TITLE: Centralizers of Matrices
QUESTION [6 upvotes]: let $A$ be a complex matrix. Denote by $J(A)$ the Jordan Canonical Form of $A$. Let $C[J(A)]$ be the centralizer of $J(A)$ in $M_n(\mathbb C)$. Can we construct a real matrix $B$, that is, $B$ has only real entries,  verifying the equality $C[J(A)]=C(B)$, in $M_n(\mathbb C)$?

REPLY [3 votes]: $J(A)$ can be written as $S+N$, where $S$ is diagonal and $N$ is nilpotent (and has only entries $0$ and $1$). Moreover, $S$ and $N$ are polynomials in $J(A)$. 
It follows that $C$ centralizes $J(A)$ if and only if it centralizes $S$ and $N$. 
The recipe for $B$ is now clear, $B = T+N$, where $T$ is obtained from $S$ by replacing the distinct eigenvalues with distinct real numbers. 
This is because $S$ and $T$ have the same centralizer, which consists of block diagonal matrices, with arbitrary blocks on each eigenspace.