右随机矩阵非正则性判定条件的证明问询

我最近在读Dobrow的《Introduction to Stochastic Processes》，书中给出了一个判断右随机矩阵（即元素非负、每行和为1的方阵）非正则的方法，原文如下：

Here is one way to tell if a right stochastic matrix (i.e., a square matrix with non-negative entries whose rows sum to 1) is not regular. If for some power $n$, all the $0$s in $P^n$ appear in the same location as all the $0$s in $P^{n+1}$, then they will appear in the same locations for all higher powers, and the matrix is not regular.

不过我翻遍了资料都找不到这个结论的证明过程。我自己猜测这应该和$0$元素对应的“经过$n$步和$n+1$步都无法到达的状态对”有关，但实在没法再深入推导下去了。有没有大佬能帮忙解释一下这个结论的证明思路呀？

备注：内容来源于stack exchange，提问作者JMontero