Editing Companion matrix (section)

{{Short description|Square matrix constructed from a monic polynomial}}
In [[linear algebra]], the  [[Ferdinand Georg Frobenius|Frobenius]]  '''companion matrix''' of the [[monic polynomial]]
<math display="block">
p(x)=c_0 + c_1 x + \cdots + c_{n-1}x^{n-1} + x^n
</math>
is the [[square matrix]] defined as

<math display="block">C(p)=\begin{bmatrix}
0 & 0 & \dots & 0 & -c_0 \\
1 & 0 & \dots & 0 & -c_1 \\
0 & 1 & \dots & 0 & -c_2 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \dots & 1 & -c_{n-1}
\end{bmatrix}.</math>

Some authors use the [[transpose]] of this matrix, <math> C(p)^T </math>, which is more convenient for some purposes such as linear [[recurrence relation]]s ([[#Linear recursive sequences|see below]]).

<math> C(p) </math> is defined from the coefficients of <math>p(x)</math>, while the [[characteristic polynomial]] as well as the [[minimal polynomial (linear algebra)|minimal polynomial]] of <math> C(p) </math> are equal to <math>p(x) </math>.<ref> {{Cite book |last=Horn |first=Roger A. |url=https://books.google.com/books?id=f6_r93Of544C&dq=%22companion+matrix%22&pg=PA147 |title=Matrix Analysis |author2=Charles R. Johnson |publisher=Cambridge University Press |year=1985 |isbn=0-521-30586-1 |location=Cambridge, UK |pages=146–147 |accessdate=2010-02-10}}</ref> In this sense, the matrix <math> C(p) </math> and the polynomial <math>p(x)</math> are "companions".