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.

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