Find the equilibrium vector for the given matrix.
First, we should determine if the transition matrix P is regular.
Since all of the entries are positive, P is regular.
Since it is regular, we can find a probability vector V where VP = V.
V will be the equilibrium vector. Let V = [x, y]. Then
VP = [x y]
= [0.3x + 0.4y 0.7x+0.6y]
Set the entries equal to each other
0.3x + 0.4y = x and 0.7x +0.6y = y
Simplify both equations.
-0.7x + 0.4y = 0 and 0.7x – 0.4y = 0
Therefore are really the same equation, so we need a second equation.
Since V is a probability vector, x + y = 1.
Now solve the system.
-0.7x + 0.4y = 0
x + y = 1
[x y ] =