0.15 unit of water, and 0.5 unit of itself.
Find the input-output matrix for this system.
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
18 / 25
Leontief input-output models
SOLUTION. The column entries show the
amounts each industry requires from the
others, and from itself, to produce one unit
of output
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
19 / 25
Leontief input-output models
SOLUTION. The column entries show the
amounts each industry requires from the
others, and from itself, to produce one unit
of output


0.5 0.1 0.2


D =  0.25 0.6 0.15 
0.25 0 0.5
The row entries show the amounts each
industry supplies to the others, and to itself,
for that industry to produce one unit of
output.
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
19 / 25
Leontief input-output models
Let the total output of the i th industry be
denoted by x i . If the economic system is
closed (that is, the economic system sells its
products only to industries within the
system), then the total output of the i th
industry is
(2)
xi = di 1 x1 + di 2 x2 + . . . + di n xn
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
20 / 25
Leontief input-output models
If the industries within the system sell
products to nonproducing groups (such as
governments or charitable organizations)
outside the system, then system is open
and the total output of the i th industry is
xi = di 1 x1 + di 2 x2 + . . . + di n xn + e 1
(3)
where e i represents the external demand for
the i th industry’s product.
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
21 / 25
Leontief input-output models
The collection of total outputs for an open
system is


x 1 = d 11 x 1 + d 12 x 2 + . . . + d 1n x n + e 1


 x = d x +d x +...+d x +e
2
21 1
22 2
2n n
2

...



x n = d n1 x 1 + d n2 x 2 + . . . + d nn x n + e n
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
22 / 25
Leontief input-output models
The collection of total outputs for an open
system is


x 1 = d 11 x 1 + d 12 x 2 + . . . + d 1n x n + e 1


 x = d x +d x +...+d x +e
2
21 1
22 2
2n n
2

...



x n = d n1 x 1 + d n2 x 2 + . . . + d nn x n + e n
The matrix form of this system is
(4)
X = DX +E,
where X is the output matrix and E is the
external demand matrix.
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
22 / 25
Leontief input-output models
EXAMPLE 2.2
An economic system composed of 3
industries has the input-output matrix


0.1 0.43 0


D =  0.15 0 0.37 
0.23 0.03 0.02
Find the output matrix
X when the external

20000


demands are E =  30000 
25000
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
23 / 25
Leontief input-output models
SOLUTION.
X = D X + E ⇒ (I − D)X = E ⇒ X = (I − D)−1 E .


46616


X ≈  51058 
38014
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
24 / 25
Leontief input-output models
SOLUTION.
X = D X + E ⇒ (I − D)X = E ⇒ X = (I − D)−1 E .


46616


X ≈  51058 
38014
To produce the given external demands, the
outputs of the 3 industries must be
approximately 46616 units for the industry I,
51058 units for industry II, and 38014 units
for industry III.
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
24 / 25
Leontief input-output models
THANK YOU FOR YOUR ATTENTION
Dr. Lê Xuân Đại (HCMUT-OISP)
APPLICATIONS OF LINEAR ALGEBRA
HCMC — 2019.
25 / 25