Let $M$ be a matrix with entries in $\mathbb C$. The SN (or Jordan-Chevalley) decomposition theorem states that we can find unique matrices $S$ and $N$ such that:
- $M=S+N$
- $S$ is diagonalizable
- $N$ is nilpotent
- $SN=NS$.
I would like to prove the uniqueness part of this theorem, since everything else is immediate from the fact that all complex matrices can be put into Jordan normal form for some choice of basis. (If $M=AJA^{-1}$, where $J$ is in Jordan normal form, write $J=J_S+J_N$, where $J_S$ consists of the diagonal part of $J$ with zeroes elsewhere, and $J_N$ consists of the line above the diagonal with zeroes elsewhere. Then let $S=AJ_SA^{-1}$ and $N=AJ_NA^{-1}$ and direct calculation verifies that these meet the criteria above.)
This post attempted to answer the same question, but unfortunately the proof is invalid because it assumes that the difference of two nilpotent matrices is another nilpotent matrix, which is not true (a counterexample is given in this post).