Related Matrices#
The companion matrix of a monic linear differential operator
\[D^n+a_{n-1}D^{n-1}+\cdots+a_0 \in F[D]\]
is defined as the \(n\times n\) matrix
\[\begin{split} \begin{pmatrix}0&1&0&0&\ldots&0\\ 0&0&1&0&\ldots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&0&\ldots&1\\ -a_0&-a_1&-a_2&-a_3&\ldots&-a_{n-1}\end{pmatrix}\end{split}\]
- CompanionMatrix(L): RngDiffOpElt -> AlgMatElt#
Returns the companion matrix of the monic differential operator \(L\).
- Example: Companion Matrix of an Operator (ex-62793f)#
> F<z> := RationalDifferentialField(Rationals()); > R<D> := DifferentialOperatorRing(F); > L := D^3-z*D^2+2*D+5; > CompanionMatrix(L); [0 1 0] [0 0 1] [-5 -2 z]