The Jordan and Frobenius normal forms of a linear map $A:\Bbb R^n \rightarrow \Bbb R^n$ seem to be maximally simple representations of $A$ in the sense that one of them contains as few nonzero entries as possible. But how do you prove that?
More precise, show that for every $A:\Bbb R^n \rightarrow \Bbb R^n$ and every Basis $B$ of $\Bbb R^n$, the transformation matrix $_B A _B$ has at least as many nonzero entries as the Jordan normal form or the Frobenius normal form of $A$ or, otherwise, give a counterexample.