Space of matrices that commute with a given matrix

Question

Let $A$ be an $n\times n$ complex matrix, and $C(A)$ be the vector space of all matrices that commute with $A$. I have to determinate if the dimension of $C(A)$ is greater or equal than $n$, or not.

If anyone can give me a hint, i think the answer is yes, but i am not sure what i have to use to prove it.

Answer 1

Hint: consider the Jordan canonical form of $A$. What can you say about matrices that commute with a Jordan block?

Answer 2

For any matrix $A$ , $\dim \mathcal{C}(A)\geq n$, moreover if $A$ is diagonalizable with $\lambda_1,\ldots,\lambda_r$ its eigenvalues and $n_i=\text{multi}(\lambda_i)$, then $\dim\mathcal{C}(A)=\displaystyle\sum_{i=1}^rn_i^2$.