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Number of perturbations of the Jordan form

I am looking for information about the number of Jordan forms that can be obtained from a given Jordan form of a small perturbation. … For example, if a Jordan form consists of a single cell 2x2 $$J=\begin{pmatrix} \lambda_0 &1\\\ 0&\lambda_0\end{pmatrix},$$ by small perturbation $\begin{pmatrix} 0&0\\\ 0&\varepsilon\end{pmatrix}$, we …
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