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# Forms

## Sesquilinear and Quadratic

1.2.9

14 October 2022

John Bamberg
Email: bamberg@maths.uwa.edu.au
Homepage: http://school.maths.uwa.edu.au/~bamberg/
Address:
School of Mathematics and Statistics
The University of Western Australia
35 Stirling Highway
CrawleyY WA 6009, Perth
Australia

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

### Copyright

© 2015-2022 by the authors

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

### Contents

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 \(\mathrm{PG}(2,8)\)
2.2 A form for \(\mathrm{W}(5,3)\)
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
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