# If all the elements in the main diagonal of a transition matrix are non-zero, is it possible that the Markov chain is not reversible?

I understand that a Markov chain is reversible if $$\pi_{i}P_{ij} = \pi_{j}P_{ji}$$ and I was looking for some examples of non-reversible Markov chains. I noticed that in all the examples I saw, at least one element in the main diagonal of the transition matrix was 0. I wonder if this is a coincidence or if there is actually a rule for this. Specifically, if all the elements in the main diagonal of a transition matrix are non-zero, is it possible that the chain is not reversible? If it is possible, could you please provide an example transition matrix?

Also, what is the physical interpretation of a Markov chain that has a stationary distribution but is not reversible? Essentially this means that the chain obeys the balance condition but not the stricter detailed balance condition, but I can't figure out an intuitive example.

$$P =\left[\matrix{ 9/10 & 0 & 1/10\\9/10 & 1/10 & 0\\ 0 & 9/10 & 1/10}\right]\\$$
This transition matrix is irreducible (so it has a stationary strictly positive distribution), but it is not reversible, simply because given the invariant distribution $$\pi$$, you have $$\pi_1P_{1,2}=0$$, while $$\pi_2 P_{2,1}\neq 0$$.
As you can tell, it would be easier just to put zeroes in the diagonal at entries 2,2 and 3,3, but I made them non-zero to answer your question. I am sure you can modify the example in such a way that all the entries are nonzero, but then you would have to compute $$\pi$$ to conclude rigorously that it is not reversible.
Off topic. I suspect that the set of irreducible, non-reversible transition matrices on a finite set is open, so that if you perturb the coefficients of $$P$$ slightly, then you obtain another irreducible, non-reversible transition matrix. So if you pick a “random” Markov chain, I guess it will be non-reversible (if you have at least 3 states).