 Companion matrix

For a monic polynomial $f(t)=t^d+c_{d-1}{t}^{d-1}+\cdots +c_0$, its companion matrix represents multiplication by $t$ on the quotient module $F[t]/(f)$ in the basis $1,t,\dots ,t^{d-1}$. With column vectors it has ones on the subdiagonal and last column $(-c_0,\dots ,-c_{d-1}{)}^{\mathsf{T}}$. For example, $C(t^2+ct+d)=\left(\begin{array}{cc}0& -d\\ 1& -c\end{array}\right)$. Transposing this matrix gives another common convention, so the subdiagonal convention should be specified when giving entries.

A companion matrix is a specific type of square matrix that is associated with a polynomial.