# Asymmetric Transition Probability Matrix with uniform stationary distribution

I am solving a discrete Markov chain problem. For this I need a Markov chain whose stationary distribution is uniform(or near to uniform distribution) and transition probability matrix is asymmetric.

[ Markov chains like Metropolis hasting has uniform stationary distribution but transition probability matrix is symmetric ]

Take the matrix $$A$$ such that for every $$i$$, we have:
1. $$A_{i,i} =0.5$$
2. $$A_{i,i+1}=0.5$$ (for $$i=n$$, set $$A_{n,0}=0.5$$)