Infinite Symmetric Matrices over Z_2 and the Lights Out Problem

TL;DR

This paper proves that the diagonal of an infinite symmetric matrix over Z_2 lies in its range and applies this to extend the Lights Out problem to infinite cases, combining linear algebra and analysis.

Contribution

It introduces a novel hybrid analysis approach to infinite matrices over Z_2 and extends the Lights Out problem to countably infinite configurations.

Findings

Diagonal vector is in the matrix range for infinite symmetric matrices over Z_2

Extension of Lights Out problem to countably infinite case

Hybrid analysis method for infinite matrix problems

Abstract

We show, using a hybrid analysis/linear algebra argument, that the diagonal vector of an infinite symmetric matrix over ${\mathbb Z}_2$ is contained in the range of the matrix. We apply this result to an extension, to the countable infinite case, of the Lights Out problem.