Forms

Sesquilinear and Quadratic

1.2.12

30 August 2024

Jan De Beule
Email: jan@debeule.eu
Homepage: http://www.debeule.eu
Address:
Department of Mathematics and Data Science
Vrije Universiteit Brussel
Pleinlaan 2
B-1050 Brussel
Belgium

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

1 Introduction

1.1 Philosophy

1.2 Overview over this manual

1.3 How to read this manual

1.4 Web resources

1.5 Release notes

2 Examples

2.1 A conic of PG(2,8)

2.2 A form for W(5,3)

2.3 What is the form preserved by this group?

3 Background Theory on Forms

3.1 Sesquilinear forms

3.1-1 Examples

3.2 Quadratic forms

3.2-1 Examples

4 Constructing forms and basic functionality

4.1 Important filters

4.1-1 Categories for forms
4.1-2 Representation for forms

4.2 Constructing forms using a matrix

  4.2-1 BilinearFormByMatrix
  4.2-2 QuadraticFormByMatrix
  4.2-3 HermitianFormByMatrix

4.3 Constructing forms using a polynomial

  4.3-1 BilinearFormByPolynomial
  4.3-2 QuadraticFormByPolynomial
  4.3-3 HermitianFormByPolynomial

4.4 Switching between bilinear and quadratic forms

  4.4-1 QuadraticFormByBilinearForm
  4.4-2 BilinearFormByQuadraticForm
  4.4-3 AssociatedBilinearForm

4.5 Evaluating forms

4.5-1 EvaluateForm

4.6 Orthogonality, totally isotropic subspaces, and totally singular subspaces

  4.6-1 OrthogonalSubspaceMat
  4.6-2 IsIsotropicVector
  4.6-3 IsSingularVector
  4.6-4 IsTotallyIsotropicSubspace
  4.6-5 IsTotallySingularSubspace

4.7 Attributes and properties of forms

  4.7-1 IsReflexiveForm
  4.7-2 IsAlternatingForm
  4.7-3 IsSymmetricForm
  4.7-4 IsOrthogonalForm
  4.7-5 IsPseudoForm
  4.7-6 IsSymplecticForm
  4.7-7 IsDegenerateForm
  4.7-8 IsSingularForm
  4.7-9 BaseField
  4.7-10 GramMatrix
  4.7-11 RadicalOfForm
  4.7-12 PolynomialOfForm
  4.7-13 DiscriminantOfForm

4.8 Recognition of forms preserved by a classical group

  4.8-1 PreservedForms
  4.8-2 PreservedSesquilinearForms
  4.8-3 PreservedQuadraticForms

4.9 The trivial form and some of its properties

5 Morphisms of forms

5.1 Morphisms of sesquilinear forms

  5.1-1 Hermitian forms
  5.1-2 Alternating forms
  5.1-3 Bilinear forms
  5.1-4 Degenerate forms

5.2 Morphisms of quadratic forms

5.2-1 Singular forms

5.3 Operations based on morphisms of forms

  5.3-1 BaseChangeToCanonical
  5.3-2 BaseChangeHomomorphism
  5.3-3 IsometricCanonicalForm
  5.3-4 ScalarOfSimilarity
  5.3-5 WittIndex
  5.3-6 IsEllipticForm
  5.3-7 IsParabolicForm
  5.3-8 IsHyperbolicForm
  5.3-9 TypeOfForm

References

Index

Forms

Sesquilinear and Quadratic

Copyright

Contents