For symmetric matrices M over F_2, diag(M) is always in the image of M and solutions satisfy a rank-parity relation, with explicit formulas for diagonal perturbations and tree-structured systems.
Batal, Parity of an odd dominating set,Commun
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Diagonal parity and loop toggling for symmetric matrices over $\mathbb F_2$
For symmetric matrices M over F_2, diag(M) is always in the image of M and solutions satisfy a rank-parity relation, with explicit formulas for diagonal perturbations and tree-structured systems.