Research Article · Preprint Mathematics
Asymptotically optimal bounds for permanental cofactor matrices
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
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.
Keywords permanent · permanental cofactors · positive semidefinite matrices · correlation matrices · asymptotically optimal bounds
Scope of the work
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.
Reproduction notes
The 6-page manuscript, editable source and citation metadata are supplied in this 8-member package. README.md gives build and execution commands. The Python standard-library checker verify.py supplies the exact finite controls specified in the manuscript alongside its general analytic proofs.
The manuscript
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The complete manuscript, as published in this edition.
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Author disclosures
AI-assisted derivation, independent internal proof checking and exact rational computational controls were used. The theorems follow from the printed analytic proofs; internal verification is not external refereeing. The isolated-row method is credited to Lavi. No exhaustive worldwide-priority claim is made.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Asymptotically optimal bounds for permanental cofactor matrices (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/asymptotically-optimal-bounds-permanental-cofactor-matrices/v/2
BibTeX
@techreport{luminary_asymptotically_optimal_bounds_permanental_cofactor_matrices_v2,
title = {Asymptotically optimal bounds for permanental cofactor matrices},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/asymptotically-optimal-bounds-permanental-cofactor-matrices/v/2}
}Loading discussion…