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
Canonical name	CompanionMatrix
Date of creation	2013-03-22 13:17:12
Last modified on	2013-03-22 13:17:12
Owner	aoh45 (5079)
Last modified by	aoh45 (5079)
Numerical id	7
Author	aoh45 (5079)
Entry type	Definition
Classification	msc 15A21