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<div class="ChapSects"><a href="chap4_mj.html#X8166C704848D128E">4 <span class="Heading">Constructing forms and basic functionality</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X83494A76866B06A5">4.1 <span class="Heading">Important filters</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7FA162E5874E8330">4.1-1 <span class="Heading">Categories for forms</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7999E38082474342">4.1-2 <span class="Heading">Representation for forms</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X78D981A67DBFCD6D">4.2 <span class="Heading">Constructing forms using a matrix</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7C9D7E517A73F02F">4.2-1 BilinearFormByMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X86B8694F782A4EE7">4.2-2 QuadraticFormByMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7C027FF77AFED321">4.2-3 HermitianFormByMatrix</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X78476EDF7B9498D7">4.3 <span class="Heading">Constructing forms using a polynomial</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X81D571077C4BCEFF">4.3-1 BilinearFormByPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X86ADE1D986CC90CB">4.3-2 QuadraticFormByPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7E21CFFA84180D0D">4.3-3 HermitianFormByPolynomial</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X843B68558283CE5F">4.4 <span class="Heading">Switching between bilinear and quadratic forms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7F13EAC17BDE228D">4.4-1 QuadraticFormByBilinearForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X812963777BBF97E3">4.4-2 BilinearFormByQuadraticForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7BF7FBCA7FF91052">4.4-3 AssociatedBilinearForm</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X8110213A7B303D1C">4.5 <span class="Heading">Evaluating forms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X808AB7B9840ABC27">4.5-1 EvaluateForm</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7A0825A987C88978">4.6 <span class="Heading">Orthogonality, totally isotropic subspaces, and totally singular subspaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7D699C077F66F3E6">4.6-1 OrthogonalSubspaceMat</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8394CCAD798053C6">4.6-2 IsIsotropicVector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X855E539185D7D3C7">4.6-3 IsSingularVector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8141325085AAC0CD">4.6-4 IsTotallyIsotropicSubspace</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X834FD9117F1DA8D0">4.6-5 IsTotallySingularSubspace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X813A02878352E9E5">4.7 <span class="Heading">Attributes and properties of forms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X80254BFD7E4B8F06">4.7-1 IsReflexiveForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7D5AB7E484CFBF63">4.7-2 IsAlternatingForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X85585B2C80413490">4.7-3 IsSymmetricForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X87E9C9A1781AB058">4.7-4 IsOrthogonalForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X861AF6EE82F4DA39">4.7-5 IsPseudoForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X86F552AE7ACC12C7">4.7-6 IsSymplecticForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7C60B9587D130DBB">4.7-7 IsDegenerateForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7A0E882F801624DA">4.7-8 IsSingularForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7BCBA564829D9E89">4.7-9 BaseField</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X847AFB4C81A90B3F">4.7-10 GramMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7855C3C07AAA1A68">4.7-11 RadicalOfForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X82E7367F817C6BD0">4.7-12 PolynomialOfForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X87B652A28534E0D2">4.7-13 DiscriminantOfForm</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X8400E22D7D51FCCE">4.8 <span class="Heading">Recognition of forms preserved by a classical group</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X784481E57E207B3D">4.8-1 PreservedForms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X84056A357E5447AF">4.8-2 PreservedSesquilinearForms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7D6F72B682E405E1">4.8-3 PreservedQuadraticForms</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X836A21687A685839">4.9 <span class="Heading">The trivial form and some of its properties</span></a>
</span>
</div>
</div>
<h3>4 <span class="Heading">Constructing forms and basic functionality</span></h3>
<p>In this chapter, all operations to construct sesquilinear and quadratic forms are listed, along with their basic attributes and properties.</p>
<p><a id="X83494A76866B06A5" name="X83494A76866B06A5"></a></p>
<h4>4.1 <span class="Heading">Important filters</span></h4>
<p><a id="X7FA162E5874E8330" name="X7FA162E5874E8330"></a></p>
<h5>4.1-1 <span class="Heading">Categories for forms</span></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>
<p>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>.</p>
<p>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>.</p>
<p>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>.</p>
<p><a id="X7999E38082474342" name="X7999E38082474342"></a></p>
<h5>4.1-2 <span class="Heading">Representation for forms</span></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>
<p>Every form is represented by a matrix, the base field and a string describing the ``type'' of the form.</p>
<p><a id="X78D981A67DBFCD6D" name="X78D981A67DBFCD6D"></a></p>
<h4>4.2 <span class="Heading">Constructing forms using a matrix</span></h4>
<p><a id="X7C9D7E517A73F02F" name="X7C9D7E517A73F02F"></a></p>
<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>
<p>Returns: a bilinear form</p>
<p>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 <em>defining field</em> 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"><span class="RefLink">3.1</span></a> for more on bilinear forms).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := IdentityMat(4, GF(9));</span>
[ [ 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 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByMatrix(mat,GF(9));</span>
< bilinear form >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(form);</span>
Bilinear form
Gram Matrix:
1 . . .
