I solved (a) but cannot find an example for (b). Could anyone show me a stochastic matrix not derived from a unitary matrix?
2026-03-26 17:31:48.1774546308
an example of stochastic matrix not derived from a unitary matrix
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this is a classical question, and the answer is well known by specialists. So let's just give a link for a paper and the example mentioned there, that is attributed to J. Hoffman: a $3\times 3$ matrix, the average of two permutation matrices, that does not come from a unitary matrix
$$\frac{P_{\sigma} + P_{\sigma^{-1}}}{2} =\frac{1}{2}\left(\begin{matrix} 0 & 1 & 1 \\ 1& 0 & 1 \\ 1& 1& 0 \end{matrix} \right) $$
I suppose checking that is a good exercise.