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.



Properties of Zero Line-Sum Matrices
Below find six suggestions.


**Case of general ZLS matrices.** 


Examples of properties are:


1. All cofactors of a *square* ZLS are equal.
2. Each *square* ZLS matrix of dimension $n$ has eigenvector $1^{n\times 1}$ with eigenvalue $0$.
3. If an $n\times n$ ZLS $A$ is moreover *symmetric*, and if $\mathrm{Sp}(\cdot)$ denotes the spectrum of a matrix, then each cofactor of $A$ equals $\frac{1}{n}\prod\_{\lambda\in\mathrm{Sp(A)\setminus\{0\}}}\lambda$.
4. Rectangular ZLS matrices over a field $K$ form a $K$-vector space in the obvious way, and according to [this thread](https://math.stackexchange.com/questions/1623551/dimension-of-vector-space-of-matrices-with-zero-row-and-column-sum/1624796) its dimension is $(m-1)(n-1)$, and if $m=n$, then according to [that thread](https://math.stackexchange.com/questions/3923/the-orthogonal-complement-of-the-space-of-row-null-and-column-null-matrices?rq=1) its orthogonal complement w.r.t. the Frobenius inner product consists solely of sums of two rank-one matrices, more precisely, this orthogonal complement consists solely of matrices of the form $v\cdot 1^{1\times n} + 1^{n\times 1}\cdot w^{\text{t}}$.
5. Some relevant Lie theoretic consideration can be found in 


> 
> Andreas Boukas, Philip Feinsilver, Anargyros Felouris, *On the Lie structure of zero sum and related matrices*. Random Oper. Stoch. Equ. 23, No. 4, 209-218 (2015)
> 
> 
>


**Case of matrices which are differences of two doubly-stochastic matrices**


6. If $A$ is the sum of two *symmetric* stochastic matrices, you may find it profitable to look in the direction of *Horn's conjecture*. Starting points could be [R Bhatia: *Algebraic Geometry Solves an Old Matrix Theorem*](http://math.univ-lyon1.fr/~ressayre/PDFs/bhatia.pdf). [Resonance](http://www.ias.ac.in/Journals/Resonance_%E2%80%93_Journal_of_Science_Education/). December 1999, or [this MO thread](https://mathoverflow.net/questions/4224/eigenvalues-of-matrix-sums).


