\contentsline {chapter}{\numberline {1}\leavevmode {\color {Chapter }Introduction}}{5}{chapter.1}
\contentsline {section}{\numberline {1.1}\leavevmode {\color {Chapter }Philosophy}}{5}{section.1.1}
\contentsline {section}{\numberline {1.2}\leavevmode {\color {Chapter }Overview over this manual}}{5}{section.1.2}
\contentsline {section}{\numberline {1.3}\leavevmode {\color {Chapter }How to read this manual}}{5}{section.1.3}
\contentsline {chapter}{\numberline {2}\leavevmode {\color {Chapter }Examples}}{6}{chapter.2}
\contentsline {section}{\numberline {2.1}\leavevmode {\color {Chapter }A conic of PG(2,8)}}{6}{section.2.1}
\contentsline {section}{\numberline {2.2}\leavevmode {\color {Chapter }A form for W(5,3)}}{7}{section.2.2}
\contentsline {section}{\numberline {2.3}\leavevmode {\color {Chapter }What is the form preserved by this group?}}{9}{section.2.3}
\contentsline {chapter}{\numberline {3}\leavevmode {\color {Chapter }Background Theory on Forms}}{10}{chapter.3}
\contentsline {section}{\numberline {3.1}\leavevmode {\color {Chapter }Sesquilinear forms}}{10}{section.3.1}
\contentsline {subsection}{\numberline {3.1.1}\leavevmode {\color {Chapter }Examples}}{11}{subsection.3.1.1}
\contentsline {section}{\numberline {3.2}\leavevmode {\color {Chapter }Quadratic forms}}{15}{section.3.2}
\contentsline {subsection}{\numberline {3.2.1}\leavevmode {\color {Chapter }Examples}}{16}{subsection.3.2.1}
\contentsline {chapter}{\numberline {4}\leavevmode {\color {Chapter }Constructing forms and basic functionality}}{20}{chapter.4}
\contentsline {section}{\numberline {4.1}\leavevmode {\color {Chapter }Important filters}}{20}{section.4.1}
\contentsline {subsection}{\numberline {4.1.1}\leavevmode {\color {Chapter }Categories for forms}}{20}{subsection.4.1.1}
\contentsline {subsection}{\numberline {4.1.2}\leavevmode {\color {Chapter }Representation for forms}}{20}{subsection.4.1.2}
\contentsline {section}{\numberline {4.2}\leavevmode {\color {Chapter }Constructing forms using a matrix}}{21}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}\leavevmode {\color {Chapter }BilinearFormByMatrix}}{21}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}\leavevmode {\color {Chapter }QuadraticFormByMatrix}}{22}{subsection.4.2.2}
\contentsline {subsection}{\numberline {4.2.3}\leavevmode {\color {Chapter }HermitianFormByMatrix}}{23}{subsection.4.2.3}
\contentsline {section}{\numberline {4.3}\leavevmode {\color {Chapter }Constructing forms using a polynomial}}{24}{section.4.3}
\contentsline {subsection}{\numberline {4.3.1}\leavevmode {\color {Chapter }BilinearFormByPolynomial}}{24}{subsection.4.3.1}
\contentsline {subsection}{\numberline {4.3.2}\leavevmode {\color {Chapter }QuadraticFormByPolynomial}}{25}{subsection.4.3.2}
\contentsline {subsection}{\numberline {4.3.3}\leavevmode {\color {Chapter }HermitianFormByPolynomial}}{26}{subsection.4.3.3}
\contentsline {section}{\numberline {4.4}\leavevmode {\color {Chapter }Switching between bilinear and quadratic forms}}{26}{section.4.4}
\contentsline {subsection}{\numberline {4.4.1}\leavevmode {\color {Chapter }QuadraticFormByBilinearForm}}{26}{subsection.4.4.1}
\contentsline {subsection}{\numberline {4.4.2}\leavevmode {\color {Chapter }BilinearFormByQuadraticForm}}{28}{subsection.4.4.2}
\contentsline {subsection}{\numberline {4.4.3}\leavevmode {\color {Chapter }AssociatedBilinearForm}}{28}{subsection.4.4.3}
\contentsline {section}{\numberline {4.5}\leavevmode {\color {Chapter }Evaluating forms}}{29}{section.4.5}
\contentsline {subsection}{\numberline {4.5.1}\leavevmode {\color {Chapter }EvaluateForm}}{29}{subsection.4.5.1}
\contentsline {section}{\numberline {4.6}\leavevmode {\color {Chapter }Orthogonality, totally isotropic subspaces, and totally singular subspaces}}{30}{section.4.6}
\contentsline {subsection}{\numberline {4.6.1}\leavevmode {\color {Chapter }OrthogonalSubspaceMat}}{30}{subsection.4.6.1}
\contentsline {subsection}{\numberline {4.6.2}\leavevmode {\color {Chapter }IsIsotropicVector}}{31}{subsection.4.6.2}
\contentsline {subsection}{\numberline {4.6.3}\leavevmode {\color {Chapter }IsSingularVector}}{31}{subsection.4.6.3}