. 1 . .
. . 1 .
. . . 1
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[0*Z(2),Z(16)^12,0*Z(2),Z(4)^2,Z(16)^13],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [Z(16)^12,0*Z(2),0*Z(2),Z(16)^11,Z(16)],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [0*Z(2),0*Z(2),0*Z(2),Z(4)^2,Z(16)^3],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [Z(4)^2,Z(16)^11,Z(4)^2,0*Z(2),Z(16)^3],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [Z(16)^13,Z(16),Z(16)^3,Z(16)^3,0*Z(2) ]];</span>
[ [ 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) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByMatrix(mat,GF(16));</span>
< bilinear form >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(form);</span>
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 .
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]*Z(7)^0;</span>
[ [ 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) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByMatrix(mat);</span>
< bilinear form >
<span class="GAPprompt">gap></span> <span class="GAPinput">WittIndex(form);</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByMatrix(mat,GF(49));</span>
< bilinear form >
<span class="GAPprompt">gap></span> <span class="GAPinput">WittIndex(form);</span>
2
</pre></div>
<p><a id="X86B8694F782A4EE7" name="X86B8694F782A4EE7"></a></p>
<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>
<p>Returns: a quadratic form</p>
<p>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 <em>defining field</em> 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"><span class="RefLink">3.2</span></a> for more on bilinear forms).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[1,0,0,0],[0,3,0,0],[0,0,0,6],[0,0,6,0]]*Z(7)^0;</span>
[ [ 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) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByMatrix(mat,GF(7));</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(form);</span>
Quadratic form
Gram Matrix:
1 . . .
. 3 . .
. . . 5
. . . .
<span class="GAPprompt">gap></span> <span class="GAPinput">gf := GF(2^2);</span>
GF(2^2)
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := InvariantQuadraticForm( SO(-1, 4, 4) )!.matrix;</span>
[ [ 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 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByMatrix( mat, gf );</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(form);</span>
Quadratic form
Gram Matrix:
z = Z(4)
. 1 . .
. . . .
. . z^2 1
. . . z^2
</pre></div>
<p>The following example shows how using the argument <var class="Arg">field</var> has influence on the properties of the constructed form.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := </span>
<span class="GAPprompt">></span> <span class="GAPinput">[[Z(2)^0,Z(2)^0,0*Z(2),0*Z(2)],[0*Z(2),Z(2)^0,0*Z(2),0*Z(2)], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [0*Z(2),0*Z(2),0*Z(2),Z(2)^0],[0*Z(2),0*Z(2),0*Z(2),0*Z(2)]];</span>
[ [ 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) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByMatrix(mat);</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">WittIndex(form);</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByMatrix(mat,GF(4));</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">WittIndex(form);</span>
2
</pre></div>
<p><a id="X7C027FF77AFED321" name="X7C027FF77AFED321"></a></p>
<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>
<p>Returns: a hermitian sesquilinear form</p>
<p>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"><span class="RefLink">3.1</span></a> for more on hermitian forms).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gf := GF(3^2);</span>
GF(3^2)
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := IdentityMat(4, gf);</span>
[ [ 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 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := HermitianFormByMatrix( mat, gf );</span>
< hermitian form >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(form);</span>
Hermitian form
Gram Matrix:
1 . . .
. 1 . .
. . 1 .
. . . 1
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[Z(11)^0,0*Z(11),0*Z(11)],[0*Z(11),0*Z(11),Z(11)],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [0*Z(11),Z(11),0*Z(11)]];</span>
[ [ 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) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := HermitianFormByMatrix(mat,GF(121));</span>
< hermitian form >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(form);</span>
Hermitian form
Gram Matrix:
1 . .
. . 2
. 2 .
