This is a well-known fact, several proofs have been outlined on Math.Se but they all rely on arguments involving Jordan normal form or modules, which are beyond my reach.
Does anybody know of a more elementary proof?
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Do you know about Smith Normal Form? – D Wiggles Jan 30 '14 at 19:51
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This is out of reach as well (it's related to modules right?) – Gabriel Romon Jan 30 '14 at 19:52
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I don't know of any proof of this that doesn't use some sort of canonical form. Jordan normal form is probably the most accessible and is definitely worth learning about. – D Wiggles Jan 31 '14 at 03:50