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Mathematics · Glossary

What is companion matrix?

Definition 3.15 University Mathematics — Year 3 · Chapter 3 — Modules over a Principal Ideal Domain

For P=Xm+am−1Xm−1+⋯+a0P = X^m + a_{m-1}X^{m-1} + \dots + a_0 monic, the companion matrix is

CP=(0−a01⋱−a1⋱0⋮1−am−1):C_P = \begin{pmatrix} 0 & & & -a_0\\ 1 & \ddots & & -a_1\\ & \ddots & 0 & \vdots\\ & & 1 & -a_{m-1} \end{pmatrix} :

the matrix of “multiplication by XX” on K[X]/(P)K[X]/(P) in the basis 1,Xˉ,…,Xˉm−11, \bar X, \dots, \bar X^{m-1}.

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