I'm trying to find a $n$-by-$m$ matrix $M$ that obeys the following constraints:
1) It is a conditional probability matrix, $$ M_{ij}\ge 0 \quad \forall i,j \qquad , \qquad \sum_j M_{ij} = 1 \quad \forall i $$ 2) It obeys the linear constraint $$ \vec{a}^T M = 0 $$ 3) It contains some vector $\vec{b}$ in its column space, i.e., $$ \exists \vec{x}\in\mathbb{R}^m\qquad \text{s.t.}\qquad M \vec{x} = \vec{b}$$
In fact, I do not so much need the matrix $M$ itself as to know whether such a matrix exists, given some specified $\vec{a}$ and $\vec{b}$. In particular, I don't know how to deal with the column space constraint. Any suggestions?