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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 :)

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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.

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