The conjugacy problem in GL (n,Z)

Abstract

We present a new algorithm that, given two matrices in GL ( n , Q ) , decides if they are conjugate in GL ( n , Z ) and, if so, determines a conjugating matrix. We also give an algorithm to construct a generating set for the centraliser in GL ( n , Z ) of a matrix in GL ( n , Q ) . We do this by reducing these problems, respectively, to the isomorphism and automorphism group problems for certain modules over rings of the form O K [ y ] / ( y l ) , where O K is the maximal order of an algebraic number field and l ∈ N , and then provide algorithms to solve the latter. The algorithms are practical and our implementations are publicly available in Magma.