</pre></div>
<p><a id="X78476EDF7B9498D7" name="X78476EDF7B9498D7"></a></p>
<h4>4.3 <span class="Heading">Constructing forms using a polynomial</span></h4>
<p>Suppose that <span class="SimpleMath">\(f\)</span> is a sesquilinear form on an <span class="SimpleMath">\(n\)</span>-dimensional vectorspace. Consider a vector <span class="SimpleMath">\(x\)</span> with coordinates <span class="SimpleMath">\(x_1,\ldots,x_{n}\)</span> with <span class="SimpleMath">\(x_i\)</span> indeterminates over the field. Then <span class="SimpleMath">\(f(x,x)\)</span> is a polynomial in <span class="SimpleMath">\(n\)</span> indeterminates. When <span class="SimpleMath">\(f\)</span> is alternating, <span class="SimpleMath">\(f(x,x)\)</span> is identically zero, but in all other cases, <span class="SimpleMath">\(f(x,x)\)</span> determines <span class="SimpleMath">\(f\)</span> completely.</p>
<p>Conversely, suppose that <span class="SimpleMath">\(Q\)</span> is a quadratic form on an <span class="SimpleMath">\(n\)</span>-dimensional vectorspace. Consider a vector <span class="SimpleMath">\(x\)</span> with coordinates <span class="SimpleMath">\(x_1,\ldots,x_{n}\)</span> with <span class="SimpleMath">\(x_i\)</span> indeterminates over the field. Then <span class="SimpleMath">\(Q(x)\)</span> is a polynomial in <span class="SimpleMath">\(n\)</span> indeterminates, and <span class="SimpleMath">\(Q(x)\)</span> determines <span class="SimpleMath">\(Q\)</span> completely.</p>
<p><strong class="pkg">Forms</strong> provides functionality to construct bilinear, hermitian and quadratic forms using an appropriate polynomial.</p>
<p><a id="X81D571077C4BCEFF" name="X81D571077C4BCEFF"></a></p>
<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>
<p>Returns: a bilinear form</p>
<p>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"><span class="RefLink">3.1</span></a> for more on bilinear forms).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">r := PolynomialRing( GF(11), 4);</span>
GF(11)[x_1,x_2,x_3,x_4]
<span class="GAPprompt">gap></span> <span class="GAPinput">vars := IndeterminatesOfPolynomialRing( r );</span>
[ x_1, x_2, x_3, x_4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">pol := vars[1]*vars[2]+vars[3]*vars[4];</span>
x_1*x_2+x_3*x_4
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByPolynomial(pol, r, 4);</span>
< bilinear form >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(form);</span>
Bilinear form
Gram Matrix:
. 6 . .
6 . . .
. . . 6
. . 6 .
Polynomial: x_1*x_2+x_3*x_4
<span class="GAPprompt">gap></span> <span class="GAPinput">r := PolynomialRing(GF(4),2);</span>
GF(2^2)[x_1,x_2]
<span class="GAPprompt">gap></span> <span class="GAPinput">pol := r.1*r.2;</span>
x_1*x_2
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByPolynomial(pol,r);</span>
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
<span class="GAPbrkprompt">brk></span> <span class="GAPinput">quit;</span>
</pre></div>
<p><a id="X86ADE1D986CC90CB" name="X86ADE1D986CC90CB"></a></p>
<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>
<p>Returns: a quadratic form</p>
<p>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"><span class="RefLink">3.2</span></a> for more on quadratic forms).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">r := PolynomialRing( GF(8), 3);</span>
GF(2^3)[x_1,x_2,x_3]
<span class="GAPprompt">gap></span> <span class="GAPinput">poly := r.1^2 + r.2^2 + r.3^2;</span>
x_1^2+x_2^2+x_3^2
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByPolynomial(poly, r);</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">RadicalOfForm(form);</span>
<vector space over GF(2^3), with 63 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">r := PolynomialRing(GF(9),4);</span>
GF(3^2)[x_1,x_2,x_3,x_4]
<span class="GAPprompt">gap></span> <span class="GAPinput">poly := Z(3)^2*r.1^2+r.2^2+r.3*r.4;</span>
x_1^2+x_2^2+x_3*x_4
<span class="GAPprompt">gap></span> <span class="GAPinput">qform := QuadraticFormByPolynomial(poly,r);</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(qform);</span>
Quadratic form
Gram Matrix:
1 . . .
. 1 . .
. . . 1
. . . .
