Re: The companion matrix of a polynomial.

[Hongyi Zhao <hongyi.zhao@gmail.com>]

On Sun, Jan 15, 2023 at 5:19 PM Bill Allombert
<Bill.Allombert@math.u-bordeaux.fr> wrote:
>
> On Sun, Jan 15, 2023 at 09:12:07AM +0800, Hongyi Zhao wrote:
> > On Sat, Jan 14, 2023 at 9:58 PM Charles Greathouse
> > <crgreathouse@gmail.com> wrote:
> > >
> > > I'm not at all sure what you're doing here. The GAP source
> > > https://github.com/gap-system/gap/blob/3d47e2bc40869ea1b232a0a658d47b1897880fec/lib/upolyirr.gi#L82
> > > seems to require that the input to CompanionMat be a polynomial or a list of coefficients of a polynomial. It doesn't seem like it can do anything meaningful to a matrix.
> >
> > I want to get the companion matrix of matrix A in GP, which can be
> > generated in GAP, as shown below:
>
> > gap> A:=[[-1,3,-1,0,-2,0,0,-2],
> > > [-1,-1,1,1,-2,-1,0,-1],
> > > [-2,-6,4,3,-8,-4,-2,1],
> > > [-1,8,-3,-1,5,2,3,-3],
> > > [0,0,0,0,0,0,0,1],
> > > [0,0,0,0,-1,0,0,0],
> > > [1,0,0,0,2,0,0,0],
> > > [0,0,0,0,4,0,1,0]];;
> > gap> P:=RationalCanonicalFormTransform(A);;
> > gap> C:=A^P;
> > [ [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0,
> > 0, 0, 0, 0, -1 ], [ 0, 0, 1, 0, 0, 0, 0, -4 ],
> >   [ 0, 0, 0, 1, 0, 0, 0, -4 ], [ 0, 0, 0, 0, 1, 0, 0, 2 ], [ 0, 0, 0,
> > 0, 0, 1, 0, 4 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ] ]
>
> This is the rational canonical form (aka the Frobenius form), not the companion matrix.

There are the following very similar concepts:

https://encyclopediaofmath.org/wiki/Frobenius_matrix
https://en.wikipedia.org/wiki/Companion_matrix

It seems to me that the explanation and form given in the links above
are no different from Frobenius form.

> In GP, it is matfrobenius. There is a flag to the the transform. Read the doc.

I calculated it as follows:

? M=[ -1,3,-1,0,-2,0,0,-2;
-1,-1,1,1,-2,-1,0,-1;-2,-6,4,3,-8,-4,-2,1;-1,8,-3,-1,5,2,3,-3;0,0,0,0,0,0,0,1;0,0,0,0,-1,0,0,0;1,0,0,0,2,0,0,0;0,0,0,0,4,0,1,0];

? [F,B]=matfrobenius(A,2);
? M==B^-1*F*B
%14 = 1
? F
%15 =
[0 0 0 0 0 -1 0 0]

[1 0 0 0 0 -4 0 0]

[0 1 0 0 0 -4 0 0]

[0 0 1 0 0  2 0 0]

[0 0 0 1 0  4 0 0]

[0 0 0 0 1  0 0 0]

[0 0 0 0 0  0 0 1]

[0 0 0 0 0  0 1 1]


As you can see, the form above is not exactly the same as the result
of GAP and I tried to verify the isomorphism between them in GAP, but
failed:

gap> P:=RationalCanonicalFormTransform(A);;
gap> C:=A^P;
[ [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0,
0, 0, 0, 0, -1 ], [ 0, 0, 1, 0, 0, 0, 0, -4 ],
  [ 0, 0, 0, 1, 0, 0, 0, -4 ], [ 0, 0, 0, 0, 1, 0, 0, 2 ], [ 0, 0, 0,
0, 0, 1, 0, 4 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ] ]
gap> F:=[[0,0,0,0,0,-1,0,0],
> [1,0,0,0,0,-4,0,0],
> [0,1,0,0,0,-4,0,0],
> [0,0,1,0,0,,2,0,0],
> [0,0,0,1,0,,4,0,0],
> [0,0,0,0,1,,0,0,0],
> [0,0,0,0,0,,0,0,1],
> [0,0,0,0,0,,0,1,1]];
[ [ 0, 0, 0, 0, 0, -1, 0, 0 ], [ 1, 0, 0, 0, 0, -4, 0, 0 ], [ 0, 1, 0,
0, 0, -4, 0, 0 ], [ 0, 0, 1, 0, 0,, 2, 0, 0 ],
  [ 0, 0, 0, 1, 0,, 4, 0, 0 ], [ 0, 0, 0, 0, 1,, 0, 0, 0 ], [ 0, 0, 0,
0, 0,, 0, 0, 1 ], [ 0, 0, 0, 0, 0,, 0, 1, 1 ] ]
gap> TransformingPermutations(F,C);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `TransformingPermutations' on 2
arguments at /home/werner/Public/repo/github.com/gap-system/gap.git/lib/methsel2.g:249
called from
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
 called from read-eval loop at *stdin*:36
type 'quit;' to quit to outer loop
brk>


Any tips for this problem?

> Cheers,
> Bill.

Best,
Zhao