The question I would like answered is the following: Given a matrix $G$ and that $G$ commutes with another matrix $X$, that is $[G, X] = 0$, what is $X$? Or more generally, what properties of $X$ may we infer?
I understand however that this question is really too vague, so here's a more specific question: If $G$ is in Jordan canonical form, does $[G, X] = 0$ imply that $X$ has the same Jordan canonical form? Or still more specific, if $G$ is diagonal with no two diagonal entries the same, does $[G, X] = 0$ imply that $X$ is diagonal?
I have convinced myself that the answer to the latter question is ‘yes’, but a simple proof eludes me.