Q:

Dimension of of the subspace formed by all matrices which commute with a given matrix

Let $A\in M_n(\mathbb{C})$ be a matirx.  Consider the subspace 
$$C(A)=\{X\in M_n(\mathbb{C})\text{ }|\text{ }AX=XA\}$$
How to prove that the dimension of $C(A)$ is at least $n$? 
The only idea I could get was that the dimension is at least $d$ where $d$ is the degree of the minimal polynomial of $A$. 

A:

Use Jordan canonical form.   A Jordan block $J = \lambda I + N$ of size $m$ commutes with the $m$ matrices $N^j$ for $j=0\ldots m-1$.

