How do I show that commuting matrices preserve generalized eigenspaces? Let T and S be two linear operators on V, such that TS = ST. How do I show that T preserves the generalized eigenspaces of S? 
There were 2 similar questions, but they were a lot more advanced than my current level, so they didn't help.

Hint: $TS = ST \implies T(S - \lambda I) = (S- \lambda I)T$ and $T(S -\lambda I)^{k} = (S - \lambda I)^{k}T$. 
