What Is the Long-Term Vector Response for Markov Transition Matrix?

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Homework Help Overview

The discussion revolves around a Markov Transition Matrix related to the movement of residents among urban, suburban, and rural areas. The original poster presents a specific matrix and initial vector values, seeking to understand the long-term vector response and the vector response at time t = 2.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the implications of the Markov Transition Matrix and its long-term behavior. There are attempts to derive equations based on the matrix and initial conditions, with some participants questioning the effectiveness of methods like Cramer's rule in this context.

Discussion Status

The discussion is ongoing, with participants providing links to external resources for further reading. There is an exploration of different mathematical approaches, but no consensus has been reached regarding the solutions to the equations posed by the original poster.

Contextual Notes

Participants note the constraints of the problem, including the specific values assigned to variables in the matrix and the requirement to find a long-term solution. There is also mention of the challenges faced with certain mathematical techniques in solving the posed equations.

oswald
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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?
 
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[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!
 

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