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<div class="ChapSects"><a href="chap4_mj.html#X8166C704848D128E">4 Constructing forms and basic functionality</a>
<div class="ContSect"> <a href="chap4_mj.html#X83494A76866B06A5">4.1 Important filters</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X7FA162E5874E8330">4.1-1 Categories for forms</a>

 <a href="chap4_mj.html#X7999E38082474342">4.1-2 Representation for forms</a>

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<div class="ContSect"> <a href="chap4_mj.html#X78D981A67DBFCD6D">4.2 Constructing forms using a matrix</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X7C9D7E517A73F02F">4.2-1 BilinearFormByMatrix</a>
 <a href="chap4_mj.html#X86B8694F782A4EE7">4.2-2 QuadraticFormByMatrix</a>
 <a href="chap4_mj.html#X7C027FF77AFED321">4.2-3 HermitianFormByMatrix</a>
</div></div>
<div class="ContSect"> <a href="chap4_mj.html#X78476EDF7B9498D7">4.3 Constructing forms using a polynomial</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X81D571077C4BCEFF">4.3-1 BilinearFormByPolynomial</a>
 <a href="chap4_mj.html#X86ADE1D986CC90CB">4.3-2 QuadraticFormByPolynomial</a>
 <a href="chap4_mj.html#X7E21CFFA84180D0D">4.3-3 HermitianFormByPolynomial</a>
</div></div>
<div class="ContSect"> <a href="chap4_mj.html#X843B68558283CE5F">4.4 Switching between bilinear and quadratic forms</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X7F13EAC17BDE228D">4.4-1 QuadraticFormByBilinearForm</a>
 <a href="chap4_mj.html#X812963777BBF97E3">4.4-2 BilinearFormByQuadraticForm</a>
 <a href="chap4_mj.html#X7BF7FBCA7FF91052">4.4-3 AssociatedBilinearForm</a>
</div></div>
<div class="ContSect"> <a href="chap4_mj.html#X8110213A7B303D1C">4.5 Evaluating forms</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X808AB7B9840ABC27">4.5-1 EvaluateForm</a>
</div></div>
<div class="ContSect"> <a href="chap4_mj.html#X7A0825A987C88978">4.6 Orthogonality, totally isotropic subspaces, and totally singular subspaces</a>

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 <a href="chap4_mj.html#X7D699C077F66F3E6">4.6-1 OrthogonalSubspaceMat</a>
 <a href="chap4_mj.html#X8394CCAD798053C6">4.6-2 IsIsotropicVector</a>
 <a href="chap4_mj.html#X855E539185D7D3C7">4.6-3 IsSingularVector</a>
 <a href="chap4_mj.html#X8141325085AAC0CD">4.6-4 IsTotallyIsotropicSubspace</a>
 <a href="chap4_mj.html#X834FD9117F1DA8D0">4.6-5 IsTotallySingularSubspace</a>
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<div class="ContSect"> <a href="chap4_mj.html#X813A02878352E9E5">4.7 Attributes and properties of forms</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X80254BFD7E4B8F06">4.7-1 IsReflexiveForm</a>
 <a href="chap4_mj.html#X7D5AB7E484CFBF63">4.7-2 IsAlternatingForm</a>
 <a href="chap4_mj.html#X85585B2C80413490">4.7-3 IsSymmetricForm</a>
 <a href="chap4_mj.html#X87E9C9A1781AB058">4.7-4 IsOrthogonalForm</a>
 <a href="chap4_mj.html#X861AF6EE82F4DA39">4.7-5 IsPseudoForm</a>
 <a href="chap4_mj.html#X86F552AE7ACC12C7">4.7-6 IsSymplecticForm</a>
 <a href="chap4_mj.html#X7C60B9587D130DBB">4.7-7 IsDegenerateForm</a>
 <a href="chap4_mj.html#X7A0E882F801624DA">4.7-8 IsSingularForm</a>
 <a href="chap4_mj.html#X7BCBA564829D9E89">4.7-9 BaseField</a>
 <a href="chap4_mj.html#X847AFB4C81A90B3F">4.7-10 GramMatrix</a>
 <a href="chap4_mj.html#X7855C3C07AAA1A68">4.7-11 RadicalOfForm</a>
 <a href="chap4_mj.html#X82E7367F817C6BD0">4.7-12 PolynomialOfForm</a>
 <a href="chap4_mj.html#X87B652A28534E0D2">4.7-13 DiscriminantOfForm</a>
</div></div>
<div class="ContSect"> <a href="chap4_mj.html#X8400E22D7D51FCCE">4.8 Recognition of forms preserved by a classical group</a>

<div class="ContSSBlock">
 <a href="chap4_mj.html#X784481E57E207B3D">4.8-1 PreservedForms</a>
 <a href="chap4_mj.html#X84056A357E5447AF">4.8-2 PreservedSesquilinearForms</a>
 <a href="chap4_mj.html#X7D6F72B682E405E1">4.8-3 PreservedQuadraticForms</a>
</div></div>
<div class="ContSect"> <a href="chap4_mj.html#X836A21687A685839">4.9 The trivial form and some of its properties</a>

</div>
</div>

<h3>4 Constructing forms and basic functionality</h3>

In this chapter, all operations to construct sesquilinear and quadratic forms are listed, along with their basic attributes and properties.

<a id="X83494A76866B06A5" name="X83494A76866B06A5"></a>

<h4>4.1 Important filters</h4>

<a id="X7FA162E5874E8330" name="X7FA162E5874E8330"></a>

<h5>4.1-1 Categories for forms</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBilinearForm</code></td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHermitianForm</code></td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSesquilinearForm</code></td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsQuadraticForm</code></td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsForm</code></td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTrivialForm</code></td><td class="tdright">( category )</td></tr></table></div>
The categories <code class="code">IsBilinearForm</code> and <code class="code">IsHermitianForm</code> are categories for bilinear and hermitian forms, respectively. They are disjoint and are both contained in the category <code class="code">IsSesquilinearForm</code>.

Quadratic forms are contained in the category <code class="code">IsQuadraticForm</code>. The categories <code class="code">IsSesquilinearForm</code> and <code class="code">IsQuadraticForm</code> are disjoint and are both contained in the category <code class="code">IsForm</code>.

The user is allowed to construct the trivial form (mapping all vectors to the zero element of the field). The trivial form is an object in the category <code class="code">IsTrivialForm</code>. This category is contained in <code class="code">IsForm</code> and disjoint from <code class="code">IsSesquilinearForm</code> and <code class="code">IsQuadraticForm</code>.

<a id="X7999E38082474342" name="X7999E38082474342"></a>

<h5>4.1-2 Representation for forms</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFormRep</code></td><td class="tdright">( representation )</td></tr></table></div>
Every form is represented by a matrix, the base field and a string describing the ``type'' of the form.

<a id="X78D981A67DBFCD6D" name="X78D981A67DBFCD6D"></a>

<h4>4.2 Constructing forms using a matrix</h4>

<a id="X7C9D7E517A73F02F" name="X7C9D7E517A73F02F"></a>

<h5>4.2-1 BilinearFormByMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BilinearFormByMatrix</code>( <var class="Arg">matrix</var>[, <var class="Arg">field</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a bilinear form

The argument <var class="Arg">matrix</var> must be a symmetric, or skew-symmetric, square matrix over the finite field <var class="Arg">field</var>. The argument <var class="Arg">field</var> is an optional argument, and if it is not given, then we assume that the defining field of the bilinear form is the smallest field containing the entries of matrix. Below we give an example where the defining field can make a difference in some applications. As it is only possible to construct reflexive bilinear forms, it is checked whether the matrix <var class="Arg">matrix</var> is symmetric or skew symmetric. If matrix <var class="Arg">matrix</var> is not symmetric nor skew symmetric, then an error message is returned. The output is a bilinear form (i.e., an object in <code class="code">IsBilinearForm</code>) with Gram matrix <var class="Arg">matrix</var> and defining field <var class="Arg">field</var>. (See <a href="chap3_mj.html#X874CD5E0802FEB50">3.1</a> for more on bilinear forms).

