companion matrix

Given a monic polynomial p(x)=xn+an-1xn-1++a1x+a0 the companion matrixMathworldPlanetmath of p(x), denoted 𝒞p(x), is the n×n matrix with 1’s down the first subdiagonal and minus the coefficients of p(x) down the last column, or alternatively, as the transposeMathworldPlanetmath of this matrix. Adopting the first convention this is simply


Regardless of which convention is used the minimal polynomialPlanetmathPlanetmath ( of 𝒞p(x) equals p(x), and the characteristic polynomialMathworldPlanetmathPlanetmath of 𝒞p(x) is just (-1)np(x). The (-1)n is needed because we have defined the characteristic polynomial to be det(𝒞p(x)-xI). If we had instead defined the characteristic polynomial to be det(xI-𝒞p(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