Sign patterns of principal minors of real symmetric matrices

2024/07/25 by Tobias Boege, Boege, Tobias, Jesse Selover +3 · 1 citation
Computer Science · Engineering · Mathematics · #05B20 (primary) 14P10 #14P25 #15A15 (seconary) #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2407.17826

openalex publication_date 2024/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We analyze a combinatorial rule satisfied by the signs of principal minors of a real symmetric matrix. The sign patterns satisfying this rule are equivalent to uniform oriented Lagrangian matroids. We first discuss their structure and symmetries and then study their asymptotics, proving that almost all of them are not representable by real symmetric matrices. We offer several conjectures and experimental results concerning representable sign patterns and the topology of their representation spaces.

Cited by

• Cholesky decomposition for symmetric matrices, Riemannian geometry, and random matrices

Related

• The sepr-sets of sign patterns
• On the signs of the principal minors of Hermitian matrices
• Eigenvalues of Real Matrices with Prescribed Principal Minors Sign and Descartes Law of Signs
• One more counterexample on sign patterns
• Sign regularity preserving linear operators