Given an arbitrary discrete probability distribution $a = (a\_1, ..., a\_n)$ and another arbitrary discrete probability distribution $b = (b\_1, ..., b\_n)$, what is the easiest known way to find a diagonalizable $n \times n$ stochastic matrix $M$ such that $M \cdot a = b$?



Diagonalizable stochastic matrix that satisfies an equation
Take $M$ to be the stochastic matrix which has $b$ in every column. Then $M$ maps every probability vector to $b$, and in particular $Ma = b$.


Concerning diagonalizability, choose any basis which contains $b$ as first basis vector and such that all other basis vectors $c$ satisfy $\sum\_i c\_i = 0$. In such a basis, $M = \mathrm{diag}(1,0,\ldots,0)$.


