Solution
= Solution
Conditioning on the initial state gives
$$
\boxed{P(X_n=i)=\sum_{j\in S}\lambda_j(P^n)_{ji}=(\lambda P^n)_i.}
$$
The row-vector convention is important: the initial distribution multiplies the <transition matrix> on the left.
= Solution
Conditioning on the initial state gives
$$
\boxed{P(X_n=i)=\sum_{j\in S}\lambda_j(P^n)_{ji}=(\lambda P^n)_i.}
$$
The row-vector convention is important: the initial distribution multiplies the <transition matrix> on the left.