Companion matrix ($C(f)$)

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.

For a monic polynomial $f(t)=t^d+c_{d-1}{t}^{d-1}+\cdots +c_0$, use the companion matrix convention with ones on the subdiagonal and last column $(-c_0,\dots ,-c_{d-1}{)}^{\mathsf{T}}$. In $tI-C(f)$, the cofactors of the last-column entries, from top to bottom, are $1,t,\dots ,t^{d-1}$. This follows by expanding their bidiagonal minors: the subdiagonal factors $-1$ cancel the cofactor signs. Laplace expansion along that column therefore gives $c_0+c_{1}t+\cdots +c_{d-2}{t}^{d-2}+(t+c_{d-1})t^{d-1}=f(t)$. The degree-one case is the same identity for the one-by-one matrix $(-c_0)$.