Relating p-adic eigenvalues and the local Smith normal form

Conditions are established under which the p-adic valuations of the invariant factors (diagonal entries of the Smith form) of an integer matrix are equal to the p-adic valuations of the eigenvalues. It is then shown that this correspondence is the typical case for “most” matrices; density counts are given for when this property holds, as well as easy transformations to this typical case.