Every square matrix is similar the sum of a diagonal and a nilpotent matrix.

Jordan-Chevalley Decomposition. Over an algebraically closed field, every square matrix $A$ can be uniquely written as:
$$A = D + N$$
where $D$ is diagonalizable, $N$ is nilpotent, and $DN = ND$.

Furthermore, both $D$ and $N$ are polynomials in $A$.

Example. For $A = \begin{pmatrix} 3 & 1 \\ 0 & 3 \end{pmatrix}$:
$$A = \underbrace{\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}}_{D} + \underbrace{\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}}_{N}$$

This decomposition is fundamental in the structure theory of linear operators and Lie algebras.