Let $ \mathcal{J}=\{A\in M_n(\mathbb{C}):\ A \text{ is a Jordan matrix}\} $ Then it is well-known that the similary orbit of $\mathcal{J}$ is all of $M_n(\mathbb{C})$.
What is the unitary orbit of $\mathcal{J}$? Is it dense?
It cannot be all of $M_n(\mathbb{C})$, because every matrix in $\mathcal{J}$ an its unitary conjugates have the property that eigenvectors corresponding to different eigenvalues are orthogonal to each other.