Polynomial: x_1^2+x_2^2+x_3*x_4
</pre></div>
<p><a id="X7E21CFFA84180D0D" name="X7E21CFFA84180D0D"></a></p>
<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>
<p>Returns: an hermitian form</p>
<p>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 <span class="SimpleMath">\(q^2\)</span> 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 <span class="SimpleMath">\(q+1\)</span>, 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"><span class="RefLink">3.1</span></a> for more on hermitian forms).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">r := PolynomialRing( GF(9), 4);</span>
GF(3^2)[x_1,x_2,x_3,x_4]
<span class="GAPprompt">gap></span> <span class="GAPinput">vars := IndeterminatesOfPolynomialRing( r );</span>
[ x_1, x_2, x_3, x_4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">poly := vars[1]*vars[2]^3+vars[1]^3*vars[2]+</span>
<span class="GAPprompt">></span> <span class="GAPinput"> vars[3]*vars[4]^3+vars[3]^3*vars[4];</span>
x_1^3*x_2+x_1*x_2^3+x_3^3*x_4+x_3*x_4^3
<span class="GAPprompt">gap></span> <span class="GAPinput">form := HermitianFormByPolynomial(poly,r);</span>
< hermitian form >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(form);</span>
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>
<p><a id="X843B68558283CE5F" name="X843B68558283CE5F"></a></p>
<h4>4.4 <span class="Heading">Switching between bilinear and quadratic forms</span></h4>
<p>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 <span class="SimpleMath">\(Q\)</span> is given, it is useful to consider the bilinear form <span class="SimpleMath">\(f\)</span> such that <span class="SimpleMath">\(f(v,v)=Q(v)\)</span>, i.e., the bilinear form which is determined by exactly the same polynomial determining the quadratic form <span class="SimpleMath">\(Q\)</span>. <strong class="pkg">Forms</strong> provides functionality to construct a bilinear form <span class="SimpleMath">\(f\)</span> from a given quadratic form <span class="SimpleMath">\(Q\)</span> such that <span class="SimpleMath">\(f(v,v)=Q(v)\)</span>. Conversely, we can extract a quadratic form from a given bilinear form.</p>
<p><a id="X7F13EAC17BDE228D" name="X7F13EAC17BDE228D"></a></p>
<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>
<p>Returns: a quadratic form</p>
<p>The argument <span class="SimpleMath">\(form\)</span> 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 <span class="SimpleMath">\(Q\)</span> (an object in <code class="code">IsQuadraticForm</code>), such that <span class="SimpleMath">\(Q(v) = form(v,v)\)</span> for all vectors <span class="SimpleMath">\(v\)</span> in a vector space equipped with <span class="SimpleMath">\(form\)</span>. An error is returned when the characteristic of the field is even, or when <span class="SimpleMath">\(form\)</span> is not orthogonal.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [ [ Z(3^2)^7, Z(3)^0, Z(3^2)^2, 0*Z(3), Z(3^2)^5 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ Z(3)^0, Z(3^2)^7, Z(3^2)^6, Z(3^2)^5, Z(3^2)^2 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ Z(3^2)^2, Z(3^2)^6, Z(3^2)^7, Z(3^2)^2, Z(3^2)^2 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0*Z(3), Z(3^2)^5, Z(3^2)^2, Z(3^2)^6, Z(3^2)^7 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ Z(3^2)^5, Z(3^2)^2, Z(3^2)^2, Z(3^2)^7, Z(3) ] ];</span>
[ [ 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) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByMatrix(mat,GF(9));</span>
< bilinear form >
<span class="GAPprompt">gap></span> <span class="GAPinput">Q := QuadraticFormByBilinearForm(form);</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(form);</span>
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
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(Q);</span>
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
<span class="GAPprompt">gap></span> <span class="GAPinput">Set(List(GF(9)^5),x->[x,x]^form=x^Q);</span>
[ true ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PolynomialOfForm(form);</span>
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
<span class="GAPprompt">gap></span> <span class="GAPinput">PolynomialOfForm(Q);</span>
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>
