Ideal Classes and Matrix Conjugation over Z

When R is a commutative ring, matrices A and B in Mn(R) are called conjugate when UAU−1 = B for some U ∈ GLn(R). The conjugacy problem in Mn(R) is: decide when two matrices in Mn(R) are conjugate. We want to look at the conjugacy problem in Mn(Z), where ideal theory and class groups make an interesting appearance. The most basic invariant for conjugacy classes of matrices is the characteristic polynomial: conjugate matrices have the same characteristic polynomial. This is not a complete invariant in general: the matrices ( 1 0 0 1 ) and ( 1 1 0 1 ), both have characteristic polynomial (T − 1)2, but ( 1 0 0 1 ) and ( 1 1 0 1 ) are not conjugate (in any M2(R)) since the identity matrix is conjugate only to itself. While there are refinements of the characteristic polynomial which settle the conjugacy problem in Mn(F ) for F a field (use the rational canonical form), we don’t pursue that direction. Instead our starting point is a special case where the characteristic polynomial is a complete invariant.