The uniform normal form of a linear mapping

Let $V$ be a finite dimensional vector space over a field $\mathrm{k}$ of characteristic $0$. Let $A$ be a linear mapping of $V$ into itself. This paper gives a normal form for $A$, which gives a better description of the structure of $A$ than the companion matrix. The computation of this normal form uses only operations from $\mathrm{k}$ and does not require finding roots of any polynomial.