Proof box 2.1

[Gregwar]

Hello, first thank you for this work!

Sorry if I misunderstood something, but in proof box 2.1 of page 39, it is stated that:

Since every entry of $P_{\pi }$ is nonnegative and no greater than one, we have that $0\le P_{\pi }^k\le 1$ for any $k$.

This would be true if $P_{\pi }^k$ were an elementwise exponent, but it appears to be a matrix exponent. For instance:

$$P_{\pi }=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$$

(Which is of course not a proper stochastic matrix), satisfies the "every entry is nonnegative and no greater than one" condition, but:

$$P_{\pi }^2=\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right]$$

Clearly does not satisfy $0\le P_{\pi }^k\le 1$.

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Shouldn't the argument be based on $P_{\pi }$ being a stochastic matrix with a spectral radius of one instead ?

[MathFoundationRL]

Yes, this sentence can be stated in a more rigorous way.