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It is known that \\[\n  \\diag(M)\\in \\Img(M). \\] Thus the affine system $Mx=\\diag(M)$ is always solvable. We strengthen this existence statement to a parity rigidity theorem: every solution satisfies \\[\n  \\diag(M)^T x\\equiv \\rank(M)\\pmod 2 . \\] For graph matrices this gives a common extension of Sutner's odd-domination theorem and Batal's parity theorem from closed-neighborhood matrices $A(G)+I$ to arbitrary partially looped graph matrices $A(G)+D$.\n  We also study how rank and nullity change when loops a","authors_text":"Mohsen Aliabadi","cross_cats":[],"headline":"Symmetric matrices over F₂ always have their diagonal vector in the column space, with solutions obeying a rank parity rule.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-11T15:57:52Z","title":"Diagonal parity and loop toggling for symmetric matrices over $\\mathbb F_2$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.11056","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-13T01:36:05.832952Z","id":"2c16aefe-1862-44f5-b475-a7028d5a575b","model_set":{"reader":"grok-4.3"},"one_line_summary":"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.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Symmetric matrices over F₂ always have their diagonal vector in the column space, with solutions obeying a rank parity rule.","strongest_claim":"We include a self-contained proof that diag(M) ∈ Img(M), and we prove that every solution of Mx=diag(M) satisfies diag(M)^T x ≡ rank(M) mod 2. 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