# Is uniqueness of steady-state vector sufficient to regular transition matrix?

If P is a transition matrix, then a steady-state vector for is a probability vector q such that $$P\mathrm{q}=\mathrm{q}$$.

A transition matrix P is regular if some power $$P^k$$ contain only strictly positive entries.

We know that the steady-state vector is the eigenvector of P associated with the eignvalue 1.

So if rank(P-I)=Dim(P)-1, then the eigenvector of P which column sum is 1 (steady-state vector) will be unique.

Is uniqueness of steady-state vector sufficient to the regularness of P?

$$P_2=\begin{pmatrix} 0 & 0.5 & 0.5 & 0\\ 0.5 & 0 & 0 & 0.5\\ 0.5 & 0 & 0 & 0.5\\ 0 & 0.5 & 0.5 & 0\\ \end{pmatrix}$$
with unique steady-state vector $$(0.25,0.25,0.25,0.25)^T$$