The CompanionMatrix(P) command returns the companion Matrix associated with the univariate polynomial $P\left(x\right)$.

If C := CompanionMatrix(P) and P is $a_0\+a_{1}x\+\cdots \+x_n$ (a monic univariate polynomial), then C is an $nxn$ Matrix where $n$ is the degree of polynomial P, $C_{i,n}=-a_{i-1}\left(i=1..n\right)$, $C_{i,i-1}=1\left(i=2..n\right)$, and $C_{i,j}=0$ for all other values of $i$ and $j$.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\:$

$p≔x^4+x^3+2x^2+3x$

$p≔x^4+x^3+2x^2+3x$

$\mathrm{CompanionMatrix}\left(p\right)$

$\left[\begin{array}{cccc}0& 0& 0& 0\\ 1& 0& 0& \mathrm{-3}\\ 0& 1& 0& \mathrm{-2}\\ 0& 0& 1& \mathrm{-1}\end{array}\right]$

$q≔\left(z+2\right)\left(z-5\right)\left(z+3\right)$

$q≔\left(z+2\right)\left(z-5\right)\left(z+3\right)$

$\mathrm{CompanionMatrix}\left(q\right)$

$\left[\begin{array}{ccc}0& 0& 30\\ 1& 0& 19\\ 0& 1& 0\end{array}\right]$