Consider a vector $p =(p_1,...p_n)$, $p_i>0$, $\sum p_i = 1$ and a matrix $M_{ij}$, which is doubly stochastic: $\sum_i M_{ij} = 1, \sum_j M_{ij} = 1, M_{ij} > 0$.

**Question 1** Just apply matrix M to a vector $p$ , i.e. $q = Mp$ (i.e. $q_i = \sum_j M_{ij} p_j$) is it true that entropy of new vector $q$ is greater then of original vector $p$ ? I.e.

$$ H(q) = -\sum_i q_i ln(q_i) > -\sum_i p_i ln(p_i) = H(p) $$

**Question 2** Is there a simple proof of it ? (It might follow from the Gibb's inequality, but it does not seem obvious for me).

**Question 3** What are generalizations,
in view of meta-principle "entropy *always* grows" that might be an example of some more general phenomena ?

**Motivation:** There is a meta-principle that *entropy grows* is some natural systems, matrix applied to a vector is probably the most simple system one can consider (Markov chain), so the question above arises. After some thinking one can restrict from all matrices to doubly stochastic, because $M^n v$ tends to a uniform distribution (which has maximal entropy of all) only for doubly stochastic $M$ , so it cannot be true for general $M$.