<p>Note that the given bilinear form <var class="Arg">form</var> is <strong class="button">not</strong> the associated bilinear form of the constructed quadratic form <span class="SimpleMath">\(Q\)</span>, according to the definition in Section <a href="chap3_mj.html#X864CAF8881067D8A"><span class="RefLink">3.2</span></a>. We can construct the associated bilinear forms by using <code class="func">AssociatedBilinearForm</code> (<a href="chap4_mj.html#X7BF7FBCA7FF91052"><span class="RefLink">4.4-3</span></a>). (See <a href="chap3_mj.html#X864CAF8881067D8A"><span class="RefLink">3.2</span></a> for more on quadratic forms).</p>
<p><a id="X812963777BBF97E3" name="X812963777BBF97E3"></a></p>
<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>
<p>Returns: a bilinear form</p>
<p>The argument <span class="SimpleMath">\(Q\)</span> must be a quadratic form (and thus it belongs to <code class="code">IsQuadraticForm</code>). The output is the orthogonal bilinear form <span class="SimpleMath">\(f\)</span> (an object in <code class="code">IsBilinearForm</code>), such that <span class="SimpleMath">\(f(v,v) = Q(v)\)</span> for all vectors <span class="SimpleMath">\(v\)</span> in a vector space equipped with <span class="SimpleMath">\(Q\)</span>. An error is returned when the characteristic of the field is even.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">r := PolynomialRing(GF(9),4);</span>
GF(3^2)[x_1,x_2,x_3,x_4]
<span class="GAPprompt">gap></span> <span class="GAPinput">poly := -r.1*r.2+Z(3^2)*r.3^2+r.4^2;</span>
-x_1*x_2+Z(3^2)*x_3^2+x_4^2
<span class="GAPprompt">gap></span> <span class="GAPinput">qform := QuadraticFormByPolynomial(poly,r);</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( qform );</span>
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
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByQuadraticForm( qform );</span>
< bilinear form >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(form);</span>
Bilinear form
Gram Matrix:
z = Z(9)
. 1 . .
1 . . .
. . z^1 .
. . . 1
<span class="GAPprompt">gap></span> <span class="GAPinput">Set(GF(9)^4, x -> [x,x]^form = x^qform);</span>
[ true ]
</pre></div>
<p>Note that the constructed bilinear form <span class="SimpleMath">\(f\)</span> is <strong class="button">not</strong> the associated bilinear form of the given quadratic form <span class="SimpleMath">\(Q\)</span>, according to the definition in Section <a href="chap3_mj.html#X864CAF8881067D8A"><span class="RefLink">3.2</span></a>. We can construct the associated bilinear forms by using <code class="func">AssociatedBilinearForm</code> (<a href="chap4_mj.html#X7BF7FBCA7FF91052"><span class="RefLink">4.4-3</span></a>). (See <a href="chap3_mj.html#X864CAF8881067D8A"><span class="RefLink">3.2</span></a> for more on quadratic forms).</p>
<p><a id="X7BF7FBCA7FF91052" name="X7BF7FBCA7FF91052"></a></p>
<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>
<p>Returns: a bilinear form</p>
<p>The argument <span class="SimpleMath">\(Q\)</span> must be a quadratic form (and thus it belongs to <code class="code">IsQuadraticForm</code>). The output is the associated bilinear form <span class="SimpleMath">\(f\)</span> (an object in <code class="code">IsBilinearForm</code>), as defined in Section <a href="chap3_mj.html#X864CAF8881067D8A"><span class="RefLink">3.2</span></a>, i.e. the bilinear form <span class="SimpleMath">\(f\)</span> such that <span class="SimpleMath">\(f(v,w) = Q(v+w)-Q(v)-Q(w)\)</span> for all vectors <span class="SimpleMath">\(v,w\)</span> in a vector space equipped with <span class="SimpleMath">\(Q\)</span>. (See <a href="chap3_mj.html#X864CAF8881067D8A"><span class="RefLink">3.2</span></a> for more on quadratic forms).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">r:= PolynomialRing(GF(121),6);</span>
GF(11^2)[x_1,x_2,x_3,x_4,x_5,x_6]
<span class="GAPprompt">gap></span> <span class="GAPinput">poly := r.1*r.5-r.2*r.6+r.3*r.4;</span>
x_1*x_5-x_2*x_6+x_3*x_4
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByPolynomial(poly,r);</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">aform := AssociatedBilinearForm(form);</span>
< bilinear form >
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(aform);</span>
Bilinear form
Gram Matrix:
. . . . 1 .
. . . . . 10
. . . 1 . .
. . 1 . . .
1 . . . . .
. 10 . . . .
