If $A \in \mathrm{M}_{n}(\mathbb{Q})$ is such that the characteristic polynomial of $A$ is irreducible over $\mathbb{Q}$, then $A$ is diagonalizable in $\mathrm{M}_{n}(\mathbb{C})$, i.e., there exists $P \in \mathrm{M}_{n}(\mathbb{C})$ such that $P$ is invertible and $P A P^{-1}$ is a diagonal matrix.
