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.
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