Re: The companion matrix of a polynomial.

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

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 ] ]

Best,
Zhao

> On Sat, Jan 14, 2023 at 8:00 AM Hongyi Zhao <hongyi.zhao@gmail.com> wrote:
>>
>> On Sat, Jan 14, 2023 at 8:48 PM Charles Greathouse
>> <crgreathouse@gmail.com> wrote:
>> >
>> > If you want the companion matrix if a polynomial in GP, pass the polynomial (not a matrix) to matcompanion.
>>
>> Is there a convenient command to generate the polynomial of a matrix
>> and then feed it to matcompanion?
>>
>> For example, in my case, I've the following matrix A, so the first
>> step is to create the polynomial of it which can be fed to
>> matcompanion directly:
>>
>>  A=[ 1, -4, 0; 1, 0, -2; 3, 1, 0 ]
>>
>> Best regards,
>> Zhao
>>
>> > On Fri, Jan 13, 2023 at 10:21 PM Hongyi Zhao <hongyi.zhao@gmail.com> wrote:
>> >>
>> >> Hi here,
>> >>
>> >> In GAP, the following command can be used to compute a companion
>> >> matrix of the polynomial:
>> >>
>> >> gap> A:=[ [ 1, -4, 0 ], [ 1, 0, -2 ], [ 3, 1, 0 ] ];;
>> >> gap> CompanionMat(A);
>> >> [ [ [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ], [ -1, 4, 0 ] ], [ [ [
>> >> 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ], [ -1, 0, 2 ] ] ]
>> >>
>> >> When I try the similar operation in GP, the corresponding error will
>> >> be triggered:
>> >>
>> >> ? A=[ 1, -4, 0; 1, 0, -2; 3, 1, 0 ]
>> >> %1 =
>> >> [1 -4  0]
>> >>
>> >> [1  0 -2]
>> >>
>> >> [3  1  0]
>> >>
>> >> ? matcompanion(A)
>> >>   ***   at top-level: matcompanion(A)
>> >>   ***                 ^---------------
>> >>   *** matcompanion: incorrect type in matcompanion (t_MAT).
>> >>   ***   Break loop: type 'break' to go back to GP prompt
>> >> break>
>> >>
>> >> Any tips for doing this in GP?
>> >>
>> >> Regards,
>> >> Zhao
>> >> --
>> >> Assoc. Prof. Hongsheng Zhao <hongyi.zhao@gmail.com>
>> >> Theory and Simulation of Materials
>> >> Hebei Vocational University of Technology and Engineering
>> >> No. 473, Quannan West Street, Xindu District, Xingtai, Hebei province
>> >>