Transition Matrices and Steady States

Picture two cities, A and B, trading residents every year. Some fraction of A's people move to B, and some fraction of B's people move to A, while everyone else stays put. Write this year's populations as a column vector, state = [a, b]. The rule for next year is linear: new a is (fraction staying in A) times a plus (fraction arriving from B) times b, and similarly for new b. Stack those two linear rules into a 2x2 matrix, and multiplying the matrix by the state vector produces next year's state in one step. Each column of the matrix tells you where that city's population goes: the entries in a column add to 1, because everyone has to end up somewhere.

To move forward several years, multiply by the same matrix repeatedly: state after year 3 is M times M times M times the original vector, or M³ times state0. Do this enough times and something striking happens: the vector stops changing. It settles into a steady state, where M times the steady vector equals the steady vector itself. That happens because the matrix keeps redistributing people between the same two bins, and as long as migration rates are positive in both directions, the system has nowhere left to drift — it balances.

This connects to inverse matrices from earlier in the unit: if you know this year's state and the transition matrix, multiplying by the inverse matrix undoes one year of migration and recovers last year's populations. Transition matrices are really just matrix multiplication applied to a story, one more place where the same bending-the-plane idea from earlier topics shows up with a real-world label attached.