</pre></div>
<p><a id="X8110213A7B303D1C" name="X8110213A7B303D1C"></a></p>
<h4>4.5 <span class="Heading">Evaluating forms</span></h4>
<p><a id="X808AB7B9840ABC27" name="X808AB7B9840ABC27"></a></p>
<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>
<p>Returns: a finite field element</p>
<p>The argument <var class="Arg">f</var> is either a sesquilinear or quadratic form defined over a finite field <span class="SimpleMath">\(\mathrm{GF}(q)\)</span>. 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 <span class="SimpleMath">\(f\)</span> is evaluated in. This operation evaluates the form on the given vector or pair of vectors and returns an element in <span class="SimpleMath">\(GF(q)\)</span>. There is also an overloading of the operation <var class="Arg">\^</var> where <code class="file">(u,v)^f</code> represents <span class="SimpleMath">\(f(u,v)\)</span> in the case that <var class="Arg">f</var> is sesquilinear, and <code class="file">u^f</code> stands for <span class="SimpleMath">\(f(u)\)</span> in the quadratic case. So for convenience, the user may use this compressed version of this operation, which we show in the following example:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[Z(8),0,0,0],[0,0,Z(8)^4,0],[0,0,0,1],[0,0,0,0]]*Z(8)^0;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByMatrix(mat,GF(8));</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">u := [ Z(2^3)^4, Z(2^3)^4, Z(2)^0, Z(2^3)^3 ];</span>
[ Z(2^3)^4, Z(2^3)^4, Z(2)^0, Z(2^3)^3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">EvaluateForm( form, u );</span>
Z(2^3)^6
<span class="GAPprompt">gap></span> <span class="GAPinput">u^form;</span>
Z(2^3)^6
<span class="GAPprompt">gap></span> <span class="GAPinput">gram := [[0,0,0,0,0,2],[0,0,0,0,2,0],[0,0,0,1,0,0],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [0,0,1,0,0,0],[0,2,0,0,0,0],[2,0,0,0,0,0]]*Z(3)^0;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByMatrix(gram,GF(3));</span>
< bilinear form >
<span class="GAPprompt">gap></span> <span class="GAPinput">u := [ [ Z(3)^0, 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0*Z(3), 0*Z(3), Z(3)^0, Z(3)^0, Z(3), 0*Z(3) ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">v := [ [ Z(3)^0, 0*Z(3), Z(3)^0, Z(3), 0*Z(3), Z(3) ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0*Z(3), Z(3)^0, 0*Z(3), Z(3), Z(3), Z(3) ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">EvaluateForm( form, u, v);</span>
[ [ Z(3)^0, Z(3)^0 ], [ 0*Z(3), 0*Z(3) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">[u,v]^form;</span>
[ [ Z(3)^0, Z(3)^0 ], [ 0*Z(3), 0*Z(3) ] ]
</pre></div>
<p><a id="X7A0825A987C88978" name="X7A0825A987C88978"></a></p>
<h4>4.6 <span class="Heading">Orthogonality, totally isotropic subspaces, and totally singular subspaces</span></h4>
<p><a id="X7D699C077F66F3E6" name="X7D699C077F66F3E6"></a></p>
<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>
<p>Returns: a base of the subspace orthogonal to the given vector or subspace with relation to the given form</p>
<p>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></p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[0,0,0,-2],[0,0,-3,0],[0,3,0,0],[2,0,0,0]]*Z(7)^0;</span>
[ [ 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) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByMatrix(mat);</span>
< bilinear form >
<span class="GAPprompt">gap></span> <span class="GAPinput">v := [0*Z(7),Z(7)^0,Z(7)^3,Z(7)^5];</span>
[ 0*Z(7), Z(7)^0, Z(7)^3, Z(7)^5 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">vperp := OrthogonalSubspaceMat(form,v);</span>
[ [ 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 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(vperp,x->[x,v]^form);</span>
[ 0*Z(7), 0*Z(7), 0*Z(7) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">sub := [[1,1,0,0],[0,0,1,2]]*Z(7)^0;</span>
[ [ Z(7)^0, Z(7)^0, 0*Z(7), 0*Z(7) ], [ 0*Z(7), 0*Z(7), Z(7)^0, Z(7)^2 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">subperp := OrthogonalSubspaceMat(form,sub);</span>
[ [ Z(7)^0, Z(7)^0, 0*Z(7), 0*Z(7) ], [ 0*Z(7), 0*Z(7), Z(7)^4, Z(7)^0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(subperp,x->List(sub,y->[x,y]^form));</span>
[ [ 0*Z(7), 0*Z(7) ], [ 0*Z(7), 0*Z(7) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[1,0,0],[0,0,1],[0,0,0]]*Z(2)^0;</span>
[ [ 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) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByMatrix(mat);</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">v := [Z(2)^0,Z(2)^0,0*Z(2)];</span>
[ Z(2)^0, Z(2)^0, 0*Z(2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">vperp := OrthogonalSubspaceMat(form,v);</span>
