I am trying to find a nice and efficient way to approach the following problem: I need to solve (for example using Solve, Reduce, or NSolve) certain type of equations involving a set of unknown square matrices, which have at most one non-zero element in each row and the sum of the $j$-th row across all matrices is equal to one. In other words, the unknown matrices have at most one most non-zero element in each row and the sum of the matrices is a row-stochastic matrix. How can I tell Mathematica to look only for these type of solutions?
Any ideas on how to elegantly enforce such constraints? Can I tell Mathematica in advance that the unknown matrices can only have one non-zero element on each row?