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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chap5_mj.html">5</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

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<h1>Forms</h1>


<h2>Sesquilinear and Quadratic</h2>

<p>
    1.2.13</p>

<p>
    5 May 2025
  </p>

</div>
<p><b>
    John Bamberg




  </b>
<br />Email: <span class="URL"><a href="mailto:bamberg@maths.uwa.edu.au">bamberg@maths.uwa.edu.au</a></span>
<br />Homepage: <span class="URL"><a href="http://school.maths.uwa.edu.au/~bamberg/">http://school.maths.uwa.edu.au/~bamberg/</a></span>
<br />Address: <br />School of Mathematics and Statistics<br /> The University of Western Australia<br /> 35 Stirling Highway<br /> Crawley WA 6009, Perth<br /> Australia<br />
</p><p><b>
    Jan De Beule




  </b>
<br />Email: <span class="URL"><a href="mailto:jan@debeule.eu">jan@debeule.eu</a></span>
<br />Homepage: <span class="URL"><a href="http://www.debeule.eu">http://www.debeule.eu</a></span>
<br />Address: <br />Department of Mathematics and Data Science<br /> Vrije Universiteit Brussel<br /> Pleinlaan 2<br /> B-1050 Brussel<br /> Belgium<br />
</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2015-2024 by the authors</p>

<p>This package may be distributed under the terms and conditions of the GNU Public License Version 2 or (at your option) any later version.</p>

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<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1_mj.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X873C99678745ABAF">1.1 <span class="Heading">Philosophy</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X786BACDB82918A65">1.2 <span class="Heading">Overview over this manual</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X8416D2657E7831A1">1.3 <span class="Heading">How to read this manual</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7B1A58BA78CC28FF">1.4 <span class="Heading">Web resources</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7FFDC142827888CA">1.5 <span class="Heading">Release notes</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2_mj.html#X7A489A5D79DA9E5C">2 <span class="Heading">Examples</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X83510E647FBB2475">2.1 <span class="Heading">A conic of 
<span class="SimpleMath">\(\mathrm{PG}(2,8)\)</span>
</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X781F69578636E8C5">2.2 <span class="Heading">A form for 
<span class="SimpleMath">\(\mathrm{W}(5,3)\)</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X78638D21797AC9A0">2.3 <span class="Heading">What is the form preserved by this group?</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap3_mj.html#X79424B627CE11FCA">3 <span class="Heading">Background Theory on Forms</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X874CD5E0802FEB50">3.1 <span class="Heading">Sesquilinear forms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A489A5D79DA9E5C">3.1-1 <span class="Heading">Examples</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X864CAF8881067D8A">3.2 <span class="Heading">Quadratic forms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A489A5D79DA9E5C">3.2-1 <span class="Heading">Examples</span></a>
</span>
</div></div>
</div>
<div class="ContChap"><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>
<div class="ContChap"><a href="chap5_mj.html#X7B9AF2E784EB8481">5 <span class="Heading">Morphisms of forms</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X784D3B338055EC9D">5.1 <span class="Heading">Morphisms of sesquilinear forms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X807E16A383D2E04C">5.1-1 <span class="Heading">Hermitian forms</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8042784984331FF4">5.1-2 <span class="Heading">Alternating forms</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7F1255F77B6874E3">5.1-3 <span class="Heading">Bilinear forms</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X79453E2B7DDE1412">5.1-4 <span class="Heading">Degenerate forms</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X87C0B98C8669A34A">5.2 <span class="Heading">Morphisms of quadratic forms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7C738FBB80F533AC">5.2-1 <span class="Heading">Singular forms</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X790B24568376AACE">5.3 <span class="Heading">Operations based on morphisms of forms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X78CCFB957A6153F5">5.3-1 BaseChangeToCanonical</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X87A6F5C979551677">5.3-2 BaseChangeHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7DFEFA2C7945A5AD">5.3-3 IsometricCanonicalForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7C7D92267EFE71DB">5.3-4 ScalarOfSimilarity</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X85FA387280DAEA69">5.3-5 WittIndex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X853AF8D97E00F1DB">5.3-6 IsEllipticForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7B73832A786FEC21">5.3-7 IsParabolicForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X85551B28798B07C7">5.3-8 IsHyperbolicForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X85F7092783AA2968">5.3-9 TypeOfForm</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapBib_mj.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd_mj.html"><span class="Heading">Index</span></a></div>
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