Let $X,Y$ be two discrete random variables. Two joint mass distributions (couplings) with marginals $X$ and $Y$ and with entries $p_{i,j}=\mathbb{P}_1(X=i,Y=j)$ and $p_{i,j}'=\mathbb{P}_2({X=i,Y=j})$ correspond to two matrices $(p_{i,j}), (p_{i,j}').$
Is there a linear transformation that maps $(p_{i,j})\mapsto(p_{i,j}')$?
If not, is there a way to move from a given coupling of $X,Y$ to any other coupling of $X,Y$?
The reason I ask is because I have a coupling of $X,Y$ and I wonder if a specific coupling exists. If there are any results that allow us to go from a given coupling to any other coupling, perhaps this can be used to provide the desired coupling or show that it doesn't exist.
There is no linear transformation in general. Let $p_{i,j}=1/4$ for each $1\le i,j\le 2$ and $p'_{i,j}=1/2$ if $i=j$, 0 otherwise. Then we would need a matrix $A$ with $$\frac14 A \begin{bmatrix}1&1\\1&1\end{bmatrix}=\frac12\begin{bmatrix}1&0\\0&1\end{bmatrix}$$ But the rank of the product of two matrices is $\le$ the minimum of the ranks, so this is impossible.