Hello,

For the steady state probability to be the final equilibrium (as you call it), you need the chain to be irreducible, aperiodic and recurrent positive.

It's written here : Markov chain - Wikipedia, the free encyclopedia (the 2nd & 3rd formulas)

You're working with stationary probabilities. So the sum of the components has to be 1. If you were working with stationaryand not some factor or fraction of the steady state probability???measures, then there is a unique solution, up to a factor.

NB : for some vocabulary problems, stationary = steady