Companion Matrix

In linear algebra, the Frobenius companion matrix of the monic polynomial

$p(t)=c_0 + c_1 t + \cdots + c_{n-1}t^{n-1} + t^n ~,$

is the square matrix defined as

$C(p)=\begin{bmatrix} 0 & 0 & \dots & 0 & -c_0 \\ 1 & 0 & \dots & 0 & -c_1 \\ 0 & 1 & \dots & 0 & -c_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -c_{n-1} \end{bmatrix}.$

With this convention, and writing the basis as, one has (for ), and generates V as a -module: C cycles basis vectors.

Some authors use the transpose of this matrix, which (dually) cycles coordinates, and is more convenient for some purposes, like linear recursive relations.

Read more about Companion Matrix: Characterization, Diagonalizability, Linear Recursive Sequences

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