Given several groups with members of each group able to transition from group to group, a transition matrix summaries the probabilities of passing from one group to any other, or remaining in the same group. Suppose a certain school has four maths group - 1 to 4, with 1 being the highest and 4 the lowest - with students able to transition between group on the basis of test scores, with probabilities summarised in the table.
| From\To |
1 |
2 |
3 |
4 |
| 1 |
0.5 |
0.3 |
0.2 |
0 |
| 2 |
0.3 |
0.4 |
0.3 |
0 |
| 3 |
0.1 |
0.2 |
0.2 |
0.5 |
| 4 |
0.1 |
0.1 |
0.1 |
0.7 |
This means for example that there is a probability of 0.2 that a student will move from group 1 to group 3.
The transition matrix is
\[\left( \begin{array}{cccc} 0.5 & 0.3 & 0.2 & 0 \\ 0.3 & 0.4 & 0.3 & 0 \\ 0.1 & 0.2 & 0.2 & 0.5 \\ 0.1 & 0.1 & 0.1 & 0.7 \end{array} \right)\]