By a Zero Line-Sum (ZLS) 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.

ZLS matrices obviously don't have full rank but, as not all rank-deficient matrices have the property of being ZLS, I wonder if ZLS matrices are also special in different aspects and also w.r.t. the mappings they define.

Question:

Have ZLS matrices already been investigated, and what are non-trivial special properties of them that have been identified? I am looking for information on matrices, that equal the difference of two doubly stochastic matrices and, on the special properties of the transformations 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 a lower-dimensional space.

Remark: I have edited this question to replace the former "0-Stochastic projection matrix" with the "Zero Line-Sum" matrix following the suggestion of Gerry Myerson; "line" in this context is the common term for row and column.

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