Topic 4.14 Notes – Matrices Modeling Contexts

What a Two-State Transition Matrix Represents

A two-state transition model tracks a system where each item is in one of two states at each step, then some fraction switches each interval.

The diagram below shows that idea in a simple two-state system, with some amount staying in the same state and some moving to the other one.

Two-state transition diagram

The state at step $k$ is written as a column vector

$$\mathbf{x}_k=\begin{bmatrix}A_k\\B_k\end{bmatrix}.$$

That means:

• $A_k$ is the amount in state $A$ after step $k$
• $B_k$ is the amount in state $B$ after step $k$

Those entries might be:

• counts like people or customers
• proportions of a whole
• probabilities of being in each state

If the model is closed, nothing enters or leaves the system, so the total stays constant. For counts, $A_k+B_k$ stays the same. For proportions or probabilities, the total stays $1$.

One detail students mix up a lot: a rate like “20% move from $A$ to $B$” means 20% of the current amount in $A$ leaves $A$. It does not mean the net amount in $A$ drops by 20%, because some amount may also enter from $B$.

Also, keep the state order fixed everywhere. If your vector is $\begin{bmatrix}A\\B\end{bmatrix}$, then every row and column must use $A$ first, $B$ second.

Building the Transition Matrix from a Context

If $\alpha$ moves from $A$ to $B$ each step, and $\beta$ moves from $B$ to $A$, then

$$T=\begin{bmatrix}1-\alpha & \beta\\ \alpha & 1-\beta\end{bmatrix}.$$

Here’s how to read it:

• first column = what happens to the amount currently in $A$
• second column = what happens to the amount currently in $B$
• first row = what contributes to the next amount in $A$
• second row = what contributes to the next amount in $B$

The diagonal entries, $1-\alpha$ and $1-\beta$, are the retention rates.

Each source column should add to $1$:

• from $A$, either stay in $A$ or move to $B$
• from $B$, either move to $A$ or stay in $B$

This setup shows up in contexts like:

• subscription plans switching between two options
• residents moving between two regions
• changing between two conditions such as healthy/sick or on/off

Common mistakes:

• putting both given rates in one row
• reversing “from” and “to”
• changing state order midway
• forgetting $20\% = 0.20$

Iterating a transition to later times

One step ahead uses

$$\mathbf{x}_{k+1}=T\mathbf{x}_k$$

which means

$$A_{k+1}=(1-\alpha)A_k+\beta B_k,\qquad B_{k+1}=\alpha A_k+(1-\beta)B_k.$$

So each new amount equals what stayed plus what came in.

For $n$ steps from the initial state,

$$\mathbf{x}_n=T^n\mathbf{x}_0.$$

That exponent means repeated matrix multiplication. $T^2=T\cdot T$, not squaring each entry.

Use:

1. $T\mathbf{x}_k$ for the next step
2. $T^n\mathbf{x}_0$ for $n$ steps from the start

Your output keeps the same units as the state vector. If the entries are counts, decimals are model predictions. Round at the end if the context needs whole people.

Quick checks:

• totals stay constant in a closed count model
• proportions or probabilities still sum to $1$

Steady State

A steady state is a vector $\mathbf{s}$ that satisfies

$$T\mathbf{s}=\mathbf{s}.$$

That means the distribution stays the same from step to step. People may still move both ways. The balance happens because the flows match:

$$\alpha A=\beta B.$$

If the total is $N$, then

$$A=\frac{\beta}{\alpha+\beta}N,\qquad B=\frac{\alpha}{\alpha+\beta}N.$$

You’ll often find or confirm steady state by repeatedly multiplying with $T$, or by using a large power $T^n\mathbf{x}_0$, until the vectors stop changing much.

Do not assume every model settles to one steady state. Some alternate forever, and repeated rounding can throw off the long-term result.

Predicting Past States and Checking Validity

If $T$ is invertible, you can work backward:

$$\mathbf{x}_k=T^{-1}\mathbf{x}_{k+1},\qquad \mathbf{x}_{k-n}=T^{-n}\mathbf{x}_k.$$

For

$$T=\begin{bmatrix}1-\alpha & \beta\\ \alpha & 1-\beta\end{bmatrix},$$

the determinant is

$$\det(T)=1-\alpha-\beta.$$

So $T^{-1}$ exists when $\alpha+\beta\ne1$.

This backward model reconstructs an earlier state. The entries of $T^{-1}$ are not reverse transition probabilities, and they can be negative.

Be suspicious if you get:

• negative amounts in a context that should stay nonnegative
• a backward calculation with a noninvertible matrix
• weird results caused by a rounded current state

On tests, keep the interval length, state labels, and state order attached to your answer.