Matrices modeling contexts

A two-state model tracks how much of a population sits in each of two categories, written as a column state vector: $\langle A, B\rangle$ means A units in state 1 and B units in state 2. The transition matrix M records the per-step movement rules, and this lesson fixes one convention and never wavers:

1. Each column of M belongs to a from-state.
2. The entries down a column are the fractions of that state's members going to each to-state (row 1 = to state 1, row 2 = to state 2).
3. Each column sums to 1, and the next state is $M \cdot (\text{current state})$, matrix on the left of a column vector.

Conventions vary between textbooks (some use rows and multiply on the other side), and either works if used consistently. What never works is building the matrix under one convention and multiplying under the other; that is the single largest error source in this topic.

"Each month, 90% of shop A's customers stay and 10% switch to B; 20% of B's customers switch to A and 80% stay." Translate column by column. From-A column: 0.9 stays (row 1), 0.1 leaves for B (row 2). From-B column: 0.2 arrives at A (row 1), 0.8 stays (row 2):

$M = [[0.9, 0.2], [0.1, 0.8]]$

Now audit: column one sums to $0.9 + 0.1 = 1$ and column two to $0.2 + 0.8 = 1$. That sum-to-1 check is not decoration; it says every member of each state is accounted for, staying or moving, nobody created and nobody destroyed. A column summing to 1.1 manufactures customers out of thin air; a column summing to 0.9 quietly deletes them. Run this audit on every matrix you build, and on every matrix you are handed.

With the matrix built, prediction is mechanical: next state $= Mv$. Start 100 customers, all with shop A: $v_0 = \langle 100, 0\rangle$.

Month 1: $Mv_0 = \langle 0.9 \cdot 100 + 0.2 \cdot 0,\; 0.1 \cdot 100 + 0.8 \cdot 0\rangle = \langle 90, 10\rangle$.

Month 2: $Mv_1 = \langle 0.9 \cdot 90 + 0.2 \cdot 10,\; 0.1 \cdot 90 + 0.8 \cdot 10\rangle = \langle 83, 17\rangle$.

Notice month 2 is not "lose 10 again": A lost 9 but recovered 2 of B's customers, netting 83, because the fractions apply to the current state, not the original one. In general, n steps means multiplying by M n times, which is multiplying once by the power $M^n$. The totals stay at 100 every step; if your total drifts, the multiplication went wrong somewhere.

Yesterday's state is recoverable: if $v_{\text{today}} = Mv_{\text{yesterday}}$, then $v_{\text{yesterday}} = M^{-1}v_{\text{today}}$. The inverse matrix is the undo map; multiplying by M again would push the system another step forward instead.

Pushed far forward, many transition systems settle. Iterating our shop matrix from $\langle 100, 0\rangle$: 90, 83, 78.1, ... approaching the split where A's inflow equals its outflow: $0.1A = 0.2B$ with $A + B = 100$ gives $A = 2B$, roughly $\langle 66.7, 33.3\rangle$. At that state the two flows balance and the mix stops changing (a steady state), even though individual customers keep switching underneath. The long run forgets the starting split; it remembers only the matrix.