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; precise density bounds are given for when the property holds, as well as easy transformations to this typical case. 
Mustafa Elsheikh, Mark Giesbrecht
math.RA cs.SC math.PR