[GAP Forum] Find rank-1 matrices in given subspace of matrices

[Benoit Jacob]

Many thanks, Alexander and Dima, for your replies!
I was missing the key idea that rank one is charactized by the nullness of
all 2x2 minors.
Thanks also Alexander for the nice implementation!
Cheers,
Benoit

2016-02-15 16:07 GMT-05:00 Alexander Hulpke <hulpke at math.colostate.edu>:

> Dear Benoit,
>
> Ah — within a subspace will give you polynomial equations (all 2x2
> subdeterminants=0) in the coefficients of a linear combination, and at
> least in principle this can be done with Groebner bases (i.e. you get rank
> <=1, but rank 0 is easily eliminated.)
>
> For example (using the appended function) for the standard basis of
> Q^{3\times 4}:
>
> gap> b:=BasisVectors(Basis(MatrixSpace(Rationals,3,4)));;
> gap> e:=Rank1Equations(b);
> [ x_1*x_6-x_2*x_5, x_1*x_7-x_3*x_5, x_1*x_8-x_4*x_5, x_2*x_7-x_3*x_6,
>   x_2*x_8-x_4*x_6, x_3*x_8-x_4*x_7, x_1*x_10-x_2*x_9, x_1*x_11-x_3*x_9,
>   x_1*x_12-x_4*x_9, x_2*x_11-x_3*x_10, x_2*x_12-x_4*x_10,
> x_3*x_12-x_4*x_11,
>   x_5*x_10-x_6*x_9, x_5*x_11-x_7*x_9, x_5*x_12-x_8*x_9, x_6*x_11-x_7*x_10,
>   x_6*x_12-x_8*x_10, x_7*x_12-x_8*x_11 ]
> gap> ReducedGroebnerBasis(e,MonomialLexOrdering());
> [ x_7*x_12-x_8*x_11, x_6*x_12-x_8*x_10, x_6*x_11-x_7*x_10,
> x_5*x_12-x_8*x_9,
>   x_5*x_11-x_7*x_9, x_5*x_10-x_6*x_9, x_3*x_12-x_4*x_11, x_3*x_8-x_4*x_7,
>   x_2*x_12-x_4*x_10, x_2*x_11-x_3*x_10, x_2*x_8-x_4*x_6, x_2*x_7-x_3*x_6,
>   x_1*x_12-x_4*x_9, x_1*x_11-x_3*x_9, x_1*x_10-x_2*x_9, x_1*x_8-x_4*x_5,
>   x_1*x_7-x_3*x_5, x_1*x_6-x_2*x_5 ]
>
> So x_7 =x_8*x_11/x_12   (and case for x_12=0) etc. and you can build an
> (ugly) parameterization from these.
>
> (Alternatively one could try to use \sum_c_i M_i=v\cdot w^T with v and w
> given by extra variables that are to be eliminated. This will yield the
> same  Groebner basi after variable elimination.)
>
> Best,
>
>   Alexander
>
>
>
> Rank1Equations:=function(mats)
> local l,f,r,vars,n,m,c,d,eqs;
>   l:=Length(mats);
>   f:=DefaultFieldOfMatrix(mats[1]);
>   r:=PolynomialRing(f,l);
>   vars:=IndeterminatesOfPolynomialRing(r);
>   n:=Length(mats[1]);
>   m:=Length(mats[1][1]);
>   eqs:=[];
>   for c in Combinations([1..n],2) do
>     for d in Combinations([1..m],2) do
>       Add(eqs,
>           Sum([1..l],x->vars[x]*mats[x][c[1]][d[1]])
>           *Sum([1..l],x->vars[x]*mats[x][c[2]][d[2]])
>           -Sum([1..l],x->vars[x]*mats[x][c[1]][d[2]])
>           *Sum([1..l],x->vars[x]*mats[x][c[2]][d[1]]));
>     od;
>   od;
>   return eqs;
> end;
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