<div class="example"><pre>
gap> mat := IdentityMat(4, GF(9));
[ [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ],
 [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ] ]
gap> form := BilinearFormByMatrix(mat,GF(9));
< bilinear form >
gap> Display(form);
Bilinear form
Gram Matrix:
1 . . .
. 1 . .
. . 1 .
. . . 1
gap> mat := [[0*Z(2),Z(16)^12,0*Z(2),Z(4)^2,Z(16)^13],
> [Z(16)^12,0*Z(2),0*Z(2),Z(16)^11,Z(16)],
> [0*Z(2),0*Z(2),0*Z(2),Z(4)^2,Z(16)^3],
> [Z(4)^2,Z(16)^11,Z(4)^2,0*Z(2),Z(16)^3],
> [Z(16)^13,Z(16),Z(16)^3,Z(16)^3,0*Z(2) ]];
[ [ 0*Z(2), Z(2^4)^12, 0*Z(2), Z(2^2)^2, Z(2^4)^13 ],
 [ Z(2^4)^12, 0*Z(2), 0*Z(2), Z(2^4)^11, Z(2^4) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2^2)^2, Z(2^4)^3 ],
 [ Z(2^2)^2, Z(2^4)^11, Z(2^2)^2, 0*Z(2), Z(2^4)^3 ],
 [ Z(2^4)^13, Z(2^4), Z(2^4)^3, Z(2^4)^3, 0*Z(2) ] ]
gap> form := BilinearFormByMatrix(mat,GF(16));
< bilinear form >
gap> Display(form);
Bilinear form
Gram Matrix:
z = Z(16)
 . z^12 . z^10 z^13
z^12 . . z^11 z^1
 . . . z^10 z^3
z^10 z^11 z^10 . z^3
z^13 z^1 z^3 z^3 .
gap> mat := [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]*Z(7)^0;
[ [ Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7) ], [ 0*Z(7), Z(7)^0, 0*Z(7), 0*Z(7) ],
 [ 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^0 ], [ 0*Z(7), 0*Z(7), Z(7)^0, 0*Z(7) ] ]
gap> form := BilinearFormByMatrix(mat);
< bilinear form >
gap> WittIndex(form);
1
gap> form := BilinearFormByMatrix(mat,GF(49));
< bilinear form >
gap> WittIndex(form);
2

</pre></div>

<a id="X86B8694F782A4EE7" name="X86B8694F782A4EE7"></a>

<h5>4.2-2 QuadraticFormByMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuadraticFormByMatrix</code>( <var class="Arg">matrix</var>[, <var class="Arg">field</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a quadratic form

The argument <var class="Arg">matrix</var> must be a square matrix over the finite field <var class="Arg">field</var>. The argument <var class="Arg">field</var> is an optional argument, and if it is not given, then we assume that the defining field of the bilinear form is the smallest field containing the entries of matrix. Below we give an example where the defining field can make a difference in some applications. Any square matrix determines a quadratic form, but the Gram matrix is recomputed so that it is an upper triangle matrix. The output is a quadratic form (i.e., an object in <code class="code">IsQuadraticForm</code>) with defining field <var class="Arg">field</var>. (See <a href="chap3_mj.html#X864CAF8881067D8A">3.2</a> for more on bilinear forms).

<div class="example"><pre>
gap> mat := [[1,0,0,0],[0,3,0,0],[0,0,0,6],[0,0,6,0]]*Z(7)^0;
[ [ Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7) ], [ 0*Z(7), Z(7), 0*Z(7), 0*Z(7) ],
 [ 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^3 ], [ 0*Z(7), 0*Z(7), Z(7)^3, 0*Z(7) ] ]
gap> form := QuadraticFormByMatrix(mat,GF(7));
< quadratic form >
gap> Display(form);
Quadratic form
Gram Matrix:
1 . . .
. 3 . .
. . . 5
. . . .
gap> gf := GF(2^2);
GF(2^2)
gap> mat := InvariantQuadraticForm( SO(-1, 4, 4) )!.matrix;
[ [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), Z(2^2)^2, Z(2)^0 ], [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2^2)^2 ] ]
gap> form := QuadraticFormByMatrix( mat, gf );
< quadratic form >
gap> Display(form);
Quadratic form
Gram Matrix:
z = Z(4)
 . 1 . .
 . . . .
 . . z^2 1
 . . . z^2

</pre></div>

The following example shows how using the argument <var class="Arg">field</var> has influence on the properties of the constructed form.

<div class="example"><pre>
gap> mat := 
> [[Z(2)^0,Z(2)^0,0*Z(2),0*Z(2)],[0*Z(2),Z(2)^0,0*Z(2),0*Z(2)], 
> [0*Z(2),0*Z(2),0*Z(2),Z(2)^0],[0*Z(2),0*Z(2),0*Z(2),0*Z(2)]];
[ [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ]
gap> form := QuadraticFormByMatrix(mat);
< quadratic form >
gap> WittIndex(form);
1
gap> form := QuadraticFormByMatrix(mat,GF(4));
< quadratic form >
gap> WittIndex(form);
2

</pre></div>

<a id="X7C027FF77AFED321" name="X7C027FF77AFED321"></a>

<h5>4.2-3 HermitianFormByMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HermitianFormByMatrix</code>( <var class="Arg">matrix</var>, <var class="Arg">field</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a hermitian sesquilinear form

The argument <var class="Arg">matrix</var> must be a hermitian square matrix over the finite field <var class="Arg">field</var>, and <var class="Arg">field</var> has square order. The field must be specified, since we can only determine the smallest field containing the entries of <var class="Arg">matrix</var>. As it is only possible to construct reflexive sesquilinear forms, it is checked whether the matrix is a hermitian matrix, and if not, an error message is returned. The output is a hermitian sesquilinear form (i.e., an object in <code class="code">IsHermitianForm</code>) with Gram matrix <var class="Arg">matrix</var> and defining field <var class="Arg">field</var>. (See <a href="chap3_mj.html#X874CD5E0802FEB50">3.1</a> for more on hermitian forms).

<div class="example"><pre>
gap> gf := GF(3^2);
GF(3^2)
gap> mat := IdentityMat(4, gf);
[ [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ],
 [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ] ]
gap> form := HermitianFormByMatrix( mat, gf );
< hermitian form >
gap> Display(form);
Hermitian form
Gram Matrix:
1 . . .
. 1 . .
. . 1 .
. . . 1
gap> mat := [[Z(11)^0,0*Z(11),0*Z(11)],[0*Z(11),0*Z(11),Z(11)],
> [0*Z(11),Z(11),0*Z(11)]];
[ [ Z(11)^0, 0*Z(11), 0*Z(11) ], [ 0*Z(11), 0*Z(11), Z(11) ],
 [ 0*Z(11), Z(11), 0*Z(11) ] ]
gap> form := HermitianFormByMatrix(mat,GF(121));
< hermitian form >
gap> Display(form);
Hermitian form
Gram Matrix:
 1 . .
 . . 2
 . 2 .

</pre></div>

<a id="X78476EDF7B9498D7" name="X78476EDF7B9498D7"></a>

<h4>4.3 Constructing forms using a polynomial</h4>

Suppose that \(f\) is a sesquilinear form on an \(n\)-dimensional vectorspace. Consider a vector \(x\) with coordinates \(x_1,\ldots,x_{n}\) with \(x_i\) indeterminates over the field. Then \(f(x,x)\) is a polynomial in \(n\) indeterminates. When \(f\) is alternating, \(f(x,x)\) is identically zero, but in all other cases, \(f(x,x)\) determines \(f\) completely.

Conversely, suppose that \(Q\) is a quadratic form on an \(n\)-dimensional vectorspace. Consider a vector \(x\) with coordinates \(x_1,\ldots,x_{n}\) with \(x_i\) indeterminates over the field. Then \(Q(x)\) is a polynomial in \(n\) indeterminates, and \(Q(x)\) determines \(Q\) completely.

Forms provides functionality to construct bilinear, hermitian and quadratic forms using an appropriate polynomial.

<a id="X81D571077C4BCEFF" name="X81D571077C4BCEFF"></a>

<h5>4.3-1 BilinearFormByPolynomial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BilinearFormByPolynomial</code>( <var class="Arg">poly</var>, <var class="Arg">r</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a bilinear form

The argument <var class="Arg">poly</var> must be a polynomial in the polynomial ring <var class="Arg">r</var>. The (optional) last argument is the dimension for the underlying vector space of the resulting form, which by default is the number of indeterminates specified by <var class="Arg">poly</var>. It is checked whether the polynomial is a homogeneous polynomial of degree two over the given field, and if not, an error message is returned. It is not possible to construct a nontrivial bilinear form from a polynomial in even characteristic. The output is a bilinear (orthogonal) form in the category <code class="code">IsBilinearForm</code>. (See <a href="chap3_mj.html#X874CD5E0802FEB50">3.1</a> for more on bilinear forms).

