[GAP Forum] Matrices that satisfy MM^T=I or MM^T=\lambda*I
Max Horn
max at quendi.de
Wed Dec 22 15:31:55 GMT 2010
Dear Katie Morrison,
you can use the "forms" package for that. Use InvariantBilinearForm(GO(n,q)) to get the invariant form GAP uses (if you want to work in char 2, you should use the quadratic form instead). Then you can use various methods of the "forms" package to convert this to a bilinear form object (BilinearFormByMatrix), then get a matrix to conjugate that to a "canonical" form; use the same for your desired target form (in your case, this is defined by the identity matrix), and presto, you get a matrix by which you can conjugate GO(n,q) to the group preserving the default scalar product.
Here is an example:
gap> n:=5;; q:=9;; G:=GO(n,q);;
gap> mat1:=BaseChangeToCanonical(BilinearFormByMatrix(InvariantBilinearForm(G).matrix,GF(q)));;
gap> mat2:=BaseChangeToCanonical(BilinearFormByMatrix(IdentityMat(n,GF(q)),GF(q)));;
gap> H:=(G^(mat1^-1))^mat2;
<matrix group of size 6886425600 with 2 generators>
gap> ForAll(GeneratorsOfGroup(H), g -> IsOne(g*TransposedMat(g)));
true
Of course in even dimension, you have to different equivalence classes of non-degenerate bilinear forms, so you also need to make sure to pick the right one.
Hope that helps,
Max
Am 22.12.2010 um 14:40 schrieb Katie Morrison:
> I understand that the general orthogonal group that GAP computes is not the
> group of matrices that satisfy MM^T=I because the GO group they compute
> actually leaves a different bilinear form fixed than the dot product. But
> is there an easy way to find the group of matrices that satisfy MM^T=I and
> or to find the generalized version of this that satisfy MM^T=\lambda*I for
> some nonzero \lambda \in GF(q)? A brute force search becomes completely
> unwieldy for matrices larger than 3 by 3, so if I can find some easy way to
> map from the general orthogonal group that GAP uses or find an efficient
> algorithm for computing this other group, that would be great.
>
> Thanks,
> Katie Morrison
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