[1m[4m[31mGAP 4 Package [1mForms[1m[4m[31m[0m [1m[4m[31mSesquilinear and Quadratic[0m 1.0 May 2007 John Bamberg Jan De Beule John Bamberg Email: [34mmailto:bamberg@cage.ugent.be[0m Homepage: [34mhttp://cage.ugent.be/~bamberg[0m Address: Department of Pure Mathematics, Ghent University, Galglaan 2, 9000 Ghent, Belgium Jan De Beule Email: [34mmailto:jdebeule@cage.ugent.be[0m Homepage: [34mhttp://cage.ugent.be/~jdebeule[0m Address: Department of Pure Mathematics, Ghent University, Galglaan 2, 9000 Ghent, Belgium ------------------------------------------------------- [1m[4m[31mCopyright[0m (C) 2007 by the authors This package may be distributed under the terms and conditions of the GNU Public License Version 2 or higher. ------------------------------------------------------- [1m[4m[31mContent (Forms)[0m 1. Introduction 1.1 Philosophy 1.2 Overview over this manual 2. Examples 2.1 A conic of PG(2,8) 2.2 A form for W(5,3) 3. Background Theory on Forms 3.1 Sesquilinear forms, dualities, and polarities 3.1-1 Example 3.2 Quadratic forms 3.2-1 Example 3.3 Morphisms of forms 3.4 An important convention 3.4-1 Example 3.5 Canonical forms 4. Functionality 4.1 Functions for creating forms 4.1-1 BilinearFormByMatrix 4.1-2 QuadraticFormByMatrix 4.1-3 HermitianFormByMatrix 4.1-4 BilinearFormByPolynomial 4.1-5 QuadraticFormByPolynomial 4.1-6 HermitianFormByPolynomial 4.2 Attributes and properties of forms 4.2-1 IsReflexiveForm 4.2-2 IsAlternatingForm 4.2-3 IsSymmetricForm 4.2-4 IsDegenerateForm 4.2-5 BaseField 4.2-6 GramMatrix 4.2-7 WittIndex 4.2-8 RadicalOfForm 4.2-9 PolynomialOfForm 4.2-10 DiscriminantOfForm 4.3 Functions for changing forms 4.3-1 BaseChangeToCanonical 4.3-2 IsometricCanonicalForm 4.4 Operations on forms 4.4-1 BaseChangeHomomorphism 4.4-2 EvaluateForm -------------------------------------------------------