On conjugacy of diagonalizable integral matrices

TL;DR

The paper proves that certain diagonalizable integral matrices with rational eigenvalues are conjugate over the integers if they are conjugate locally, and applies this to adjacency matrices of specific graphs.

Contribution

It establishes a criterion for conjugacy of diagonalizable integral matrices over Z based on local conjugacy, and applies it to Paley and Peisert graphs.

Findings

Matrices with rational eigenvalues are conjugate over Z if locally conjugate.

Paley and Peisert graph adjacency matrices are conjugate in GL(p^2,Z) for primes p ≡ 3 mod 4.

Answers a question by Peter Sin regarding graph matrix conjugacy.

Abstract

It is shown that under some additional assumption two diagonalizable integral matrices X and Y with only rational eigenvalues are conjugate in GL(n,Z) if and only if they are conjugate over all localizations. This is used to prove that for a prime p == 3 (mod 4) the adjacency matrices of the Paley graph and the Peisert graph on p^2 vertices are conjugate in GL(p^2,Z), answering a question by Peter Sin.