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gap> # test_forms5: hermitian forms
gap> f := GF(9);
GF(3^2)
gap> gram := [[0,0,0,Z(9)^2],[0,0,1,0],[0,1,0,0],[-Z(9)^2,0,0,0]]*Z(9)^0;
[ [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3^2)^2 ], [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ],
[ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ], [ Z(3^2)^6, 0*Z(3), 0*Z(3), 0*Z(3) ] ]
gap> form := HermitianFormByMatrix(gram,f);
< hermitian form >
gap> TypeOfForm(form);
1/2
gap> BaseChangeToCanonical(form);
[ [ Z(3)^0, 0*Z(3), 0*Z(3), Z(3^2)^3 ], [ Z(3^2), 0*Z(3), 0*Z(3), Z(3^2)^2 ],
[ 0*Z(3), Z(3^2), Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3^2)^2, Z(3^2)^3, 0*Z(3) ] ]
gap> iso := IsometricCanonicalForm(form);
< hermitian form >
gap> Display(form);
Hermitian form
Gram Matrix:
z = Z(9)
. . . z^2
. . 1 .
. 1 . .
z^6 . . .
Witt Index: 2
gap> Display(iso);
Hermitian form
Gram Matrix:
1 . . .
. 1 . .
. . 1 .
. . . 1
Witt Index: 2
gap> quit;
[ Dauer der Verarbeitung: 0.14 Sekunden
(vorverarbeitet)
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