Finding values of a matrix with specific distribution

Question

Suppose I have a MxN matrix.I want to choose values such that:
1. If I choose the values of each column, they have the same distribution (Dist(c1) = Dist(c2) = ... = Dist(cN))
2. If I choose the values of each row, they have the same distribution (Dist(r1) = Dist(r2) = ... = Dist(rM)
3. All of the M*N values have a truncated Gaussian distribution with specific mean and standard deviation.

Is it possible? If so, I was wondering how I can approach this problem.

Answer 1

There is a first positive answer with matrices having their entries in the form $c_id_j,$ i.e., rank-one matrices :

$$\begin{pmatrix}c_1\\c_2\\ \vdots \\ c_M \end{pmatrix}\begin{pmatrix}d_1&d_2&\cdots&d_N\end{pmatrix}$$

where the $c_k$s obey a certain distribution $D_1$, and the $d_{\ell}$s, another one, $D_2$.

Edit : This case is by no means anecdotal : it is the case where distributions are independent. In all other cases (rank > 1), I think you will have non-independent distributions.