Suppose there are three major centers for Move-It-Yourself trucks. Every month half
of those in Boston and in Los Angeles go to Chicago, the other half stay here they
are, and the trucks in Chicago are split equally between Boston and Los Angeles Set
up the 3 by 3 transition matrix A, and find the steady state u∞ corresponding to the
eigenvalue λ = 1.
To set up the transition matrix A, we can consider the three major centers as states: Boston, Los Angeles, and Chicago. The rows in the matrix represent the starting states, and the columns represent the ending states.
Let's label the states as follows:
1. Boston
2. Los Angeles
3. Chicago
According to the given information, every month, half of the trucks in Boston and half in Los Angeles go to Chicago. So, the transition probabilities from Boston to Chicago and Los Angeles to Chicago are both 0.5. The trucks that don't go to Chicago stay in their respective cities, so the transition probabilities from Boston to Boston and Los Angeles to Los Angeles are both 0.5.
The trucks in Chicago are split equally between Boston and Los Angeles. Therefore, the transition probability from Chicago to Boston is 0.5, and the transition probability from Chicago to Los Angeles is also 0.5.
Using this information, we can construct the 3 by 3 transition matrix A:
A = [[0.5, 0, 0.5],
[0, 0.5, 0.5],
[0.5, 0.5, 0]]
To find the steady state u∞ corresponding to the eigenvalue λ = 1, we need to solve the equation A*u = λ*u, where u is the steady state vector.
In this case, since λ = 1, we need to solve the equation A*u = u.
To do this, we can set up the matrix equation:
(A - I)*u = 0
where I is the identity matrix.
(A - I) = [[-0.5, 0, 0.5],
[0, -0.5, 0.5],
[0.5, 0.5, -1]]
Solving for u requires finding the null space or kernel of (A - I). This can be done using various methods such as row reduction or finding the eigenvalues and eigenvectors of (A - I).
Once you have the null space, you can normalize the vector to obtain the steady state vector u∞.