Math Problem Statement
Solve this
Let the migration matrix between cities A, B, and C be the one given by A = [[0, 1/3, 2/3], [1/2, 1/6, 1/3], [1/2, 1/2, 0]]. Is there a limit distribution of the population? If there is, find it.
Solution
as np
Define the migration matrix
A = np.array([[0, 1/3, 2/3], [1/2, 1/6, 1/3], [1/2, 1/2, 0]])
Calculate the steady-state distribution by solving (A.T - I)x = 0
eigvals, eigvecs = np.linalg.eig(A.T) steady_state = eigvecs[:, np.isclose(eigvals, 1)]
Normalize the steady-state vector so that it sums to 1
steady_state = steady_state / steady_state.sum()
steady_state = steady_state.realYes, there is a limit distribution of the population
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Markov Chains
Eigenvalues and Eigenvectors
Steady-State Distribution
Formulas
Ax = x, where A is the transition matrix and x is the steady-state vector
Normalization condition: sum(x) = 1
Theorems
Perron-Frobenius Theorem for stochastic matrices
Suitable Grade Level
Undergraduate Level