I have a constrained optimization problem for a matrix $A \in \mathbb{R}^{m \times n}$. The objective doesn't matter, and the constraint is that for some scalars $\alpha, \beta \in \mathbb{R}$, we have $$ \begin{cases} A_{i,j} &\ge 0, \forall i, j \\ \sum_{j=1}^n A_{i,j} &= \alpha, \forall i \\ \sum_{i=1}^m A_{i,j} &= \beta, \forall j \\ \end{cases} $$
If there are only the nonnegativity and row sum constraints, I can use the parameterization $A = \operatorname{Softmax}(B)$ where $B$ is any matrix in $\mathbb{R}^{m \times n}$, and it reduces to an unconstrained optimization problem. However, the additional column sum constraint makes everything hard.
How can I parameterize $A$ as an unconstrained matrix?