Symmetric Matrices over F_2 and the Lights Out Problem

TL;DR

This paper proves a property of symmetric matrices over F_2 and applies it to generalize the Lights Out problem on graphs, revealing new insights into the problem's structure.

Contribution

The paper establishes a fundamental theorem about symmetric matrices over F_2 and extends the Lights Out problem to broader graph settings.

Findings

Range of symmetric matrices over F_2 contains diagonal vector

Generalization of Lights Out problem on graphs

New structural insights into Lights Out problem

Abstract

We prove that the range of a symmetric matrix over F_2 contains the vector of its diagonal elements. We apply the theorem to a generalization of the "Lights Out" problem on graphs.