I have this problem:
Let $K$ be a field. Let $J\in M_n(K)$ a Jordan matrix. Prove that there exists a diagonal matrix $D$ and a nilpotent matrix $N$ such that $J=D+N$ and $DN=ND$.
I saw that this was similar to the Jordan-Chevalley decomposition, but this theorem is for every endomorphism. If I understood my problem, I only have to say that I can separate the diagonal matrix and the matrix with $1$ from the jordan block, and prove that the diagonal and the nilpotent matrix commute. Is that right, or I have to prove it like in jordan-chevalley decomposition?
Thank you!