<div class="example"><pre>
gap> r := PolynomialRing( GF(11), 4);
GF(11)[x_1,x_2,x_3,x_4]
gap> vars := IndeterminatesOfPolynomialRing( r );
[ x_1, x_2, x_3, x_4 ]
gap> pol := vars[1]*vars[2]+vars[3]*vars[4];
x_1*x_2+x_3*x_4
gap> form := BilinearFormByPolynomial(pol, r, 4);
< bilinear form >
gap> Display(form);
Bilinear form
Gram Matrix:
 . 6 . .
 6 . . .
 . . . 6
 . . 6 .
Polynomial: x_1*x_2+x_3*x_4
gap> r := PolynomialRing(GF(4),2);
GF(2^2)[x_1,x_2]
gap> pol := r.1*r.2;
x_1*x_2
gap> form := BilinearFormByPolynomial(pol,r);
Error, No orthogonal form can be associated with a quadratic polynomial in even cha\
ra
cteristic at ./pkg/forms/lib/forms.gi:371 called from
BilinearFormByPolynomial( pol, pring, n ) at ./pkg/forms/lib/forms.gi:391 called fr\
om
<function "BilinearFormByPolynomial no dimension">( <arguments> )
called from read-eval loop at *stdin*:9
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;

</pre></div>

<a id="X86ADE1D986CC90CB" name="X86ADE1D986CC90CB"></a>

<h5>4.3-2 QuadraticFormByPolynomial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuadraticFormByPolynomial</code>( <var class="Arg">poly</var>, <var class="Arg">r</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a quadratic form

The argument <var class="Arg">poly</var> must be a polynomial in the polynomial ring <var class="Arg">r</var>. The (optional) last argument is the dimension for the underlying vector space of the resulting form, which by default is the number of indeterminates specified by <var class="Arg">poly</var>. It is checked whether the polynomial is a homogeneous polynomial of degree two over the given field, and if not, an error message is returned. The output is a quadratic form in the category <code class="code">IsQuadraticForm</code>. (See <a href="chap3_mj.html#X864CAF8881067D8A">3.2</a> for more on quadratic forms).

<div class="example"><pre>
gap> r := PolynomialRing( GF(8), 3);
GF(2^3)[x_1,x_2,x_3]
gap> poly := r.1^2 + r.2^2 + r.3^2;
x_1^2+x_2^2+x_3^2
gap> form := QuadraticFormByPolynomial(poly, r);
< quadratic form >
gap> RadicalOfForm(form);
<vector space over GF(2^3), with 63 generators>
gap> r := PolynomialRing(GF(9),4);
GF(3^2)[x_1,x_2,x_3,x_4]
gap> poly := Z(3)^2*r.1^2+r.2^2+r.3*r.4;
x_1^2+x_2^2+x_3*x_4
gap> qform := QuadraticFormByPolynomial(poly,r);
< quadratic form >
gap> Display(qform);
Quadratic form
Gram Matrix:
1 . . .
. 1 . .
. . . 1
. . . .
Polynomial: x_1^2+x_2^2+x_3*x_4

</pre></div>

<a id="X7E21CFFA84180D0D" name="X7E21CFFA84180D0D"></a>

<h5>4.3-3 HermitianFormByPolynomial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HermitianFormByPolynomial</code>( <var class="Arg">poly</var>, <var class="Arg">r</var>[, <var class="Arg">n</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: an hermitian form

The argument <var class="Arg">poly</var> must be a polynomial in the polynomial ring <var class="Arg">r</var> defined over a finite field of square order \(q^2\) The (optional) last argument is the dimension for the underlying vector space of the resulting form, which by default is the number of indeterminates specified by <var class="Arg">poly</var>. It is checked whether the polynomial is a homogeneous polynomial of degree \(q+1\), and if not, an error message is returned. The output is a hermitian form in the category <code class="code">IsHermitianForm</code>. (See <a href="chap3_mj.html#X874CD5E0802FEB50">3.1</a> for more on hermitian forms).

<div class="example"><pre>
gap> r := PolynomialRing( GF(9), 4);
GF(3^2)[x_1,x_2,x_3,x_4]
gap> vars := IndeterminatesOfPolynomialRing( r );
[ x_1, x_2, x_3, x_4 ]
gap> poly := vars[1]*vars[2]^3+vars[1]^3*vars[2]+
> vars[3]*vars[4]^3+vars[3]^3*vars[4];
x_1^3*x_2+x_1*x_2^3+x_3^3*x_4+x_3*x_4^3
gap> form := HermitianFormByPolynomial(poly,r);
< hermitian form >
gap> Display(form);
Hermitian form
Gram Matrix:
. 1 . .
1 . . .
. . . 1
. . 1 .
Polynomial: x_1^3*x_2+x_1*x_2^3+x_3^3*x_4+x_3*x_4^3

</pre></div>

<a id="X843B68558283CE5F" name="X843B68558283CE5F"></a>

<h4>4.4 Switching between bilinear and quadratic forms</h4>

When the characteristic of the field is odd, a homogeneous quadratic polynomial determines a bilinear form, and a quadratic form. In some situations, when a quadratic form \(Q\) is given, it is useful to consider the bilinear form \(f\) such that \(f(v,v)=Q(v)\), i.e., the bilinear form which is determined by exactly the same polynomial determining the quadratic form \(Q\). Forms provides functionality to construct a bilinear form \(f\) from a given quadratic form \(Q\) such that \(f(v,v)=Q(v)\). Conversely, we can extract a quadratic form from a given bilinear form.

<a id="X7F13EAC17BDE228D" name="X7F13EAC17BDE228D"></a>

<h5>4.4-1 QuadraticFormByBilinearForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuadraticFormByBilinearForm</code>( <var class="Arg">form</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a quadratic form

The argument \(form\) is an orthogonal bilinear form (and thus it belongs to <code class="code">IsBilinearForm</code>), otherwise a ``No method found'' error is returned. The output is the quadratic form \(Q\) (an object in <code class="code">IsQuadraticForm</code>), such that \(Q(v) = form(v,v)\) for all vectors \(v\) in a vector space equipped with \(form\). An error is returned when the characteristic of the field is even, or when \(form\) is not orthogonal.

<div class="example"><pre>
gap> mat := [ [ Z(3^2)^7, Z(3)^0, Z(3^2)^2, 0*Z(3), Z(3^2)^5 ], 
> [ Z(3)^0, Z(3^2)^7, Z(3^2)^6, Z(3^2)^5, Z(3^2)^2 ], 
> [ Z(3^2)^2, Z(3^2)^6, Z(3^2)^7, Z(3^2)^2, Z(3^2)^2 ], 
> [ 0*Z(3), Z(3^2)^5, Z(3^2)^2, Z(3^2)^6, Z(3^2)^7 ], 
> [ Z(3^2)^5, Z(3^2)^2, Z(3^2)^2, Z(3^2)^7, Z(3) ] ];
[ [ Z(3^2)^7, Z(3)^0, Z(3^2)^2, 0*Z(3), Z(3^2)^5 ],
 [ Z(3)^0, Z(3^2)^7, Z(3^2)^6, Z(3^2)^5, Z(3^2)^2 ],
 [ Z(3^2)^2, Z(3^2)^6, Z(3^2)^7, Z(3^2)^2, Z(3^2)^2 ],
 [ 0*Z(3), Z(3^2)^5, Z(3^2)^2, Z(3^2)^6, Z(3^2)^7 ],
 [ Z(3^2)^5, Z(3^2)^2, Z(3^2)^2, Z(3^2)^7, Z(3) ] ]
gap> form := BilinearFormByMatrix(mat,GF(9));
< bilinear form >
gap> Q := QuadraticFormByBilinearForm(form);
< quadratic form >
gap> Display(form);
Bilinear form
Gram Matrix:
z = Z(9)
z^7 1 z^2 . z^5
 1 z^7 z^6 z^5 z^2
z^2 z^6 z^7 z^2 z^2
 . z^5 z^2 z^6 z^7
z^5 z^2 z^2 z^7 2
gap> Display(Q);
Quadratic form
Gram Matrix:
z = Z(9)
z^7 2 z^6 . z^1
 . z^7 z^2 z^1 z^6
 . . z^7 z^6 z^6
 . . . z^6 z^3
 . . . . 2
gap> Set(List(GF(9)^5),x->[x,x]^form=x^Q);
[ true ]
gap> PolynomialOfForm(form);
Z(3^2)^7*x_1^2-x_1*x_2+Z(3^2)^6*x_1*x_3+Z(3^2)*x_1*x_5+Z(3^2)^7*x_2^2+Z(3^2)^2
*x_2*x_3+Z(3^2)*x_2*x_4+Z(3^2)^6*x_2*x_5+Z(3^2)^7*x_3^2+Z(3^2)^6*x_3*x_4+Z(3^2
)^6*x_3*x_5+Z(3^2)^6*x_4^2+Z(3^2)^3*x_4*x_5-x_5^2
gap> PolynomialOfForm(Q);
Z(3^2)^7*x_1^2-x_1*x_2+Z(3^2)^6*x_1*x_3+Z(3^2)*x_1*x_5+Z(3^2)^7*x_2^2+Z(3^2)^2
*x_2*x_3+Z(3^2)*x_2*x_4+Z(3^2)^6*x_2*x_5+Z(3^2)^7*x_3^2+Z(3^2)^6*x_3*x_4+Z(3^2
)^6*x_3*x_5+Z(3^2)^6*x_4^2+Z(3^2)^3*x_4*x_5-x_5^2