\contentsline {subsection}{\numberline {4.6.4}\leavevmode {\color {Chapter }IsTotallyIsotropicSubspace}}{32}{subsection.4.6.4}
\contentsline {subsection}{\numberline {4.6.5}\leavevmode {\color {Chapter }IsTotallySingularSubspace}}{33}{subsection.4.6.5}
\contentsline {section}{\numberline {4.7}\leavevmode {\color {Chapter }Attributes and properties of forms}}{33}{section.4.7}
\contentsline {subsection}{\numberline {4.7.1}\leavevmode {\color {Chapter }IsReflexiveForm}}{33}{subsection.4.7.1}
\contentsline {subsection}{\numberline {4.7.2}\leavevmode {\color {Chapter }IsAlternatingForm}}{33}{subsection.4.7.2}
\contentsline {subsection}{\numberline {4.7.3}\leavevmode {\color {Chapter }IsSymmetricForm}}{34}{subsection.4.7.3}
\contentsline {subsection}{\numberline {4.7.4}\leavevmode {\color {Chapter }IsOrthogonalForm}}{34}{subsection.4.7.4}
\contentsline {subsection}{\numberline {4.7.5}\leavevmode {\color {Chapter }IsPseudoForm}}{34}{subsection.4.7.5}
\contentsline {subsection}{\numberline {4.7.6}\leavevmode {\color {Chapter }IsSymplecticForm}}{34}{subsection.4.7.6}
\contentsline {subsection}{\numberline {4.7.7}\leavevmode {\color {Chapter }IsDegenerateForm}}{34}{subsection.4.7.7}
\contentsline {subsection}{\numberline {4.7.8}\leavevmode {\color {Chapter }IsSingularForm}}{35}{subsection.4.7.8}
\contentsline {subsection}{\numberline {4.7.9}\leavevmode {\color {Chapter }BaseField}}{35}{subsection.4.7.9}
\contentsline {subsection}{\numberline {4.7.10}\leavevmode {\color {Chapter }GramMatrix}}{35}{subsection.4.7.10}
\contentsline {subsection}{\numberline {4.7.11}\leavevmode {\color {Chapter }RadicalOfForm}}{35}{subsection.4.7.11}
\contentsline {subsection}{\numberline {4.7.12}\leavevmode {\color {Chapter }PolynomialOfForm}}{36}{subsection.4.7.12}
\contentsline {subsection}{\numberline {4.7.13}\leavevmode {\color {Chapter }DiscriminantOfForm}}{36}{subsection.4.7.13}
\contentsline {section}{\numberline {4.8}\leavevmode {\color {Chapter }Recognition of sesquilinear forms preserved by a classical group}}{36}{section.4.8}
\contentsline {subsection}{\numberline {4.8.1}\leavevmode {\color {Chapter }PreservedSesquilinearForms}}{37}{subsection.4.8.1}
\contentsline {chapter}{\numberline {5}\leavevmode {\color {Chapter }Morphisms of forms}}{38}{chapter.5}
\contentsline {section}{\numberline {5.1}\leavevmode {\color {Chapter }Morphisms of sesquilinear forms}}{38}{section.5.1}
\contentsline {subsection}{\numberline {5.1.1}\leavevmode {\color {Chapter }Hermitian forms}}{39}{subsection.5.1.1}
\contentsline {subsection}{\numberline {5.1.2}\leavevmode {\color {Chapter }Alternating forms}}{39}{subsection.5.1.2}
\contentsline {subsection}{\numberline {5.1.3}\leavevmode {\color {Chapter }Bilinear forms}}{39}{subsection.5.1.3}
\contentsline {subsection}{\numberline {5.1.4}\leavevmode {\color {Chapter }Degenerate forms}}{41}{subsection.5.1.4}
\contentsline {section}{\numberline {5.2}\leavevmode {\color {Chapter }Morphisms of quadratic forms}}{41}{section.5.2}
\contentsline {section}{\numberline {5.3}\leavevmode {\color {Chapter }Operations based on morphisms of forms}}{42}{section.5.3}
\contentsline {subsection}{\numberline {5.3.1}\leavevmode {\color {Chapter }BaseChangeToCanonical}}{42}{subsection.5.3.1}
\contentsline {subsection}{\numberline {5.3.2}\leavevmode {\color {Chapter }BaseChangeHomomorphism}}{43}{subsection.5.3.2}
\contentsline {subsection}{\numberline {5.3.3}\leavevmode {\color {Chapter }IsometricCanonicalForm}}{44}{subsection.5.3.3}
\contentsline {subsection}{\numberline {5.3.4}\leavevmode {\color {Chapter }ScalarOfSimilarity}}{44}{subsection.5.3.4}
\contentsline {subsection}{\numberline {5.3.5}\leavevmode {\color {Chapter }WittIndex}}{45}{subsection.5.3.5}
\contentsline {subsection}{\numberline {5.3.6}\leavevmode {\color {Chapter }IsEllipticForm}}{45}{subsection.5.3.6}
\contentsline {subsection}{\numberline {5.3.7}\leavevmode {\color {Chapter }IsParabolicForm}}{45}{subsection.5.3.7}
\contentsline {subsection}{\numberline {5.3.8}\leavevmode {\color {Chapter }IsHyperbolicForm}}{45}{subsection.5.3.8}
