Let $J$ be a $k \times k$ jordan block, prove that any matrix which commutes with $J$ is a polynomial in $J$ Let $J$ be a $k \times k$ jordan block, prove that any matrix which commutes with $J$ is a polynomial in $J$.
I appreciate your hints, Thanks

2 by 2 only. i really hope you will try the 3 by 3 case by hand, analogous to this:
Aright, eigenvalue $w,$ 
$$ J =
\left(
\begin{array}{rr}
w & 1 \\
0 & w
\end{array}
\right),
$$
trial
$$ M =
\left(
\begin{array}{rr}
a & b \\
c & d
\end{array}
\right),
$$
Next
$$ JM =
\left(
\begin{array}{rr}
wa + c & wb+d \\
wc & wd
\end{array}
\right),
$$
but
$$ MJ =
\left(
\begin{array}{rr}
wa  & wb+a \\
wc & wd+ c
\end{array}
\right).
$$
We find that $JM = MJ$ precisely when
$$  c = 0 \; \; \mbox{AND} \; \; a=d.   $$ Notice that $w$ does not appear.
Under these conditions,
$$ M =
\left(
\begin{array}{rr}
a & b \\
0 & a
\end{array}
\right).
$$
That means that $$  M = (a-bw)I + b J,  $$ that is a polynomial in $J.$
