companion matrix

Given a monic polynomial $p(x)=x^n+a_{n-1}{x}^{n-1}+\mathrm{\dots }+a_{1}x+a_0$ the companion matrix of $p(x)$, denoted ${𝒞}_{p(x)}$, is the $n\times n$ matrix with $1$’s down the first subdiagonal and minus the coefficients of $p(x)$ down the last column, or alternatively, as the transpose of this matrix. Adopting the first convention this is simply

Regardless of which convention is used the minimal polynomial (http://planetmath.org/MinimalPolynomialEndomorphism) of ${𝒞}_{p(x)}$ equals $p(x)$, and the characteristic polynomial of ${𝒞}_{p(x)}$ is just ${(-1)}^{n}p(x)$. The${(-1)}^n$ is needed because we have defined the characteristic polynomial to be $det({𝒞}_{p(x)}-xI)$. If we had instead defined the characteristic polynomial to be $det(xI-{𝒞}_{p(x)})$ then this would not be needed.