Matrices Modeling Contexts
Introduced| English | Meaning |
|---|---|
| state vector/steɪt ˈvektə/ | state vector |
| state/steɪt/ | A possible category or condition occupied by an item in a transition model. |
| transition matrix/trænˈsɪʃn ˈmeɪtrɪks/ | transition matrix |
Predicting the future with a matrix
- Every year, some people move from the city to the suburbs, and some move back.
- Given this year's populations, can you predict next year's?
- A matrix can store the movement rates and do the prediction for you.
- This is how matrices model populations, markets, weather, and more.
State vectors: the current picture
- Collect the current amounts into a state vector 状态向量 — one entry per category.
- For a city and its suburbs: $\begin{bmatrix}\text{city}\\ \text{suburb}\end{bmatrix}$, say $\langle 600, 400\rangle$ (thousands).
- Each state 状态 is one thing being tracked; the vector is a snapshot in time.
- Next year's snapshot will be a new state vector.
A state vector in this kind of model holds…
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A state vector lists how much is in each category right now — one entry per state.
The transition matrix: the rules
- A transition matrix 转移矩阵 stores the fraction moving between every pair of states.
- Each number is a rate: e.g. $5\%$ of city dwellers move to the suburbs each year.
- The diagram below shows two states with the movement rates on the arrows.
- Those same rates become the entries of the matrix.
The matrix that stores the rates of movement between states is called the ____ matrix.
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The transition matrix holds the fraction moving from each state to each other state.
Multiplying the transition matrix by the current state vector gives the previous state.
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False. One matrix-times-vector step advances the model by one period, predicting the next state.
Stepping forward in time
- Multiply the transition matrix by the current state vector to get next year's.
- One matrix-times-vector = one year forward.
- Want two years? Multiply again. Ten years? Multiply ten times.
- The model marches forward, one step per multiplication.
A good use of a matrix?
Matrices model transformations, systems and transitions. Sort each task.
To predict the state two periods ahead, you…
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Each multiplication advances one period, so two periods need two multiplications (or multiply by the matrix squared).
Select all true statements about matrix models.
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The same transition matrix is reused each period — you do not redraw it. The other three are correct.
The long run
- Repeated steps often settle toward a steady state that stops changing.
- At the steady state, as many people leave the city as arrive — populations hold constant.
- Matrices let you find this long-run balance without simulating every single year.
- The same idea powers web-page ranking, genetics, and epidemic models.
Set the model up consistently: decide once whether the state vector is a column multiplied on the right of the matrix, and make each column's (or row's) rates add to $1$ so no one is lost or invented. A mismatched setup silently gives nonsense predictions.
City $= 600$, suburb $= 400$ (thousands). Each year $5\%$ of city → suburb, $2\%$ of suburb → city.
- Leaving the city: $0.05 \times 600 = 30$; arriving: $0.02 \times 400 = 8$.
- New city $= 600 - 30 + 8 = 578$; new suburb $= 400 + 30 - 8 = 422$.
- The matrix does exactly this arithmetic in a single multiplication.
A state vector holds the current amount in each state, and a transition matrix stores the rates of movement between them. Multiplying the matrix by the state vector steps the model one period forward; repeating it predicts the long-run behaviour. This is how matrices model real-world change.
Write from a next state of (4, 6)
- The transition matrix has rows (1, 0) and (0, 2).
- The current state is (4, 3).
- Find the next state.
A transition matrix with rows (1, 0) and (0, 2) multiplies the state vector (4, 3). What is the next state vector?
2 times 3 is 6.
What is the next state?
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Only the second state is scaled by 2.