Matrix Canonical Forms
Let $V$ be an $F$-vector space of dimension $n$. Recall that if we have a linear operator $T \in \mathrm{End}(V)$, we can represent it by the matrix
\[\begin{align*} A = \begin{pmatrix} [T(b_1)]_\mathcal{B} & \cdots & [T(b_n)]_\mathcal{B} \; \end{pmatrix} \end{align*}\]where $\mathcal{B} = \{b_1, \ldots, b_n\}$ is a basis of $V$. Ideally, if we choose the appropriate basis, we get a matrix whose structure is very easy to work with. One such example is a diagonal matrix which is an ideal scenario for a variety of reasons. Unfortunately, this can very well fail and so we resort to other canonical forms which have their own respective theoretical advantages.