the initial distribution
A transition matrix tells you how a chain MOVES, but not where it BEGINS. To turn the machinery into actual probabilities you must also say where the chain starts. The initial distribution, usually written pi_0, is the probability distribution over states at time 0: pi_0(i) = P(X_0 = i). It might be certainty about one starting state, or genuine uncertainty spread over several.
Once you fix the initial distribution and the transition matrix, the chain's entire future is pinned down. The distribution at time n is found by pushing the start forward: pi_n = pi_0 times P^n, a single row-vector-times-matrix calculation. Each step multiplies by one more copy of P. This pi_n is the marginal distribution at time n — the probabilities of being in each state at that moment, having averaged over all the random paths that could have led there.
Two chains with the same transition matrix but different starting distributions are different probabilistic objects in the short run — yet for well-behaved (irreducible, aperiodic) chains they forget their start: pi_n converges to the same stationary distribution no matter where they began. So the initial distribution governs the transient, early behaviour and is crucial for short-horizon predictions, even though it washes out in the long run.
Weather chain, start known sunny: pi_0 = (1, 0). Then pi_1 = (1, 0) P = (0.8, 0.2), pi_2 = pi_1 P = (0.72, 0.28), and so on toward (2/3, 1/3). Start instead at the equilibrium pi_0 = (2/3, 1/3): then pi_1 = pi_0 P = (2/3, 1/3) again — at equilibrium the distribution never changes.
The state distribution at time n is pi_0 times P^n; starting at the stationary distribution freezes it forever.
The initial distribution is part of the model's specification, not something the transition matrix determines. A chain is the pair (initial distribution, transition matrix); change either and you change the process.