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 γ_n in C(B)≼γ_n(per B)I. An elementary isolated-row induction and an explicit positive-definite equicorrelation family give, for every n≥2, ((n+1)^2)/(4n)<γ_n≤(n^2+2n+3)/(4n). Consequently γ_n=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 and does not require finite polynomial certificates.
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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
For complex correlation matrices of order n≥2, the optimal raw permanental cofactor constant lies strictly above (n+1)²/(4n) and at most (n²+2n+3)/(4n), giving sharp linear and constant asymptotic terms and the stated weighted PSD and order-six consequences.
Disclosures
AI tools assisted with derivation, exposition and code preparation. The author is responsible for the mathematical claims, proofs and references. The analytic proofs are supplied in full; exact finite controls accompany the source. The isolated-row method is credited to Lavi.
Reproduction
The six-page manuscript, complete editable source, fixed-timestamp Tectonic build helper, exact Python checker and citation metadata are supplied in a nine-member package. README.md gives the build and execution commands.