Use transition matrices and an initial state vector to predict future states and to find the long-term steady state of a system.

What this dot point is asking

You must build a transition matrix from movement data, predict the state after one or more steps, and find or interpret the long-term steady state.

Setting up a transition matrix

A system moves objects (customers, animals, voters) between a fixed set of categories at constant rates each period. The transition matrix $T$ records these rates. Using the convention $S_{n+1} = T S_n$, the entry in row $i$, column $j$ is the proportion moving from category $j$ into category $i$. Each column therefore sums to 1, since everything currently in a category must go somewhere.

Predicting future states

To advance one week, multiply the current state by $T$. Repeat, or use $S_n = T^n S_0$ to jump ahead.

The long-term steady state

As $n$ grows, the state vector usually settles to a fixed steady state $S$ that satisfies $S = TS$ and stops changing from week to week. You can find it by computing high powers $T^n S_0$ until the numbers stabilise, or by solving $S = TS$ with the constraint that the categories add to the total.

Why a steady state exists

The reason the system settles is that the transition matrix gently pulls the state toward a balance point. Whenever more leave a category than join it, that category shrinks; whenever more join than leave, it grows. The steady state is precisely where these flows balance for every category at once, so $TS = S$. For a regular transition matrix - one where some power has all positive entries - this balance point is unique and is reached from any starting distribution, which is why the long-run market share does not depend on where the shoppers began. This is the key insight examiners want you to articulate when interpreting a steady state.

Interpreting the result in context

The steady state predicts the eventual market share regardless of where shoppers started, provided the transition rates stay constant. Examiners reward a sentence interpreting the numbers, for example "store B eventually holds about two-thirds of the market."