Definition:Companion Matrix

Definition

Let $P$ be the polynomial of degree $n$ presented in the form:

$\map P x = x^n - a_{n - 1} x^{n - 1} - \cdots - a_1 x - a_0$

The companion matrix of $P$ is the square matrix of order $n$:

$C = \begin {pmatrix} a_{n - 1} & a_{n - 2} & \cdots & a_2 & a_1 & a_0 \\

1 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 1 & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & 0 & 0 \\ 0 & 0 & \cdots & 0 & 1 & 0 \\ \end {pmatrix}$

Also see

• Results about companion matrices can be found here.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): companion matrix