Companion matrix

Let \(k\) be a field and let \(p(x) = a_0 + a_1 x + \cdots + a_{n-1} x^{n-1} + x^n\in k[x]\) be a monic polynomial of degree \(n\geq 1\). The companion matrix of \(p\) is \[ C_p \coloneqq \begin{bmatrix} 0 & 0 & \cdots & 0 &-a_0 \\ 1 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & \cdots & 0 & -a_2 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \cdots & 1 & -a_{n-1} \end{bmatrix} \in\Mat_{n\times n}(k). \]