compan

Companion matrix

Introduction

A companion matrix $C\in\mathbb{C}^{n\times n}$ has the following form

Roots of the polynomial $p(x)$​​ are eigenvalues of $C$, where

This can be seen by expressing $p(x)$ as the determinant:

Roots of $\det(x I - C)=0$​ are eigenvalues of $C$, which are also roots of $p$.

Usage

compan(p)

• p is a vector containing coefficients of a polynomial, where p(1) and p(end) correspond to the highest and zeroth order terms, respectively.
• p(2)/p(1), p(3)/p(1), ..., p(end)/p(1) correspond to $a_{n-1}, a_{n-2}, \ldots,a_0$​ of Equation (1), respectively.
• It returns the companion matrix shown in Equation (1).

Example

• Let $p(x)=x^2+2x+3$. The following shows the companion matrix C, the roots of $p(x)$​ and the eigenvalues of C. The roots and the eigenvalues are identical.

Input

p = [1 2 3];
c = compan(p)
eig(c)
roots(p)

Output

c = 
-2.0000  -3.0000
 1.0000   0.0000

ans = 
-1.0000 + 1.4142i
-1.0000 - 1.4142i

ans = 
-1.0000 - 1.4142i
-1.0000 + 1.4142i