By a doubly stochastic projection matrix I mean matrices with the property, that each row sum and each column sum equals zero:

$$A\in\mathbb{R}^{m\times n}:\ \sum_{i=1}^{n}a_{ij}=\sum_{j=1}^{m}a_{ij}=0 $$ 

these can be thought of as being the difference of two "ordinary" doubly stochastic matrices.

As not all projection matrices have the property of being doubly 0-stochastic, I wonder if they are also special in different aspects and also w.r.t. the projections they define.  

>**Question:**  
have 0-stochastic matrices already been investigated and, what are non-trivial special properties that have been identified?
I am looking for information on matrices, that are resemble the difference of two doubly stochastic matrices and, on the special properties of the projections they define.