Sign patterns of principal minors of real symmetric matrices

TL;DR

This paper investigates the combinatorial sign patterns of principal minors in real symmetric matrices, linking them to oriented Lagrangian matroids, and explores their structure, symmetries, and representability.

Contribution

It establishes a connection between principal minor sign patterns and uniform oriented Lagrangian matroids, and studies their asymptotic properties and representability.

Findings

Most sign patterns are not realizable by real symmetric matrices

The structure and symmetries of these sign patterns are characterized

Experimental results and conjectures about their representation spaces are presented

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.