Generating all invertible matrices by row operations

TL;DR

This paper demonstrates a Gray code listing for all invertible matrices over finite fields, where consecutive matrices differ by simple row operations, and proves Hamilton connectivity of the flip graph, advancing Lovász's conjecture.

Contribution

It introduces a Gray code for invertible matrices over finite fields using elementary row operations, and establishes Hamilton connectivity of the associated flip graph.

Findings

All invertible matrices can be listed with minimal row operation differences.

The flip graph on invertible matrices is Hamilton connected, except when it forms a cycle.

This work confirms a special case of Lovász's conjecture on Hamiltonicity of vertex-transitive graphs.

Abstract

We show that all invertible $n \times n$ matrices over any finite field $\mathbb{F}_q$ can be generated in a Gray code fashion. More specifically, there exists a listing such that (1) each matrix appears exactly once, and (2) two consecutive matrices differ by adding or subtracting one row from a previous or subsequent row, or by multiplying or diving a row by the generator of the multiplicative group of $\mathbb{F}_q$. This even holds if the addition and subtraction of each row is allowed to some specific rows satisfying a certain mild condition. Moreover, we can prescribe the first and the last matrix if $n\ge 3$, or $n=2$ and $q>2$. In other words, the corresponding flip graph on all invertible $n \times n$ matrices over $\mathbb{F}_q$ is Hamilton connected if it is not a cycle. This solves yet another special case of Lov\'{a}sz conjecture on Hamiltonicity of vertex-transitive graphs.