Definition of canonical form unclear

[fingolfin]

Consider this example:

gap> gram:=DiagonalMat([2,1,1,1,1,1]*Z(5)^0);;
gap> form := BilinearFormByMatrix( gram, GF(5) );
< bilinear form >
gap> Display(form);
Bilinear form
Gram Matrix:
 2 . . . . .
 . 1 . . . .
 . . 1 . . .
 . . . 1 . .
 . . . . 1 .
 . . . . . 1
gap> Display(IsometricCanonicalForm(form));
Elliptic bilinear form
Gram Matrix:
 1 . . . . .
 . 2 . . . .
 . . . 3 . .
 . . 3 . . .
 . . . . . 3
 . . . . 3 .
Witt Index: 2

I don't understand how this form fits with the description of "canonical forms" in chapter 5 of the forms manual. Based on that, I would have expected this Gram matrix

 2 . . . . .
 . 1 . . . .
 . . . 1 . .
 . . 1 . . .
 . . . . . 1
 . . . . 1 .

Granted, these two forms are similar, but I thought we really get the canonical form?

What am I missing?

[fingolfin]

Case in point: the manual includes examples/include/basechangehom.include as an example. There we see this example:

gl:=GL(3,3);
go:=GO(3,3);
form := PreservedSesquilinearForms(go)[1]; 
gram := GramMatrix( form );  
b := BaseChangeToCanonical(form);;
hom := BaseChangeHomomorphism(b, GF(3));
newgo := Image(hom, go); 
gens := GeneratorsOfGroup(newgo);;
canonical := b * gram * TransposedMat(b);

In the pre-generated .include file and hence in the manual, it claims this is the output:

 1 . .
 . . 2
 . 2 .

But when actually running the code, we get

 2 . .
 . . 1
 . 1 .

Well, actually... running the code multiple times, I sometimes get the one and sometimes the other... so much for "canonical"...

[fingolfin]

@johnbamberg @jdebeule if you next look at forms, perhaps you could also briefly consider this issue...

[fingolfin]

So what I described here so far is only about a scalar multiple. We can't kill scalars in BaseChangeToCanonical, which only can work up to similarity.

Alas for IsometricCanonicalForm I would expect it to produce "the" canonical form?

But beyond that, after the (false) issue #65 I now believe that the description of canonical forms in section 5.1-3 is wrong: in the case where $U$ has dimension 2, the value of $\mu$ should not always be a non-square. Instead I think it should be a non-square for $n\equiv 1 \mod 4$ and a square (so e.g. $1$) when $n\equiv 3\mod 4$.

@johnbamberg @jdebeule can you confirm?