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

[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?

 

[EnumaElish]

Here is one: http://en.wikipedia.org/wiki/Stochastic_matrix

And another: http://mathworld.wolfram.com/StochasticMatrix.html

 

[oswald]

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!

 

[oswald]

answer
http://www.yaronet.com/posts.php?s=108496