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

Question

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 $A$.

Possibly helpful, though I haven't come up with a solution yet, is that all the matrices in $C(A)$ are simultaneously triangulable. Thus WLOG, they are all upper triangular. — Christian Sykes, Aug 15 '17 at 22:29

score 4 · Answer 1 · answered Aug 14 '17 at 23:20

4

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

answered Aug 14 '17 at 23:20

Robert Israel

1 Answers1