Let $P=\left(P_{i j}\right)$ be a transition matrix. What does it mean to say that $P$ is (a) irreducible, $(b)$ recurrent?

Suppose that $P$ is irreducible and recurrent and that the state space contains at least two states. Define a new transition matrix $\tilde{P}$ by

$$\tilde{P}_{i j}=\left\{\begin{array}{lll} 0 & \text { if } & i=j \\ \left(1-P_{i i}\right)^{-1} P_{i j} & \text { if } & i \neq j \end{array}\right.$$

Prove that $\tilde{P}$ is also irreducible and recurrent.