Asymptotically optimal bounds for permanental cofactor matrices
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Abstract
For a complex correlation matrix B of order n, let C(B)ij = per B(i|j) be its raw permanental cofactor matrix. We bound the optimal constant γn in C(B) ≼ γn(per B)I. An elementary isolated-row induction and an explicit positive-definite equicorrelation family give, for every n ≥ 2, (n+1)²/(4n) < γn ≤ (n²+2n+3)/(4n). Consequently γn = n/4 + 1/2 + Θ(1/n): both the leading coefficient and the constant term are sharp. The upper estimate extends, with the necessary diagonal weights, to all Hermitian positive semidefinite matrices, including zero diagonals. We also obtain an explicit nonnegative reserve for the standard natural-representation block in order six, and give an exact order-six obstruction to the raw cofactor bound with constant two. The proof is analytic.
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent, permanental cofactors, positive semidefinite matrices, correlation matrices, asymptotically optimal bounds
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
All-order raw-cofactor norm bounds on complex correlation matrices, with diagonal-weighted extension to every Hermitian positive semidefinite matrix; explicit lower family, sharp leading growth and constant term, quantified order-six natural block.
Disclosures
AI-assisted derivation, independent internal proof checking and exact rational computational controls were used. The theorems follow from the printed analytic proofs. The isolated-row method is credited to Lavi.
Reproduction
The supplementary ZIP includes the complete LaTeX source, standard-library Python exact checker and output, reader/build instructions, citation metadata and a complete SHA-256 manifest. Compile main.tex with Tectonic or an ordinary LaTeX installation; run python verify.py and python -O verify.py. Finite controls corroborate the analytic proofs.