</pre></div>

Note that the given bilinear form <var class="Arg">form</var> is not the associated bilinear form of the constructed quadratic form \(Q\), according to the definition in Section <a href="chap3_mj.html#X864CAF8881067D8A">3.2</a>. We can construct the associated bilinear forms by using <code class="func">AssociatedBilinearForm</code> (<a href="chap4_mj.html#X7BF7FBCA7FF91052">4.4-3</a>). (See <a href="chap3_mj.html#X864CAF8881067D8A">3.2</a> for more on quadratic forms).

<a id="X812963777BBF97E3" name="X812963777BBF97E3"></a>

<h5>4.4-2 BilinearFormByQuadraticForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BilinearFormByQuadraticForm</code>( <var class="Arg">Q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a bilinear form

The argument \(Q\) must be a quadratic form (and thus it belongs to <code class="code">IsQuadraticForm</code>). The output is the orthogonal bilinear form \(f\) (an object in <code class="code">IsBilinearForm</code>), such that \(f(v,v) = Q(v)\) for all vectors \(v\) in a vector space equipped with \(Q\). An error is returned when the characteristic of the field is even.

<div class="example"><pre>
gap> r := PolynomialRing(GF(9),4);
GF(3^2)[x_1,x_2,x_3,x_4]
gap> poly := -r.1*r.2+Z(3^2)*r.3^2+r.4^2;
-x_1*x_2+Z(3^2)*x_3^2+x_4^2
gap> qform := QuadraticFormByPolynomial(poly,r);
< quadratic form >
gap> Display( qform );
Quadratic form
Gram Matrix:
z = Z(9)
 . 2 . .
 . . . .
 . . z^1 .
 . . . 1
Polynomial: -x_1*x_2+Z(3^2)*x_3^2+x_4^2
gap> form := BilinearFormByQuadraticForm( qform );
< bilinear form >
gap> Display(form);
Bilinear form
Gram Matrix:
z = Z(9)
 . 1 . .
 1 . . .
 . . z^1 .
 . . . 1
gap> Set(GF(9)^4, x -> [x,x]^form = x^qform);
[ true ]

</pre></div>

Note that the constructed bilinear form \(f\) is not the associated bilinear form of the given quadratic form \(Q\), according to the definition in Section <a href="chap3_mj.html#X864CAF8881067D8A">3.2</a>. We can construct the associated bilinear forms by using <code class="func">AssociatedBilinearForm</code> (<a href="chap4_mj.html#X7BF7FBCA7FF91052">4.4-3</a>). (See <a href="chap3_mj.html#X864CAF8881067D8A">3.2</a> for more on quadratic forms).

<a id="X7BF7FBCA7FF91052" name="X7BF7FBCA7FF91052"></a>

<h5>4.4-3 AssociatedBilinearForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AssociatedBilinearForm</code>( <var class="Arg">Q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a bilinear form

The argument \(Q\) must be a quadratic form (and thus it belongs to <code class="code">IsQuadraticForm</code>). The output is the associated bilinear form \(f\) (an object in <code class="code">IsBilinearForm</code>), as defined in Section <a href="chap3_mj.html#X864CAF8881067D8A">3.2</a>, i.e. the bilinear form \(f\) such that \(f(v,w) = Q(v+w)-Q(v)-Q(w)\) for all vectors \(v,w\) in a vector space equipped with \(Q\). (See <a href="chap3_mj.html#X864CAF8881067D8A">3.2</a> for more on quadratic forms).

<div class="example"><pre>
gap> r:= PolynomialRing(GF(121),6);
GF(11^2)[x_1,x_2,x_3,x_4,x_5,x_6]
gap> poly := r.1*r.5-r.2*r.6+r.3*r.4;
x_1*x_5-x_2*x_6+x_3*x_4
gap> form := QuadraticFormByPolynomial(poly,r);
< quadratic form >
gap> aform := AssociatedBilinearForm(form);
< bilinear form >
gap> Display(aform);
Bilinear form
Gram Matrix:
 . . . . 1 .
 . . . . . 10
 . . . 1 . .
 . . 1 . . .
 1 . . . . .
 . 10 . . . .

</pre></div>

<a id="X8110213A7B303D1C" name="X8110213A7B303D1C"></a>

<h4>4.5 Evaluating forms</h4>

<a id="X808AB7B9840ABC27" name="X808AB7B9840ABC27"></a>

<h5>4.5-1 EvaluateForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EvaluateForm</code>( <var class="Arg">f</var>, <var class="Arg">u</var>[, <var class="Arg">v</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a finite field element

The argument <var class="Arg">f</var> is either a sesquilinear or quadratic form defined over a finite field \(\mathrm{GF}(q)\). The other argument is a pair of vectors or matrices, or a single vector or matrix. In case that <var class="Arg">u</var> (and <var class="Arg">v</var> when using three arguments) is a matrix, its rows represent a basis for the subspace (or subspaces) where \(f\) is evaluated in. This operation evaluates the form on the given vector or pair of vectors and returns an element in \(GF(q)\). There is also an overloading of the operation <var class="Arg">\^</var> where <code class="file">(u,v)^f</code> represents \(f(u,v)\) in the case that <var class="Arg">f</var> is sesquilinear, and <code class="file">u^f</code> stands for \(f(u)\) in the quadratic case. So for convenience, the user may use this compressed version of this operation, which we show in the following example:

<div class="example"><pre>
gap> mat := [[Z(8),0,0,0],[0,0,Z(8)^4,0],[0,0,0,1],[0,0,0,0]]*Z(8)^0;;
gap> form := QuadraticFormByMatrix(mat,GF(8));
< quadratic form >
gap> u := [ Z(2^3)^4, Z(2^3)^4, Z(2)^0, Z(2^3)^3 ];
[ Z(2^3)^4, Z(2^3)^4, Z(2)^0, Z(2^3)^3 ]
gap> EvaluateForm( form, u );
Z(2^3)^6
gap> u^form;
Z(2^3)^6
gap> gram := [[0,0,0,0,0,2],[0,0,0,0,2,0],[0,0,0,1,0,0],
> [0,0,1,0,0,0],[0,2,0,0,0,0],[2,0,0,0,0,0]]*Z(3)^0;;
gap> form := BilinearFormByMatrix(gram,GF(3));
< bilinear form >
gap> u := [ [ Z(3)^0, 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0 ], 
> [ 0*Z(3), 0*Z(3), Z(3)^0, Z(3)^0, Z(3), 0*Z(3) ] ];;
gap> v := [ [ Z(3)^0, 0*Z(3), Z(3)^0, Z(3), 0*Z(3), Z(3) ], 
> [ 0*Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3), Z(3) ] ];;
gap> EvaluateForm( form, u, v);
[ [ Z(3)^0, Z(3)^0 ], [ 0*Z(3), 0*Z(3) ] ]
gap> [u,v]^form;
[ [ Z(3)^0, Z(3)^0 ], [ 0*Z(3), 0*Z(3) ] ]

</pre></div>

<a id="X7A0825A987C88978" name="X7A0825A987C88978"></a>

<h4>4.6 Orthogonality, totally isotropic subspaces, and totally singular subspaces</h4>

<a id="X7D699C077F66F3E6" name="X7D699C077F66F3E6"></a>

<h5>4.6-1 OrthogonalSubspaceMat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrthogonalSubspaceMat</code>( <var class="Arg">form</var>, <var class="Arg">v</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrthogonalSubspaceMat</code>( <var class="Arg">form</var>, <var class="Arg">mat</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a base of the subspace orthogonal to the given vector or subspace with relation to the given form

