Stochastic matrix question

Question

A stochastic matrix is one which each column sum equal one.

$$P= \begin{bmatrix} P_{11} & P_{12} & \ldots & P_{1N} \\ P_{21} & P_{22} & \ldots & P_{2N} \\ \ldots & \ldots & \ldots & \ldots \\ P_{N1} & P_{N2} & \ldots & P_{NN} \end{bmatrix}$$

Show that is $P$ is a stochastic matrix. Then $P^2$ is a stochastic matrix.

Then show $P^n$ is stochastic for all postive integer N.

I am not sure how to do this I think by induction. I know stochastic is that $\forall i $ $\sum_{j=1}^{N}p_{ij}=1$

Answer 1

Let $e$ be the all one-matrix, the condition for $P$ to be a stochastic matrix can be rewritten as $$e^TP=e^T.$$

Now, let me try to show that $P^2$ is a stochastic matrix.

$$e^TP^2=(e^TP)P=(e^T)P=e^TPe=e^T$$

Hence $P^2$ is stochastic.

I will leave the case for $P^n$ as an exercise.

Answer 2

Check any column of $P^2$, it's linear combination of columns of $P$, where the coefficients are again the entries of columns of $P$. Hence the sum of the entries of any column of $P^2$ is 1.