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.

Clarification in response to Jochen Glueck's correct remarks: I use the term "projection" in a formally not correct way, namely meaning any mapping to lower-dimensional space.