The argument <var class="Arg">form</var> is a sesquilinear or quadratic form. For a given vector <var class="Arg">v</var>, this operation returns a base of the subspace orthogonal to <var class="Arg">v</var> with relation to the sesquilinear <var class="Arg">form</var> or with relation to the associated bilinear form of the quadratic form <var class="Arg">form</var>. For a given matrix <var class="Arg">mat</var>, this operation returns a base of the subspace orthogonal to the subspace spanned by the rows of <var class="Arg">mat</var> with relation to the sesquilinear <var class="Arg">form</var> or with relation to the associated bilinear form of the quadratic form <var class="Arg">form</var>

<div class="example"><pre>
gap> mat := [[0,0,0,-2],[0,0,-3,0],[0,3,0,0],[2,0,0,0]]*Z(7)^0;
[ [ 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^5 ], [ 0*Z(7), 0*Z(7), Z(7)^4, 0*Z(7) ],
 [ 0*Z(7), Z(7), 0*Z(7), 0*Z(7) ], [ Z(7)^2, 0*Z(7), 0*Z(7), 0*Z(7) ] ]
gap> form := BilinearFormByMatrix(mat);
< bilinear form >
gap> v := [0*Z(7),Z(7)^0,Z(7)^3,Z(7)^5];
[ 0*Z(7), Z(7)^0, Z(7)^3, Z(7)^5 ]
gap> vperp := OrthogonalSubspaceMat(form,v);
[ [ Z(7)^0, Z(7)^0, 0*Z(7), 0*Z(7) ], [ Z(7)^0, 0*Z(7), Z(7)^0, 0*Z(7) ],
 [ 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^0 ] ]
gap> List(vperp,x->[x,v]^form);
[ 0*Z(7), 0*Z(7), 0*Z(7) ]
gap> sub := [[1,1,0,0],[0,0,1,2]]*Z(7)^0;
[ [ Z(7)^0, Z(7)^0, 0*Z(7), 0*Z(7) ], [ 0*Z(7), 0*Z(7), Z(7)^0, Z(7)^2 ] ]
gap> subperp := OrthogonalSubspaceMat(form,sub);
[ [ Z(7)^0, Z(7)^0, 0*Z(7), 0*Z(7) ], [ 0*Z(7), 0*Z(7), Z(7)^4, Z(7)^0 ] ]
gap> List(subperp,x->List(sub,y->[x,y]^form));
[ [ 0*Z(7), 0*Z(7) ], [ 0*Z(7), 0*Z(7) ] ]
gap> mat := [[1,0,0],[0,0,1],[0,0,0]]*Z(2)^0;
[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2) ] ]
gap> form := QuadraticFormByMatrix(mat);
< quadratic form >
gap> v := [Z(2)^0,Z(2)^0,0*Z(2)];
[ Z(2)^0, Z(2)^0, 0*Z(2) ]
gap> vperp := OrthogonalSubspaceMat(form,v);
[ <an immutable GF2 vector of length 3>, <an immutable GF2 vector of length
 3> ]
gap> bil_form := AssociatedBilinearForm(form);
< bilinear form >
gap> List(vperp,x->[x,v]^bil_form);
[ 0*Z(2), 0*Z(2) ]
gap> sub := [[1,0,1],[1,0,0]]*Z(2)^0;
[ [ Z(2)^0, 0*Z(2), Z(2)^0 ], [ Z(2)^0, 0*Z(2), 0*Z(2) ] ]
gap> subperp := OrthogonalSubspaceMat(form,sub);
[ <an immutable GF2 vector of length 3>, <an immutable GF2 vector of length
 3> ]
gap> List(subperp,x->List(sub,y->[x,y]^bil_form));
[ [ 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2) ] ]

</pre></div>

<a id="X8394CCAD798053C6" name="X8394CCAD798053C6"></a>

<h5>4.6-2 IsIsotropicVector</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsIsotropicVector</code>( <var class="Arg">form</var>, <var class="Arg">v</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: true or false

The operation return <var class="Arg">true</var> if and only if <var class="Arg">v</var> is isotropic with relation to the sesquilinear or quadratic form <var class="Arg">form</var>.

<div class="example"><pre>
gap> mat := [[1,0,0,0],[0,-1,0,0],[0,0,0,1],[0,0,1,0]]*Z(41)^0;
[ [ Z(41)^0, 0*Z(41), 0*Z(41), 0*Z(41) ],
 [ 0*Z(41), Z(41)^20, 0*Z(41), 0*Z(41) ],
 [ 0*Z(41), 0*Z(41), 0*Z(41), Z(41)^0 ],
 [ 0*Z(41), 0*Z(41), Z(41)^0, 0*Z(41) ] ]
gap> form := BilinearFormByMatrix(mat);
< bilinear form >
gap> v := [1,1,0,0]*Z(41)^0;
[ Z(41)^0, Z(41)^0, 0*Z(41), 0*Z(41) ]
gap> IsIsotropicVector(form,v);
true
gap> mat := [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,0,0],[0,0,1,0,0],[0,0,0,0,0]]*Z(8)^0;
[ [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ]
gap> form := QuadraticFormByMatrix(mat);
< quadratic form >
gap> v1 := [1,0,0,0,0]*Z(8)^0;
[ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ]
gap> v2 := [0,1,0,0,0]*Z(8)^0;
[ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ]
gap> IsIsotropicVector(form,v1);
true
gap> IsIsotropicVector(form,v2);
true

</pre></div>

<a id="X855E539185D7D3C7" name="X855E539185D7D3C7"></a>

<h5>4.6-3 IsSingularVector</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSingularVector</code>( <var class="Arg">form</var>, <var class="Arg">v</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: true or false

The operation return <var class="Arg">true</var> if and only if <var class="Arg">v</var> is singular with relation to the quadratic form <var class="Arg">form</var>. Note that only when the characteristic of the field is odd, the singular vectors with relation to a quadratic form are the isotropic vectors with relation to its associated form.

<div class="example"><pre>
gap> mat := [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,0,0],[0,0,1,0,0],[0,0,0,0,0]]*Z(8)^0;
[ [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ]
gap> form := QuadraticFormByMatrix(mat);
< quadratic form >
gap> v1 := [1,0,0,0,0]*Z(8)^0;
[ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ]
gap> v2 := [0,1,0,0,0]*Z(8)^0;
[ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ]
gap> IsSingularVector(form,v1);
false
gap> IsSingularVector(form,v2);
true
gap> IsIsotropicVector(form,v1);
true
gap> IsIsotropicVector(form,v2);
true

</pre></div>

<a id="X8141325085AAC0CD" name="X8141325085AAC0CD"></a>

<h5>4.6-4 IsTotallyIsotropicSubspace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTotallyIsotropicSubspace</code>( <var class="Arg">form</var>, <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: true or false

The operation return <var class="Arg">true</var> if and only if the subspace spanned by the vectors in the list <var class="Arg">sub</var> is totally isotropic with relation to the sesquilinear or quadratic form <var class="Arg">form</var>. Note that when <var class="Arg">form</var> is a quadratic form, it is checked whether <var class="Arg">sub</var> generates a subspace that is totally isotropic with relation to the associated bilinear form of <var class="Arg">form</var>.

<div class="example"><pre>
gap> mat := [[1,0,0,0],[0,-1,0,0],[0,0,0,1],[0,0,1,0]]*Z(7)^0;
[ [ Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7) ], [ 0*Z(7), Z(7)^3, 0*Z(7), 0*Z(7) ],
 [ 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^0 ], [ 0*Z(7), 0*Z(7), Z(7)^0, 0*Z(7) ] ]
gap> form := BilinearFormByMatrix(mat);
< bilinear form >
gap> sub:= [[Z(7)^0,0*Z(7),Z(7)^0,Z(7)],[0*Z(7),Z(7)^0,Z(7)^0,Z(7)^4]];
[ [ Z(7)^0, 0*Z(7), Z(7)^0, Z(7) ], [ 0*Z(7), Z(7)^0, Z(7)^0, Z(7)^4 ] ]
gap> IsTotallyIsotropicSubspace(form,sub);
true
gap> mat := IdentityMat(6,GF(2));
[ <a GF2 vector of length 6>, <a GF2 vector of length 6>,
 <a GF2 vector of length 6>, <a GF2 vector of length 6>,
 <a GF2 vector of length 6>, <a GF2 vector of length 6> ]
gap> form := HermitianFormByMatrix(mat,GF(4));
< hermitian form >
gap> sub := [[Z(2)^0,0*Z(2),0*Z(2),Z(2)^0,Z(2)^0,Z(2)^0], 
> [0*Z(2),Z(2)^0,0*Z(2),Z(2^2)^2,Z(2^2),Z(2)^0], 
> [0*Z(2),0*Z(2),Z(2)^0,Z(2)^0,Z(2^2),Z(2^2)^2]];
[ [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ],
 [ 0*Z(2), Z(2)^0, 0*Z(2), Z(2^2)^2, Z(2^2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2^2), Z(2^2)^2 ] ]
gap> IsTotallyIsotropicSubspace(form,sub);
true

</pre></div>

<a id="X834FD9117F1DA8D0" name="X834FD9117F1DA8D0"></a>

<h5>4.6-5 IsTotallySingularSubspace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTotallySingularSubspace</code>( <var class="Arg">form</var>, <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: true or false

The operation return <var class="Arg">true</var> if and only if the subspace spanned by the vectors in the list <var class="Arg">sub</var> is totally singular with relation to quadratic form <var class="Arg">form</var>. Note that only when the characteristic of the field is odd, the totally singular subspaces of given dimension \(n\) with relation to a quadratic form are exactly the totally isotropic subspaces of dimension \(n\) with relation to its associated form.

