Considering a matrix $A\in gl(n, \mathbb{C})$, and $M\in GL(n, \mathbb{C})$ such that $ A = M J M^{-1} $ with $J \in gl(n, \mathbb{C})$ in normal Jordan form, I'm looking for an estimate of $\Vert M \Vert$ and $\Vert M^{-1} \Vert$...
The norm considered is $\Vert M \Vert = \sup_{x \in \mathbb{C}^n \backslash \{ 0\}} \frac{\Vert Mx \Vert_2}{\Vert x \Vert_2}$ where $\Vert u \Vert_2 = \sqrt{\vert u_1 \vert^2 + \cdots + \vert u_n \vert^2}$.
I guess it depends on $\Vert A \Vert$ and the distance between the couple of eigenvalues, but I can't find any result on it. Would you have some references ?
Thank you !