If you have probability distribution transition function, it makes sense to want to diagonalize it. And you can't do that without algebraic closure of complex numbers. Even a simple transition matrix of a 3 cycle has complex eigenvalues (the 3rd roots of unity).
The stationary probability distribution of some transition matrix is the eigenvectors, so the stationary "probability distribution" of even very simple matrices (like cycle of 3 values) are complex. It's not as magical as they make it out to be.
The stationary probability distribution of some transition matrix is the eigenvectors, so the stationary "probability distribution" of even very simple matrices (like cycle of 3 values) are complex. It's not as magical as they make it out to be.