<div class="example"><pre>
gap> mat := [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,0,0],[0,0,1,0,0],[0,0,0,0,0]]*Z(8)^0;
[ [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
 [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ]
gap> form := QuadraticFormByMatrix(mat);
< quadratic form >
gap> sub := [[Z(2)^0,0*Z(2),Z(2^3)^6,Z(2^3),Z(2^3)^3],
> [0*Z(2),Z(2)^0,Z(2^3)^6,Z(2^3)^2,Z(2^3)]];
[ [ Z(2)^0, 0*Z(2), Z(2^3)^6, Z(2^3), Z(2^3)^3 ],
 [ 0*Z(2), Z(2)^0, Z(2^3)^6, Z(2^3)^2, Z(2^3) ] ]
gap> IsTotallySingularSubspace(form,sub);
true

</pre></div>

<a id="X813A02878352E9E5" name="X813A02878352E9E5"></a>

<h4>4.7 Attributes and properties of forms</h4>

<a id="X80254BFD7E4B8F06" name="X80254BFD7E4B8F06"></a>

<h5>4.7-1 IsReflexiveForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsReflexiveForm</code>( <var class="Arg">f</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: true or false.

A sesquilinear form \(f\) on a vector space \(V\) is reflexive if \(f(v,w)=0 \Rightarrow f(w,v)=0\) for all \(v,w \in V\). The argument \(f\) must be a sesquilinear form (and thus it belongs to <code class="code">IsSesquilinearForm</code>). A sesquilinear form \(f\) is reflexive if whenever we have \(f(u,v)=0\), for two vectors \(u,v\) in the associated vector space, then we also have \(f(v,u)=0\). This attribute simply returns <var class="Arg">true</var> or <var class="Arg">false</var> according to whether <var class="Arg">f</var> is reflexive or not, and is stored as a property of <var class="Arg">f</var>. It is not possible in this version of Forms to construct non-reflexive forms. (See <a href="chap3_mj.html#X874CD5E0802FEB50">3.1</a> for more on reflexive sesquilinear forms).

<a id="X7D5AB7E484CFBF63" name="X7D5AB7E484CFBF63"></a>

<h5>4.7-2 IsAlternatingForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAlternatingForm</code>( <var class="Arg">f</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: true or false.

A sesquilinear form \(f\) on a vector space \(V\) is alternating if \(f(v,v)=0\) for all \(v \in V\). The argument \(f\) must be a sesquilinear form (and thus it belongs to <code class="code">IsSesquilinearForm</code>). A bilinear form \(f\) is alternating if \(f(v,v)=0\) for all \(v\). This method simply returns <var class="Arg">true</var> or <var class="Arg">false</var> according to whether <var class="Arg">f</var> is alternating or not, and is stored as a property of <var class="Arg">f</var>. (See <a href="chap3_mj.html#X874CD5E0802FEB50">3.1</a> for more on alternating sesquilinear forms).

<a id="X85585B2C80413490" name="X85585B2C80413490"></a>

<h5>4.7-3 IsSymmetricForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSymmetricForm</code>( <var class="Arg">f</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: true or false.

A sesquilinear form \(f\) on a vector space \(V\) is symmetric if \(f(v,w)=f(w,v)\) for all \(v,w \in V\). The argument \(f\) must be a sesquilinear form (and thus it belongs to <code class="code">IsSesquilinearForm</code>). A bilinear form \(f\) is symmetric if \(f(u,v)=f(v,u)\) for all pairs of vectors \(u\) and \(v\). This attribute simply returns <var class="Arg">true</var> or <var class="Arg">false</var> according to whether <var class="Arg">f</var> is symmetric or not, and is stored as a property of <var class="Arg">f</var>. (See <a href="chap3_mj.html#X874CD5E0802FEB50">3.1</a> for more on symmetric sesquilinear forms).

<a id="X87E9C9A1781AB058" name="X87E9C9A1781AB058"></a>

<h5>4.7-4 IsOrthogonalForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsOrthogonalForm</code>( <var class="Arg">f</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: true or false.

The argument \(f\) must be a sesquilinear form (and thus it belongs to <code class="code">IsSesquilinearForm</code>). A bilinear form \(f\) is called orthogonal if the characteristic of the underlying field is odd, and \(f\) is a symmetric form. (See <a href="chap3_mj.html#X874CD5E0802FEB50">3.1</a> for more on bilinear forms). This operation simply returns <var class="Arg">true</var> or <var class="Arg">false</var> according to whether <var class="Arg">f</var> is an orthogonal bilinear form or not, and is stored as a property of <var class="Arg">f</var>.

<a id="X861AF6EE82F4DA39" name="X861AF6EE82F4DA39"></a>

<h5>4.7-5 IsPseudoForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPseudoForm</code>( <var class="Arg">f</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: true or false.

When the characteristic of the field is odd, we call a form \(f\) orthogonal if and only \(f\) is symmetric, and when the characteristic of the field is even, we call a form \(f\) pseudo if and only if \(f\) is symmetric but not alternating. The argument \(f\) must be a sesquilinear form (and thus it belongs to <code class="code">IsSesquilinearForm</code>). (See <a href="chap3_mj.html#X874CD5E0802FEB50">3.1</a> for more on pseudo forms). This method simply returns <var class="Arg">true</var> or <var class="Arg">false</var> according to whether <var class="Arg">f</var> is a pseudo form or not, and is stored as a property of <var class="Arg">f</var>.

<a id="X86F552AE7ACC12C7" name="X86F552AE7ACC12C7"></a>

<h5>4.7-6 IsSymplecticForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSymplecticForm</code>( <var class="Arg">f</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: true or false.

We call a bilinear form \(f\) symplectic if and only if \(f\) is alternating. The argument \(f\) must be a sesquilinear form (and thus it belongs to <code class="code">IsSesquilinearForm</code>). (See <a href="chap3_mj.html#X874CD5E0802FEB50">3.1</a> for more on symplectic forms). This method simply returns <var class="Arg">true</var> or <var class="Arg">false</var> according to whether <var class="Arg">f</var> is symplectic or not, and is stored as a property of <var class="Arg">f</var>.

<a id="X7C60B9587D130DBB" name="X7C60B9587D130DBB"></a>

<h5>4.7-7 IsDegenerateForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsDegenerateForm</code>( <var class="Arg">f</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: true or false.

The argument \(f\) must be a form (and thus it belongs to <code class="code">IsForm</code>). A sesquilinear form \(f\) is degenerate if its radical is non-trivial. A quadratic form is degenerate if and only if the radical of the associated bilinear form is non-trivial. Note that degeneracy for quadratic forms is too restrictive if the characteristic is even. See also <code class="func">IsSingularForm</code> (<a href="chap4_mj.html#X7A0E882F801624DA">4.7-8</a>). This attribute simply returns <var class="Arg">true</var> or <var class="Arg">false</var> according to whether <var class="Arg">f</var> is degenerate or not, and is stored as a property of <var class="Arg">f</var>.

<a id="X7A0E882F801624DA" name="X7A0E882F801624DA"></a>

<h5>4.7-8 IsSingularForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSingularForm</code>( <var class="Arg">f</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns: true or false.

