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**Majialin** I stumbled across this question, and I'm not quite sure how to go about and find the Jordan form of this kind of question. Any help on how to get started?

Let N be an element of L(P4(C)), defined by N(p(z)) = zp'''(z)+p'(z).Show that N is nilpotent and find the Jordan form of N.

Any help is greatly appreciated!

Find the matricial representation of N wrt the basis $\displaystyle \{1,z,z^2,z^3,z^4\}$ of $\displaystyle P_4(\mathbb{C})[z]$ . From here it follows at once that N is nilpotent, and check that $\displaystyle N^5=0 $ but $\displaystyle N^4\ne 0$, so that its Jordan Form is...

Tonio