# Markov Transition Matrix

• oswald
In summary, the conversation discusses a Markov Transition Matrix that shows the movement of residents between urban, suburban, and rural areas. It also provides values for A, O, B, Q, Y, and Z which represent the probabilities of people moving between areas. The problem then asks for the vector response for t=2 and in the long run, as well as any suggested bibliographies related to stochastic matrices. The conversation ends with a new question about finding the values for q1, q2, and q3 using a set of equations. Cramer's rule does not work for this problem.
oswald
A dynamic interurban of people shows the following Markov Transition Matrix of residents to urban, suburban and rural areas:

__________Urban___Suburban____Rural
Urban ... a...b...y
suburban... o.....q.....z
Rural ... 1-a-o ...1-b-q ... 1-y-z

A = 0.9
O = 0.05
B = 0.1
Q = 0.7
Y = 0.1
Z = 0.1

Solve the problem knowing that the vector answer today is
Uo ... 10
(SUo) = (... 40)
Ro ... 50

What vector response to t = 2 vector and what is the answer in the long run. Remember that
Ut
(SUt) = Aλ1 ^ tX1 + Bλ2 ^ tX2 + Cλ3 ^ tX3
Rt

any bibliography suggestion?

New question

[q1 q2 q3] x [ 0,1 -0,05 -0,05] = [ 0 0 0 ]
......-0,7..1...-0,3
......-0,8..0...0,8

0,1q1 - 0,7q2 -0,8q3 =0
-0,5q1 +q2 = 0
-0,5q1 -0,3q2+0,8q3=0

q1+q2+q3=1

q1=?
q2=?
q3=?
Cramer doesn't work!

## 1. What is a Markov Transition Matrix?

A Markov Transition Matrix is a mathematical tool used to describe the probability of transitioning from one state to another in a discrete event or system. It is used in various fields, including economics, biology, and computer science.

## 2. How is a Markov Transition Matrix constructed?

A Markov Transition Matrix is constructed using a square matrix, where the rows and columns represent the possible states of the system. The values in the matrix represent the probabilities of transitioning from one state to another. The sum of each row in the matrix must equal 1.

## 3. What is the importance of a Markov Transition Matrix?

A Markov Transition Matrix is an essential tool in analyzing and predicting the behavior of a system over time. It allows for a mathematical representation of the system and helps to understand the probabilities of future states and events.

## 4. Can a Markov Transition Matrix be used to model complex systems?

Yes, a Markov Transition Matrix can be used to model complex systems, but it is limited to systems that meet certain assumptions. These assumptions include a finite number of states, the Markov property (future state depends only on the current state), and a constant transition probability over time.

## 5. How is a Markov Transition Matrix different from a regular transition matrix?

A Markov Transition Matrix differs from a regular transition matrix in that it follows the Markov property, meaning that future states only depend on the current state and not on previous states. In a regular transition matrix, the probabilities may depend on the entire history, making it more complex to analyze and predict future states.

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