Jordan Canonical Form
Let $V$ be a finite-dimensional vector space over $F$ and $T\in\mathrm{End}(V)$ be a linear operator. We will assume that the minimal polynomial $m_T(x) \in F[x]$ of $T$ splits over $F$. In an ideal world, we would be able to represent $T$ by a diagonal matrix. In a certain sense, however, we can still get pretty close to a diagonal matrix.
Suppose that
\[\begin{align*} m_T(x) = (x - \lambda_1)^{e_1} \cdots (x - \lambda_n)^{e_n}. \end{align*}\]Then we know that $V$ decomposes into the direct sum
\[\begin{align*} V \cong V_1 \oplus \cdots \oplus V_n \end{align*}\]as an $F[x]$-module where $V_i = \{v \in V \mid (T - \lambda_i)^{e_i}v = 0\}$. Furthermore, $V_i$ decomposes into