The argument \(f\) must be a quadratic form (and thus it belongs to <code class="code">IsQuadraticForm</code>). A quadratic form \(f\) is singular if its radical is non-trivial. When the characteristic of the field is odd, a quadratic form is singular if and only if it is degenerate. This is not the case when the characteristic of the field is even. This method simply returns <var class="Arg">true</var> or <var class="Arg">false</var> according to whether <var class="Arg">f</var> is singular or not, and is stored as a property of <var class="Arg">f</var>.

<a id="X7BCBA564829D9E89" name="X7BCBA564829D9E89"></a>

<h5>4.7-9 BaseField</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BaseField</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: the underlying field of <var class="Arg">f</var>.

The argument \(f\) must be a form (and thus it belongs to <code class="code">IsForm</code>). The method returns the field which is stored as the defining field of \(f\). We sometimes stipulate in Forms that a form have a defining field, for mathematical reasons. Clearly, to define a hermitian form one needs to specify the field of scalars for the vector space that you wish your hermitian form to act on. The default, if the user has not specified a field on creation of a form, is the smallest field containing the entries or coefficients of the input (a matrix or polynomial). Having a particular defining field for a form can be very useful, for example, when one wants to find a change of basis from one form to another (isometric) form. In this case, one needs to know in which \(\mathrm{GL}(d,q)\) the base-transition matrix should be taken.

<a id="X847AFB4C81A90B3F" name="X847AFB4C81A90B3F"></a>

<h5>4.7-10 GramMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GramMatrix</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: the Gram matrix of <var class="Arg">f</var>.

The argument \(f\) must be a form (and thus it belongs to <code class="code">IsForm</code>). This method returns the Gram matrix of \(f\) (see <a href="chap3_mj.html#X874CD5E0802FEB50">3.1</a> and <a href="chap3_mj.html#X864CAF8881067D8A">3.2</a>).

<a id="X7855C3C07AAA1A68" name="X7855C3C07AAA1A68"></a>

<h5>4.7-11 RadicalOfForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RadicalOfForm</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: The radical of the form <var class="Arg">f</var>

The argument \(f\) must be a form (and thus it belongs to <code class="code">IsForm</code>) on some vector space \(V\). The radical of a sesquilinear form \(f\) is the subspace consisting of vectors which are orthogonal to every vector, i.e.,

\[\mathrm{Rad}(f) = \{v \in V | f(v,w) = 0,\, \forall w \in V\}.\]

The radical of the quadratic form \(Q\), is the intersection of the set of all singular vectors with relation to \(Q\) and the radical of the associated bilinear form \(f\) (see <code class="func">AssociatedBilinearForm</code> (<a href="chap4_mj.html#X7BF7FBCA7FF91052">4.4-3</a>)), i.e.

\[\mathrm{Rad}(Q) = \{v \in V | Q(v) = 0\,\, \mathrm{and}\,\, v \in \mathrm{Rad}(f)\}.\]

<div class="example"><pre>
gap> r := PolynomialRing( GF(8), 3 );
GF(2^3)[x_1,x_2,x_3]
gap> poly := r.1^2 + r.2 * r.3;
x_1^2+x_2*x_3
gap> form := QuadraticFormByPolynomial( poly, r );
< quadratic form >
gap> r := RadicalOfForm( form );
<vector space of dimension 0 over GF(2^3)>
gap> Dimension(r);
0

</pre></div>

<a id="X82E7367F817C6BD0" name="X82E7367F817C6BD0"></a>

<h5>4.7-12 PolynomialOfForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PolynomialOfForm</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: the polynomial associated with <var class="Arg">f</var>.

The argument <var class="Arg">f</var> must be a form (and thus it belongs to <code class="code">IsForm</code>). All forms, except for bilinear forms in even characteristic, have an associated polynomial defining a quadratic or hermitian form (see <a href="chap3_mj.html#X874CD5E0802FEB50">3.1</a> and <a href="chap3_mj.html#X864CAF8881067D8A">3.2</a>). This method returns the polynomial associated with <var class="Arg">f</var>, and if not already bound, stores it as a property of <var class="Arg">f</var>.

<div class="example"><pre>
gap> mat := [ [ Z(8) , 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
> [ 0*Z(2), Z(2)^0, Z(2^3)^5, 0*Z(2), 0*Z(2) ], 
> [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
> [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], 
> [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ];;
gap> form := QuadraticFormByMatrix(mat,GF(8));
< quadratic form >
gap> PolynomialOfForm(form);
Z(2^3)*x_1^2+x_2^2+Z(2^3)^5*x_2*x_3+x_4*x_5

</pre></div>

<a id="X87B652A28534E0D2" name="X87B652A28534E0D2"></a>

<h5>4.7-13 DiscriminantOfForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DiscriminantOfForm</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: a string

The argument \(f\) must be a form (and thus it belongs to <code class="code">IsForm</code>). Given a quadratic or bilinear form <var class="Arg">f</var> of even dimension, this operation returns a string: ``square'' or ``nonsquare''. More specifically, let \(f\) be a from over \(\mathrm{GF}(q)\), and let \(M\) be the Gram matrix of \(f\). Define the discriminant of \(Q\) (n.b., quasideterminant in <a href="chapBib_mj.html#biBAtlas">[CCN+85]</a>) as `square' if \(\mathrm{det}(M)\) is a square of \(\mathrm{GF}(q)\), and `non-square' otherwise. The discriminant is an invariant of nondegenerate orthogonal spaces over finite fields of odd order, up to isometry. Thus, discriminants can be used to delineate the isometry type of an orthogonal form in even (algebraic) dimension. The discriminant of a hermitian form is not defined, and applying this operation on a hermitian form, will result in an error message.

<div class="example"><pre>
gap> gram := InvariantQuadraticForm(GO(-1,4,5))!.matrix;;
gap> qform := QuadraticFormByMatrix(gram, GF(5));
< quadratic form >
gap> DiscriminantOfForm( qform );
"nonsquare"

</pre></div>

<a id="X8400E22D7D51FCCE" name="X8400E22D7D51FCCE"></a>

<h4>4.8 Recognition of forms preserved by a classical group</h4>

In this section, we describe a function that was initially developed by Frank Celler (and which has now been adapted to Forms) for the recognition of quadratic and sesquilinear forms left invariant by a matrix group. More importantly, we should stress that this routine differs to that already offered by the MeatAxe in that it finds sesquilinear forms preserved up to scalars. Eventually, the procedure used for finding preserved sesquilinear forms does use the MeatAxe but in some cases it can rule out the existence of preserved forms without calling the MeatAxe. For more information on the algorithm, please see <a href="chapBib_mj.html#biBCLGNNPP">[CLN+08]</a>.

<a id="X784481E57E207B3D" name="X784481E57E207B3D"></a>

<h5>4.8-1 PreservedForms</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PreservedForms</code>( <var class="Arg">group</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a list of forms

The argument <var class="Arg">group</var> is a matrix group. The function uses random methods to find all of the bilinear, unitary or quadratic forms preserved by <var class="Arg">group</var> (the trivial form is also a possibility) up to a scalar. Since the procedure relies on a pseudo-random generator, the user may need to execute the operation more than once to find all invariant sesquilinear forms. Note that when possible, a quadratic form will be given.

<div class="example"><pre>
gap> group := GO(-1,4,4);
GO(-1,4,4)
gap> pres_forms := PreservedForms(group);
[ < quadratic form > ]
gap> group := GO(1,6,9);
GO(+1,6,9)
gap> pres_forms := PreservedForms(group);
[ < quadratic form > ]
gap> group := SU(4,16);
SU(4,16)
gap> pres_forms := PreservedForms(group);
[ < hermitian form > ]

</pre></div>

The next example demonstrates the impact of randomized methods on the number of preserved forms returned. For the particular matrix group, two preserved forms are returned after four calls of the operation.

