Important: The linalg package has been deprecated. Use the superseding command LinearAlgebra[CompanionMatrix], instead.

- For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.

The call companion(p, x) will return a companion matrix associated with the polynomial p(x).

If C := companion(p, x) and p is $\mathrm{a0}+\mathrm{a1}x+...+x^n$ (a monic univariate polynomial) then $C_{i,n}=-\mathrm{coeff}\left(p\,x\,i-1\right)$ ($i=1..n$), $C_{i,i-1}=1$ ($i=2..n$), and $C_{i,j}=0$ for other values of i and j.

The command with(linalg,companion) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{linalg}\right)\:$

$p≔x^4+9x^3+2x^2+17x+5$

$p≔x^4+9x^3+2x^2+17x+5$

$\mathrm{companion}\left(p\,x\right)$

$\left[\begin{array}{cccc}0& 0& 0& \mathrm{-5}\\ 1& 0& 0& \mathrm{-17}\\ 0& 1& 0& \mathrm{-2}\\ 0& 0& 1& \mathrm{-9}\end{array}\right]$

$q≔\mathrm{expand}\left(\left(z+2\right)\left(z-5\right)\left(z+3\right)\right)$

$q≔z^3-19z-30$

$\mathrm{companion}\left(q\,z\right)$

$\left[\begin{array}{ccc}0& 0& 30\\ 1& 0& 19\\ 0& 1& 0\end{array}\right]$