Small Matrices with Small Inverses: Unimodular Zerofree Cases

TL;DR

This paper studies special unimodular matrices that are small and invertible with both the matrix and its inverse containing no zero entries, classifying these rare balanced cases.

Contribution

It classifies small unimodular matrices with small inverses that are zero-free, a rare and balanced case contrasting typical trade-offs.

Findings

Identifies and classifies unimodular matrices with both the matrix and inverse being small and zero-free.

Provides symmetry-based classification of these matrices.

Discusses properties and implications of these balanced matrices.

Abstract

We consider unimodular matrices $M$ such that neither $M$ nor $M^{-1}$ contain zero entries. Matrices typically exhibit a trade-off: small $M$ imply large $M^{-1}$. We investigate rare cases where both remain small, classify these matrices up to symmetry, and discuss aspects of this balanced setting.