  
  
                                    [1X Forms [101X
  
  
                          [1X Sesquilinear and Quadratic [101X
  
  
                                     1.2.9
  
  
                                14 October 2022
  
  
                                  John Bamberg
  
                                  Jan De Beule
  
  
  
  John Bamberg
      Email:    [7Xmailto:bamberg@maths.uwa.edu.au[107X
      Homepage: [7Xhttp://school.maths.uwa.edu.au/~bamberg/[107X
      Address:  [33X[0;14YSchool of Mathematics and Statistics[133X
                [33X[0;14YThe University of Western Australia[133X
                [33X[0;14Y35 Stirling Highway[133X
                [33X[0;14YCrawleyY WA 6009, Perth[133X
                [33X[0;14YAustralia[133X
  
  
  Jan De Beule
      Email:    [7Xmailto:jan@debeule.eu[107X
      Homepage: [7Xhttp://www.debeule.eu[107X
      Address:  [33X[0;14YDepartment of Mathematics and Data Science[133X
                [33X[0;14YVrije Universiteit Brussel[133X
                [33X[0;14YPleinlaan 2[133X
                [33X[0;14YB-1050 Brussel[133X
                [33X[0;14YBelgium[133X
  
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2015-2022 by the authors[133X
  
  [33X[0;0YThis  package  may  be distributed under the terms and conditions of the GNU
  Public License Version 2 or (at your option) any later version.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (Forms)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YPhilosophy[133X
    1.2 [33X[0;0YOverview over this manual[133X
    1.3 [33X[0;0YHow to read this manual[133X
    1.4 [33X[0;0YWeb resources[133X
    1.5 [33X[0;0YRelease notes[133X
  2 [33X[0;0YExamples[133X
    2.1 [33X[0;0YA conic of [22XPG(2,8)[122X[133X
    2.2 [33X[0;0YA form for [22XW(5,3)[122X[133X
    2.3 [33X[0;0YWhat is the form preserved by this group?[133X
  3 [33X[0;0YBackground Theory on Forms[133X
    3.1 [33X[0;0YSesquilinear forms[133X
      3.1-1 [33X[0;0YExamples[133X
    3.2 [33X[0;0YQuadratic forms[133X
      3.2-1 [33X[0;0YExamples[133X
  4 [33X[0;0YConstructing forms and basic functionality[133X
    4.1 [33X[0;0YImportant filters[133X
      4.1-1 [33X[0;0YCategories for forms[133X
      4.1-2 [33X[0;0YRepresentation for forms[133X
    4.2 [33X[0;0YConstructing forms using a matrix[133X
      4.2-1 BilinearFormByMatrix
      4.2-2 QuadraticFormByMatrix
      4.2-3 HermitianFormByMatrix
    4.3 [33X[0;0YConstructing forms using a polynomial[133X
      4.3-1 BilinearFormByPolynomial
      4.3-2 QuadraticFormByPolynomial
      4.3-3 HermitianFormByPolynomial
    4.4 [33X[0;0YSwitching between bilinear and quadratic forms[133X
      4.4-1 QuadraticFormByBilinearForm
      4.4-2 BilinearFormByQuadraticForm
      4.4-3 AssociatedBilinearForm
    4.5 [33X[0;0YEvaluating forms[133X
      4.5-1 EvaluateForm
    4.6 [33X[0;0YOrthogonality, totally isotropic subspaces, and totally singular
    subspaces[133X
      4.6-1 OrthogonalSubspaceMat
      4.6-2 IsIsotropicVector
      4.6-3 IsSingularVector
      4.6-4 IsTotallyIsotropicSubspace
      4.6-5 IsTotallySingularSubspace
    4.7 [33X[0;0YAttributes and properties of forms[133X
      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 [33X[0;0YRecognition of forms preserved by a classical group[133X
      4.8-1 PreservedForms
      4.8-2 PreservedSesquilinearForms
      4.8-3 PreservedQuadraticForms
    4.9 [33X[0;0YThe trivial form and some of its properties[133X
  5 [33X[0;0YMorphisms of forms[133X
    5.1 [33X[0;0YMorphisms of sesquilinear forms[133X
      5.1-1 [33X[0;0YHermitian forms[133X
      5.1-2 [33X[0;0YAlternating forms[133X
      5.1-3 [33X[0;0YBilinear forms[133X
      5.1-4 [33X[0;0YDegenerate forms[133X
    5.2 [33X[0;0YMorphisms of quadratic forms[133X
      5.2-1 [33X[0;0YSingular forms[133X
    5.3 [33X[0;0YOperations based on morphisms of forms[133X
      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
  
  
  [32X
