Translate transition descriptions, diagrams and matrices

Translate transition representations

A transition matrix records the proportion moving from each current state to each next state. Read columns as sources and rows as destinations, then use the same direction when drawing labelled arrows

1. Read the column as the source The column index names the current state from which the transition starts \(\displaystyle t_{21}:1\to2\)
2. Read the row as the destination The row index names the next state reached by the transition \(\displaystyle t_{ij}:j\to i\)
3. Complete each column to 1 The proportions leaving one current state must add to 1 \(\displaystyle \sum_i t_{i1}=1\)
4. Draw arrows from source to destination Every positive entry becomes a directed arrow from its column state to its row state \(\displaystyle t_{21}>0\Rightarrow1\to2\)

Worked example

Use the transition matrix below to list every directed labelled transition in the order state 1, state 2, state 3

1. \(\displaystyle t_{11}=0.6,\quad t_{21}=0.3,\quad t_{31}=0.1\Rightarrow1\to1\left(0.6\right),\ 1\to2\left(0.3\right),\ 1\to3\left(0.1\right)\)Read down column 1 to list transitions from state 1
2. \(\displaystyle t_{12}=0.2,\quad t_{22}=0.5,\quad t_{32}=0.3\Rightarrow2\to1\left(0.2\right),\ 2\to2\left(0.5\right),\ 2\to3\left(0.3\right)\)Read down column 2 to list transitions from state 2
3. \(\displaystyle t_{13}=0.1,\quad t_{23}=0.4,\quad t_{33}=0.5\Rightarrow3\to1\left(0.1\right),\ 3\to2\left(0.4\right),\ 3\to3\left(0.5\right)\)Read down column 3 to list transitions from state 3
4. \(\displaystyle 0.6+0.3+0.1=1,\quad0.2+0.5+0.3=1,\quad0.1+0.4+0.5=1\)Each source-state column totals 1, confirming the transition proportions

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