[ <an immutable GF2 vector of length 3>, <an immutable GF2 vector of length
3> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">bil_form := AssociatedBilinearForm(form);</span>
< bilinear form >
<span class="GAPprompt">gap></span> <span class="GAPinput">List(vperp,x->[x,v]^bil_form);</span>
[ 0*Z(2), 0*Z(2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">sub := [[1,0,1],[1,0,0]]*Z(2)^0;</span>
[ [ Z(2)^0, 0*Z(2), Z(2)^0 ], [ Z(2)^0, 0*Z(2), 0*Z(2) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">subperp := OrthogonalSubspaceMat(form,sub);</span>
[ <an immutable GF2 vector of length 3>, <an immutable GF2 vector of length
3> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(subperp,x->List(sub,y->[x,y]^bil_form));</span>
[ [ 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2) ] ]
</pre></div>
<p><a id="X8394CCAD798053C6" name="X8394CCAD798053C6"></a></p>
<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>
<p>Returns: true or false</p>
<p>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>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[1,0,0,0],[0,-1,0,0],[0,0,0,1],[0,0,1,0]]*Z(41)^0;</span>
[ [ 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) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByMatrix(mat);</span>
< bilinear form >
<span class="GAPprompt">gap></span> <span class="GAPinput">v := [1,1,0,0]*Z(41)^0;</span>
[ Z(41)^0, Z(41)^0, 0*Z(41), 0*Z(41) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIsotropicVector(form,v);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">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;</span>
[ [ 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) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByMatrix(mat);</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">v1 := [1,0,0,0,0]*Z(8)^0;</span>
[ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">v2 := [0,1,0,0,0]*Z(8)^0;</span>
[ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIsotropicVector(form,v1);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIsotropicVector(form,v2);</span>
true
</pre></div>
<p><a id="X855E539185D7D3C7" name="X855E539185D7D3C7"></a></p>
<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>
<p>Returns: true or false</p>
<p>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.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">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;</span>
[ [ 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) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByMatrix(mat);</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">v1 := [1,0,0,0,0]*Z(8)^0;</span>
[ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">v2 := [0,1,0,0,0]*Z(8)^0;</span>
[ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSingularVector(form,v1);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSingularVector(form,v2);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIsotropicVector(form,v1);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIsotropicVector(form,v2);</span>
true
</pre></div>
<p><a id="X8141325085AAC0CD" name="X8141325085AAC0CD"></a></p>
<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>
<p>Returns: true or false</p>
<p>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>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[1,0,0,0],[0,-1,0,0],[0,0,0,1],[0,0,1,0]]*Z(7)^0;</span>
[ [ 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) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByMatrix(mat);</span>
< bilinear form >
<span class="GAPprompt">gap></span> <span class="GAPinput">sub:= [[Z(7)^0,0*Z(7),Z(7)^0,Z(7)],[0*Z(7),Z(7)^0,Z(7)^0,Z(7)^4]];</span>
[ [ Z(7)^0, 0*Z(7), Z(7)^0, Z(7) ], [ 0*Z(7), Z(7)^0, Z(7)^0, Z(7)^4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTotallyIsotropicSubspace(form,sub);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := IdentityMat(6,GF(2));</span>
[ <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> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := HermitianFormByMatrix(mat,GF(4));</span>
< hermitian form >
<span class="GAPprompt">gap></span> <span class="GAPinput">sub := [[Z(2)^0,0*Z(2),0*Z(2),Z(2)^0,Z(2)^0,Z(2)^0], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [0*Z(2),Z(2)^0,0*Z(2),Z(2^2)^2,Z(2^2),Z(2)^0], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [0*Z(2),0*Z(2),Z(2)^0,Z(2)^0,Z(2^2),Z(2^2)^2]];</span>
[ [ 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 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTotallyIsotropicSubspace(form,sub);</span>
true
