If a transition matrix $A$ is regular, prove that $A^∞$ has all the same columns and that the columns are the steady state vector.

Question

I know that this is quite an elementary theorem, but I have yet to see a proof of this except for quoting that $A^∞$ is going to be of rank 1, so that all the columns must be the same. Any help is appreciated!

Answer 1

Note that for each $i$, $$ \lim_{n\to\infty} A^n e_i =A^\infty e_i $$ is the $i$-th column vector of $A^\infty$, where $(e_i)_{i=1}^n$ denotes the standard basis. Each $e_i$ represents initial state starting from $i$, and under the regularity assumption, state vector converges to the stationary state vector $$\pi=\lim_{n\to\infty}\pi^{(n)}=\lim_{n\to\infty}A^n \pi^{(0)}$$ regardless of the initial state $\pi^{(0)}$. Thus, in particular, every column vector of $A^\infty$ is equal to $\pi$.