Companion matrix

Definition

Let $F$ be a field, let $n \ge 1$, and let $p(t) = t^n + c_{n-1} t^{n-1} + \cdots + c_1 t + c_0$ be a monic polynomial over $F$. The companion matrix of $p$ is the $n \times n$ matrix $$C(p) = \begin{pmatrix} 0 & 0 & \cdots & 0 & -c_0 \\ 1 & 0 & \cdots & 0 & -c_1 \\ 0 & 1 & \cdots & 0 & -c_2 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \cdots & 1 & -c_{n-1} \end{pmatrix}.$$

Remarks

The characteristic polynomial and the minimal polynomial of $C(p)$ both equal $p$. In the basis $e_1, \dots, e_n$, the map of $C(p)$ sends $e_i$ to $e_{i+1}$ for $i < n$. Some authors use the transpose of this matrix as the companion matrix. Companion matrices are the blocks of the rational canonical form.

Explanations

L1 human

For each monic polynomial, the companion matrix is a matrix with exactly that characteristic polynomial. It shifts each basis vector to the next one, and the polynomial fixes the image of the last one. So questions about roots of polynomials become questions about eigenvalues of matrices.