# Smith normal form of powers of a matrix

I have been trying to compute the cokernels of powers of integer matrices using Smith normal form. When is $\text{Smith}(A^2) = (\text{Smith}(A))^2$ and more generally when is $\text{Smith}(A^n) = (\text{Smith}(A))^n$? Some numerical experiments showed that this is not always true. Is there at least a good sufficient condition for determining when such a formula holds? Thanks!

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I'm wondering of you can get something out of property III on pp.2 of this paper: Axioms for invariant factors. Queiró, João Filipe. Portugaliae Mathematica (1997) Volume: 54, Issue: 3, page 263-269 PDF link. – user2468 Aug 19 '12 at 2:34