  
  
                              [1XGAP 4 Package [5XForms[0m[1X[0m
  
  
                           [1XSesquilinear and Quadratic[0m
  
  
                                     1.2.1
  
  
                                 November 2009
  
  
                                  John Bamberg
  
                                  Jan De Beule
  
  
  
  John Bamberg
      Email:    [7Xmailto:bamberg@maths.uwa.edu.au[0m
      Homepage: [7Xhttp://school.maths.uwa.edu.au/portal_memberdata/bamberg[0m
      Address:  School  of  Mathematics  and  Statistics,  The  University  of
                Western   Australia,  35  Stirling  Highway,  Crawley,  Perth,
                Western Australia 6009
  
  
  Jan De Beule
      Email:    [7Xmailto:jdebeule@cage.ugent.be[0m
      Homepage: [7Xhttp://cage.ugent.be/~jdebeule[0m
      Address:  Department  of  Pure  Mathematics  and Computer Algebra, Ghent
                University, Krijgslaan 281 -- S22, 9000 Ghent, Belgium
  
  
  
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  [1XCopyright[0m
  © 2009 by the authors
  
  This  package  may  be distributed under the terms and conditions of the GNU
  Public License Version 2 or higher.
  
  
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  [1XContents (Forms)[0X
  
  1 Introduction
    1.1 Philosophy
    1.2 Overview over this manual
    1.3 How to read this manual
    1.4 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 sesquilinear forms preserved by a classical group
      4.8-1 PreservedSesquilinearForms
    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
    5.3 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
  
  
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