This is the chart associated with a Markov matrix
The equivalence(communication) classes are:
- {1,2,3,4} - transitive
- {5,6,7} - transitive
- {8} - ergotic
My teacher said that "all equivalence classes are either ergotic or transitive". I agree. However he said, that this was complicated to proof.
I said that suppose there was a class that was both transitive and ergotic. Being ergotic means that once getting in, you can't get out. But if it is also transitive means that you can get out. So, it is only transitive. As a consequence it must be either transitive or ergotic.
My teacher told me that there was a complicated proof, but than what is wrong with mine? Is my proof wrong?
Thanks