<div class="example"><pre>
gap> gens := [ [ [ Z(5)^0, 0*Z(5), 0*Z(5), 0*Z(5) ], 
> [ 0*Z(5), 0*Z(5), Z(5)^3, Z(5^2)^21 ],
> [ 0*Z(5), Z(5), Z(5), Z(5^2)^3 ], 
> [ 0*Z(5), Z(5^2)^21, Z(5^2)^15, Z(5)^2 ] ], 
> [ [ Z(5)^3, Z(5^2)^7, Z(5^2)^16, Z(5^2)^15 ], 
> [ 0*Z(5), Z(5)^0, Z(5^2)^4, Z(5)^3 ], 
> [ Z(5^2)^22, Z(5^2)^10, Z(5^2)^21, Z(5)^2 ], 
> [ Z(5^2)^7, Z(5^2)^23, Z(5^2)^9, Z(5^2)^11 ] ], 
> [ [ 0*Z(5), Z(5^2), 0*Z(5), 0*Z(5) ], 
> [ Z(5^2)^5, 0*Z(5), 0*Z(5), 0*Z(5) ], 
> [ 0*Z(5), 0*Z(5), Z(5)^0, Z(5^2)^4 ], 
> [ 0*Z(5), 0*Z(5), Z(5^2)^20, Z(5)^2 ] ] ];
[ [ [ Z(5)^0, 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^3, Z(5^2)^21 ],
 [ 0*Z(5), Z(5), Z(5), Z(5^2)^3 ],
 [ 0*Z(5), Z(5^2)^21, Z(5^2)^15, Z(5)^2 ] ],
 [ [ Z(5)^3, Z(5^2)^7, Z(5^2)^16, Z(5^2)^15 ],
 [ 0*Z(5), Z(5)^0, Z(5^2)^4, Z(5)^3 ],
 [ Z(5^2)^22, Z(5^2)^10, Z(5^2)^21, Z(5)^2 ],
 [ Z(5^2)^7, Z(5^2)^23, Z(5^2)^9, Z(5^2)^11 ] ],
 [ [ 0*Z(5), Z(5^2), 0*Z(5), 0*Z(5) ], [ Z(5^2)^5, 0*Z(5), 0*Z(5), 0*Z(5) ],
 [ 0*Z(5), 0*Z(5), Z(5)^0, Z(5^2)^4 ],
 [ 0*Z(5), 0*Z(5), Z(5^2)^20, Z(5)^2 ] ] ]
gap> group := Group(gens);
<matrix group with 3 generators>
gap> PreservedForms(group);
[ < quadratic form >, < hermitian form > ]
gap> PreservedForms(group);
[ < quadratic form >, < hermitian form > ]
gap> PreservedForms(group);
[ < quadratic form >, < hermitian form > ]
gap> PreservedForms(group);
[ < quadratic form >, < hermitian form > ]

</pre></div>

<a id="X84056A357E5447AF" name="X84056A357E5447AF"></a>

<h5>4.8-2 PreservedSesquilinearForms</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PreservedSesquilinearForms</code>( <var class="Arg">group</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a list of forms

The argument <var class="Arg">group</var> is a matrix group. The function uses random methods to find all of the bilinear or unitary forms preserved by <var class="Arg">group</var> (the trivial form is also a possibility) up to a scalar. Since the procedure relies on a pseudo-random generator, the user may need to execute the operation more than once to find all invariant sesquilinear forms.

<div class="example"><pre>
gap> g := SU(4,3);
SU(4,3)
gap> forms := PreservedSesquilinearForms(g);
[ < hermitian form > ]
gap> Display( forms[1] );
Hermitian form
Gram Matrix:
. . . 1
. . 1 .
. 1 . .
1 . . .

</pre></div>

Here is another example which shows that this procedure is suitable in some cases where using the MeatAxe is not applicable. Here, our matrix group is the group of similarities preserving a (hyperbolic) bilinear form on a 6-dimensional vector space over \(\mathrm{GF}(3)\) .

<div class="example"><pre>
gap> a := [ [ -1, 0, 0, -1, 0, 1 ], [ 0, -1, -1, 0, 0, 1 ], 
> [ -1, 0, 0, 1, 0, 0 ], [ 0, -1, 1, 0, 0, -1 ], 
> [ 0, 0, 0, 0, 0, -1 ], [ 0, -1, -1, 1, 1, 1 ] ] * One(GF(3));;
gap> b := [ [ 1, -1, 1, -1, 1, -1 ], [ 1, 1, -1, 1, 1, 0 ], 
> [ -1, 0, 1, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 1 ], 
> [ 1, 1, 1, 1, 1, 1 ], [ -1, 1, 1, 1, -1, 0 ] ] * One(GF(3));;
gap> g := Group( a, b );
<matrix group with 2 generators>
gap> forms := PreservedSesquilinearForms( g );
[ < bilinear form > ]
gap> Display( forms[1] );
Bilinear form
Gram Matrix:
. 1 . . . .
1 . . . . .
. . . 1 . .
. . 1 . . .
. . . . . 1
. . . . 1 .
gap> m := GModuleByMats( [a,b], GF(3) );;
gap> usemeataxe := MTX.InvariantBilinearForm(m);
fail

</pre></div>

<a id="X7D6F72B682E405E1" name="X7D6F72B682E405E1"></a>

<h5>4.8-3 PreservedQuadraticForms</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PreservedQuadraticForms</code>( <var class="Arg">group</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a list of forms

The argument <var class="Arg">group</var> is a matrix group. The function uses random methods to find all of the quadratic forms preserved by <var class="Arg">group</var> up to a scalar. Since the procedure relies on a pseudo-random generator, the user may need to execute the operation more than once to find all invariant sesquilinear forms.

<div class="example"><pre>
gap> group := GO(-1,4,8);
GO(-1,4,8)
gap> pres_forms := PreservedQuadraticForms(group);
[ < quadratic form > ]
gap> group := GO(1,6,9);
GO(+1,6,9)
gap> pres_forms := PreservedQuadraticForms(group);
[ < quadratic form > ]

</pre></div>

<a id="X836A21687A685839" name="X836A21687A685839"></a>

<h4>4.9 The trivial form and some of its properties</h4>

It can be useful to work with trivial a quadratic or sesquilinear form, i.e. a form mapping all vectors, couples of vectors respectively, to the zero element of their basefield. As mentioned in Section <a href="chap4_mj.html#X83494A76866B06A5">4.1</a>, Forms allows the construction of an object in the Category <code class="code">IsTrivialForm</code>.

<div class="example"><pre>
gap> mat := [[0,0,0],[0,0,0],[0,0,0]]*Z(7)^0;
[ [ 0*Z(7), 0*Z(7), 0*Z(7) ], [ 0*Z(7), 0*Z(7), 0*Z(7) ],
 [ 0*Z(7), 0*Z(7), 0*Z(7) ] ]
gap> form1 := BilinearFormByMatrix(mat,GF(7));
< trivial form >
gap> form2 := QuadraticFormByMatrix(mat,GF(7));
< trivial form >
gap> form1 = form2;
true
gap> IsQuadraticForm(form1);
false
gap> IsSesquilinearForm(form1);
false
gap> mat := [[0,0],[0,0]]*Z(4)^0;
[ [ 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2) ] ]
gap> form3 := BilinearFormByMatrix(mat,GF(4));
< trivial form >
gap> form3 = form1;
false

</pre></div>

As we have seen by the above example, there is only one trivial form for a given vector space over a finite field, and such a trivial form can result from the construction of a quadratic form or a sesquilinear form, but the trivial form itself is none of these, although it can behave as a sesquilinear or a quadratic form, depending on its arguments.

<div class="example"><pre>
gap> mat := [[0,0,0,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]]*Z(3)^0;
[ [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ],
 [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ] ]
gap> form := BilinearFormByMatrix(mat,GF(3));
< trivial form >
gap> v := Random(GF(3)^4);
[ Z(3), Z(3), 0*Z(3), Z(3) ]
gap> [v,v]^form;
0*Z(3)
gap> v^form;
0*Z(3)

</pre></div>

The attributes and properties described in Section <a href="chap4_mj.html#X813A02878352E9E5">4.7</a> are all applicable to trivial forms.

<div class="example"><pre>
gap> mat := [[0,0,0],[0,0,0],[0,0,0]]*Z(11)^0;
[ [ 0*Z(11), 0*Z(11), 0*Z(11) ], [ 0*Z(11), 0*Z(11), 0*Z(11) ],
 [ 0*Z(11), 0*Z(11), 0*Z(11) ] ]
gap> form := QuadraticFormByMatrix(mat,GF(121));
< trivial form >
gap> IsReflexiveForm(form);
true
gap> IsAlternatingForm(form);
true
gap> IsSymmetricForm(form);
true
gap> IsOrthogonalForm(form);
false
gap> IsPseudoForm(form);
false
gap> IsSymplecticForm(form);
true
gap> IsDegenerateForm(form);
true
gap> IsSingularForm(form);
true
gap> BaseField(form);
GF(11^2)
gap> GramMatrix(form);
[ [ 0*Z(11), 0*Z(11), 0*Z(11) ], [ 0*Z(11), 0*Z(11), 0*Z(11) ],
 [ 0*Z(11), 0*Z(11), 0*Z(11) ] ]
gap> RadicalOfForm(form);
<vector space of dimension 3 over GF(11^2)>

</pre></div>

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