Find all matrices which commute with a given matrix I wonder how to find all matrices $B$ which satisfy $AB=BA$, where
$A=\left( {\begin{array}{*{20}{c}}
  1&{}&{}&{}&{} \\ 
  1&1&{}&{}&{} \\ 
  1&1&1&{}&{} \\ 
   \vdots & \vdots & \vdots & \ddots &{} \\ 
  1&1&1&1&1 
\end{array}} \right)$
(the coefficients above the diagonal of the matrix $A$ are zero). I tried Jordan form of $A$, but still couldn't see what to do next...
Any help will be appreciated :)

Note that $A - I$ is nilpotent with rank $n-1$.  As such, $A$ is non-derogatory (in particular, $A$ is similar to the Jordan block of size $n$ with eigenvalue $1$).  It follows that $B$ commutes with $A$ if and only if $B = p(A)$ for some polynomial $p$ (see Horn and Johnson's Matrix Analysis for a reference on this).
The end result is that $B$ will commute with $A$ if and only if $B$ is lower-triangular and Toeplitz.
