# companion matrix

Given a monic polynomial $p(x)=x^{n}+a_{n-1}x^{n-1}+\dots+a_{1}x+a_{0}$ the companion matrix of $p(x)$, denoted $\mathcal{C}_{p(x)}$, 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

 $\mathcal{C}_{p(x)}=\begin{pmatrix}0&0&\ldots&\ldots&\ldots&-a_{0}\\ 1&0&\ldots&\ldots&\ldots&-a_{1}\\ 0&1&\ldots&\ldots&\ldots&-a_{2}\\ 0&0&\ddots&&&\vdots\\ \vdots&\vdots&&\ddots&&\vdots\\ 0&0&\ldots&\ldots&1&-a_{n-1}\end{pmatrix}.$

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

Title companion matrix CompanionMatrix 2013-03-22 13:17:12 2013-03-22 13:17:12 aoh45 (5079) aoh45 (5079) 7 aoh45 (5079) Definition msc 15A21