Key Idea: A transition matrix shows the chance of moving between states each step, like patches of a field going from infested to clear. Powers of the matrix predict the states many steps ahead.
| Idea | Rule |
|---|---|
| Build T | Column = start, row = end |
| Check | Every column adds to 1 |
| One step | s₁ = T s₀ |
| n steps | sₙ = Tⁿ s₀ (the GDC does the power) |
Pest-control example, T = [[0.6, 0.1], [0.4, 0.9]]
- Start — 400 infested, 600 clear.
- Compute — T³ s₀ on the GDC.
- Result — About 225 infested after 3 months.
- Trend — 400 → 300 → 250 → 225: falling each month.
Read T from column to row: from the start state into the end state.
A column of T reads 0.7 and 0.4. Is that allowed? No. It must add to 1.
To go 5 steps ahead, do you use 5T? No. Use T⁵ s₀.
In this T, what share of infested patches stays infested each month? 0.6, or 60%.
How many patches are clear after 3 months? 775.