A topological criterion of diagonalizability

For $A \in \mathcal{M}_n(\mathbb{C})$ , let $S(A):=\{PAP^{-1}, P\in GL_n(\mathbb{C})\}$ denote the similarity class of $A$ in $\mathcal{M}_n(\mathbb{C})$ .

Assume $A$ is diagonalizable.

1) a) Characterize $S(A)$ using the characteristic polynomial and the minimal polynomial of $A$ .

1) b) Deduce that $S(A)$ is closed.

Now assume that $A$ is not diagonalizable.

Show that there exists a diagonalizable matrix in the closure of $S(A)$ .

Deduce a criterion of diagonalizability for $A$ .

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