</pre></div>
<p><a id="X834FD9117F1DA8D0" name="X834FD9117F1DA8D0"></a></p>
<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>
<p>Returns: true or false</p>
<p>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 <span class="SimpleMath">\(n\)</span> with relation to a quadratic form are exactly the totally isotropic subspaces of dimension <span class="SimpleMath">\(n\)</span> with relation to its associated form.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">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;</span>
[ [ 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) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByMatrix(mat);</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">sub := [[Z(2)^0,0*Z(2),Z(2^3)^6,Z(2^3),Z(2^3)^3],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [0*Z(2),Z(2)^0,Z(2^3)^6,Z(2^3)^2,Z(2^3)]];</span>
[ [ 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) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTotallySingularSubspace(form,sub);</span>
true
</pre></div>
<p><a id="X813A02878352E9E5" name="X813A02878352E9E5"></a></p>
<h4>4.7 <span class="Heading">Attributes and properties of forms</span></h4>
<p><a id="X80254BFD7E4B8F06" name="X80254BFD7E4B8F06"></a></p>
<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>
<p>Returns: true or false.</p>
<p>A sesquilinear form <span class="SimpleMath">\(f\)</span> on a vector space <span class="SimpleMath">\(V\)</span> is <em>reflexive</em> if <span class="SimpleMath">\(f(v,w)=0 \Rightarrow f(w,v)=0\)</span> for all <span class="SimpleMath">\(v,w \in V\)</span>. The argument <span class="SimpleMath">\(f\)</span> must be a sesquilinear form (and thus it belongs to <code class="code">IsSesquilinearForm</code>). A sesquilinear form <span class="SimpleMath">\(f\)</span> is <em>reflexive</em> if whenever we have <span class="SimpleMath">\(f(u,v)=0\)</span>, for two vectors <span class="SimpleMath">\(u,v\)</span> in the associated vector space, then we also have <span class="SimpleMath">\(f(v,u)=0\)</span>. 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 <strong class="pkg">Forms</strong> to construct non-reflexive forms. (See <a href="chap3_mj.html#X874CD5E0802FEB50"><span class="RefLink">3.1</span></a> for more on reflexive sesquilinear forms).</p>
<p><a id="X7D5AB7E484CFBF63" name="X7D5AB7E484CFBF63"></a></p>
<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>
<p>Returns: true or false.</p>
<p>A sesquilinear form <span class="SimpleMath">\(f\)</span> on a vector space <span class="SimpleMath">\(V\)</span> is <em>alternating</em> if <span class="SimpleMath">\(f(v,v)=0\)</span> for all <span class="SimpleMath">\(v \in V\)</span>. The argument <span class="SimpleMath">\(f\)</span> must be a sesquilinear form (and thus it belongs to <code class="code">IsSesquilinearForm</code>). A bilinear form <span class="SimpleMath">\(f\)</span> is <em>alternating</em> if <span class="SimpleMath">\(f(v,v)=0\)</span> for all <span class="SimpleMath">\(v\)</span>. 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"><span class="RefLink">3.1</span></a> for more on alternating sesquilinear forms).</p>
<p><a id="X85585B2C80413490" name="X85585B2C80413490"></a></p>
<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>
<p>Returns: true or false.</p>
<p>A sesquilinear form <span class="SimpleMath">\(f\)</span> on a vector space <span class="SimpleMath">\(V\)</span> is <em>symmetric</em> if <span class="SimpleMath">\(f(v,w)=f(w,v)\)</span> for all <span class="SimpleMath">\(v,w \in V\)</span>. The argument <span class="SimpleMath">\(f\)</span> must be a sesquilinear form (and thus it belongs to <code class="code">IsSesquilinearForm</code>). A bilinear form <span class="SimpleMath">\(f\)</span> is <em>symmetric</em> if <span class="SimpleMath">\(f(u,v)=f(v,u)\)</span> for all pairs of vectors <span class="SimpleMath">\(u\)</span> and <span class="SimpleMath">\(v\)</span>. 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"><span class="RefLink">3.1</span></a> for more on symmetric sesquilinear forms).</p>
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<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>
<p>Returns: true or false.</p>
<p>The argument <span class="SimpleMath">\(f\)</span> must be a sesquilinear form (and thus it belongs to <code class="code">IsSesquilinearForm</code>). A bilinear form <span class="SimpleMath">\(f\)</span> is called <em>orthogonal</em> if the characteristic of the underlying field is odd, and <span class="SimpleMath">\(f\)</span> is a symmetric form. (See <a href="chap3_mj.html#X874CD | |
