We know that the square of a stochastic matrix is also stochastic, because the two-step transition matrix of a Markov chain is necessarily stochastic. However, in there another way to independently prove this fact?
2026-03-26 06:30:06.1774506606
Proof that the square of a stochastic matrix is stochastic
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I will do for column-stochastic matrix. Let $X=(x_{ij})$. Then the sum of the entries of the first column of $X^2$ is $(x_{11}+...+x_{n1})x_{11}+(x_{12}+...+x_{n2})x_{21}+...=x_{11}+...+x_{n1}=1$. Of course, this works for any column.