Let $A\in M_n(\mathbb{C})$. Let us consider the conjugacy action of $GL_n(\mathbb{C})$ on $M_n(\mathbb{C})$ and let us denote $\mathcal{O}_A$ the orbit of $A$ under this action. We suppose that $\mathcal{O}_A$ is not closed in $M_n(\mathbb{C})$ and we want to determine $\text{cl}(\mathcal{O}_A)$.
I already proved that if $\mathcal{O}_A$ is not closed in $M_n(\mathbb{C})$ hence $A$ is not diagonalizable in $\mathbb{C}$ (let us say $A \not \in \mathcal{D}_n(\mathbb{C}))$.
Hence wlog we can suppose that $A$ is for instance triangularizable on $\mathbb{C}$. Then we know that $\text{cl}(\mathcal{O}_A)$ contains a diagonalizable matrix on $\mathbb{C}$.
How could we go further ?
Thanks in advance !