### 20.4.24 Markov chains

The markov command finds characteristic features of a Markov chain.

- markov takes M, a transition matrix for a Markov process.
- markov(M) returns a sequence consisting of
  - the list of the positive recurrent states.
  - the list of corresponding invariant probabilities.
  - the list of other strong connected components.
  - the list of probabilities of ending up in the sequence of recurrent states.

#### Example

markov([[0,0,1/2,0,1/2],[0,0,1,0,0],[1/4,1/4,0,1/4,1/4],[0,0,1/2,0,1/2],[0,0,0,0,1]])

⎡ ⎣ |
4
| ⎤ ⎦
| , |
⎡ ⎣ |
0 | 0 | 0 | 0 | 1
| ⎤ ⎦
| , |
⎡ ⎣ |
3 | 1 | 2 | 0
| ⎤ ⎦
| , |
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ |
1
1
1
1
1
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