Transition matrices and state vectors | Year 12

01 / LEARN

The idea, step by step

States and transitions

A state vector records how a population or stock is distributed at one time. Here it is a column vector. In a closed model, its entries sum to the fixed total. The next state is found from $x_{n + 1} = T x_{n}$ .

For these transition matrices, columns are current states and rows are next states. The entry $t_{i j}$ is the fraction moving from current state $j$ to next state $i$ . This differs from our adjacency convention, so state it clearly. Every column must sum to 1, and every entry must lie between 0 and 1.

Proportions act on current amounts

The next amount in A is the current A amount times its A-to-A fraction, plus the current B amount times its B-to-A fraction. That is precisely the first row of the matrix product. Fractional predicted counts are expected amounts, not a claim that a fraction of a person literally moves.

If people enter or leave, a closed matrix alone is inadequate. Model the extra flow or redefine the states. Do not fix a deficient column by arbitrary rounding; investigate which destination is missing and report uncertainty in the estimates.

x_{n + 1} = T x_{n}

Columns=current; rows=next.
Next / current	A	B
A	0.8	0.3
B	0.2	0.7

A common trap

Checking row sums for a column-stochastic model.

Instead: Each starting-state column contains all possible destinations and must sum to 1.

02 / WORKED EXAMPLES

See the thinking

Example 1

A one-period forecast

Apply the displayed matrix to the state (100,50).

Next A: $0.8 (100) + 0.3 (50) = 95$ .
Next B: $0.2 (100) + 0.7 (50) = 55$ .
Check $95 + 55 = 150$ .

State column (95,55).

Example 2

Missing transition entries

The first column is (0.65, $p$ ), the second ( $q$ ,0.9). Find $p, q$ .

Each column sums to 1.
$p = 1 - 0.65 = 0.35$ ; $q = 1 - 0.9 = 0.1$ .

p = 0.35, q = 0.10

Check your understanding

For rows (0.7,0.2),(0.3,0.8), what fraction goes B to A?

Show answer

0.2, or 20%.

Use column B and row A.

03 / WORKSHEET

Your turn

6 questions · 20 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Use rows (0.7,0.2),(0.3,0.8) to update (80,20).
3 marks
Rows of a proposed closed transition matrix are (0.4,0.3),(0.5,0.7). Is it valid?
3 marks
Find $a, b$ so rows (0.75, $a$ ),( $b$ ,0.6) form a column-stochastic matrix.
3 marks
A state vector is (120,80). What total must a closed transition preserve?
3 marks
Each day 90% of North bikes stay North; 40% of South bikes move North.
5 marks
1. (a)
  Write the North,South column-stochastic matrix.
  2 marks
2. (b)
  Predict tomorrow from (20,30).
  3 marks
A forecast is (34.6,15.4) for 50 bikes. Explain its meaning.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Use rows (0.7,0.2),(0.3,0.8) to update (80,20).
3 marks
A: $0.7 (80) + 0.2 (20) = 60$ .
B: $0.3 (80) + 0.8 (20) = 40$ .
(60,40).
Rows of a proposed closed transition matrix are (0.4,0.3),(0.5,0.7). Is it valid?
3 marks
It does not account for 10% of those initially in A.
No; the first column totals 0.9.
Find $a, b$ so rows (0.75, $a$ ),( $b$ ,0.6) form a column-stochastic matrix.
3 marks
$0.75 + b = 1$ and $a + 0.6 = 1$ .
$a = 0.4, b = 0.25$ .
A state vector is (120,80). What total must a closed transition preserve?
3 marks
The starting entries sum to 200.
Column sums of 1 preserve this total.
200.
Each day 90% of North bikes stay North; 40% of South bikes move North.
5 marks
a) Each column describes destinations from one depot.
b) North: $0.9 (20) + 0.4 (30) = 30$ .
b) South: $0.1 (20) + 0.6 (30) = 20$ .
1. (a)
  Write the North,South column-stochastic matrix.
  2 marks
  Each column describes destinations from one depot.
  Rows (0.9,0.4),(0.1,0.6).
2. (b)
  Predict tomorrow from (20,30).
  3 marks
  North: $0.9 (20) + 0.4 (30) = 30$ .
  South: $0.1 (20) + 0.6 (30) = 20$ .
  (30,20).
a) Rows (0.9,0.4),(0.1,0.6).; b) (30,20).
A forecast is (34.6,15.4) for 50 bikes. Explain its meaning.
3 marks
An operational integer plan should still total 50.
Do not interpret fractional counts literally.
It is an expected distribution; actual whole-bike counts vary.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics matrix-models option. This is the offered fifth-topic option, alongside the four compulsory topics; a school-designed Open Topic is an alternative, not prescribed content in this chapter.

Official curriculum source ↗

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