Quelle  theory.xml   Sprache: XML

<!-- 

  theory.xml    Forms package documentation
                                                                   John Bamberg
                                                               and Jan De Beule
                                                           
  Copyright (C) 2021, Vrije Universiteit Brussel
  Copyright (C) 2021, The University of Western Australia

This is the chapter of the documentation describing background theory of forms and preliminary examples.
-->

<Chapter Label="theory">
<Heading>Background Theory on                                                               and Jan De Beule

Injava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
of =><> if <M>(w= Rightarrow)<>
texts: Cameron  Mv \ <M.  form EsymmetricE>Index=Form
Aschbacher< KeyAschbacher>orKleidman Liebeck C Key/java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91

<SectionM(+ww<M,  that alternating.the
<Heading>Sesquilinear  the   ,  form><M  also
Esesquilinear<E<ndexKeyF =""Form/java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
 vector spaceMV/>over  <>/>  a java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
Mf/  <VtimesV/> M<M   linearin first, 
semilinear<Index>Semilinear</Index>Alt=""<fv )   w{alphaT/>/java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
 <\</>!-< NotHTMLalpha>>AltOnlyHTML#4;Alt>
(the <E>companion automorphism</E><Index>Companion
Automorphism</Index> of <M>f</M>) such that
<Alt Only="Text"><Display>f(v,\lambda w)=\lambda^\alpha f(v,w),</Display></Alt>
<Alt Only="LaTeX"><Display>f(v,\lambda w)=\lambda^\alpha f(v,w),</Display></Alt>
<Alt Only="HTML MathJax"><Display>f(v,\lambda w)=\lambda^\alpha f(v,<P/>
<Alt Only="HTML noMathJax"><Display>f(v,_This     EGramE  M/.
for v,i <M and>lambda <M.
If <M>\alpha</M> is the identity, then <M>f</M> is <E>bilinear</E><Index
Key=" ""java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
/
Abilinear  Ereflexive/Index=Form
Subkey="reflexive">Form>
for <>w  <M.   is><E> =Form
Subkey="symmetric">Form</Index> if Alt=TextMM_f/M></AltAlt=LaTeX<>_<M<AltAlt=""<>_/>/>
. is  symmetricbilinear  reflexive   form
is <E>alternating</E><Index Key="Form" symmetric matrix, i.e., <Alt Only="Text"><M>M_fAltAlt =LaTeXMM_fTM_f<<Alt
<f,v)=</>for <Mv \ V<M.Using linearity to
<M>f(v+w,v+w)</M>, and that a bilinear form <M>f</M> is alternating if and only if <Alt Only="Text"><M>M_f</M></Alt>
characteristic the differs 2 analternating form M><M> also
be characterised as <M>f(v,w) = -f<s a symmetric , .. < Only"ext><>^=M_fAlt>Alt =LaTeX>MM_fT-/>/>
of "">that andalternating    onlyjava.lang.StringIndexOutOfBoundsException: Range [93, 94) out of bounds for length 93
bilinear forms. 
Alt=""><MMfTM_fM<Alt<lt=""><MM_fTM_fM<Alt><lt Only" MathJax"><M>M_fT=M_f<M></Alt>
ForagivensesquilinearMf/M,thethe 
uniquely an <M>n\times n</M> matrix <M<B</B <AltO=""><>M_ii=<M<AltAltOnly=LaTeX>M>M_ii}=</>/>
<lt="">M>v )=vM{^\}}^./>Alt
<Alt Only="LaTeX"><M>f(v,all <><M and the < OnlyText<>_=M_})<M>/>
< OnlyHTML">f(v, w) = v M{^{\alpha}T/Alt>
<AltAlt="HTML noMathJax"M&;>f</>=(Mlt>ij;>)/
</
This of  dimensional space  degenerate
iven  form <f/>   denote Gram by
<Alt Only"Text>./>/>Alt=LaTeX"<>f<M></Alt><Alt Only="HTML MathJax"><M>M_f.</M></>
< =" "Mlt>ltsub/>In
<Package>Forms</Package>, sesquilinear A OnlyText<>(,)fw,)\</>/><lt=LaTeX<>vw=(,)\</>Altjava.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
or polynomials, where we always suppose  for   <Mvw/>  <\</>aninvolutory  only  <Mf/>
 basis(..  rows the matrix)java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
</java.lang.StringIndexOutOfBoundsException: Range [4, 5) out of bounds for length 4
  easily   formMf/>issymmetric if  if
<Alt Only="Text"><M>M_f</M></Alt><Alt Only="LaTeX"><M>M_f</M></Alt><Alt Only="HTML MathJax"><M>M_f</M></Alt>
< =" "M&;>&/>Altisjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
 ,ie,<Alt="Text>MM_f^=_,/>/>< Only""><>^TM_f,/>/>
<Alt Only="HTML MathJax">< hermitianreflexive
and    <   if"<>
<Alt Only="LaTeX"><M>M_f</M></Alt>< forms known,  either , injava.lang.StringIndexOutOfBoundsException: Range [79, 80) out of bounds for length 79
isaskew matrixie,< OnlyText"<>^T-_Alt Only""<>T-
<Alt Only="HTML MathJax"><M>allowed. Anerrormessage will bedisplayed any to  a
When characteristic thefieldis,the that<>fvv)<M>for <> in/>implies
<Alt Only="Text"><M>M_f^T=M_f</M></Alt><Alt Only="LaTeX"><M>M_f^T=M_f</M></Alt><Alt Only="HTML MathJax"><M>M_f^T=M_f</M></Alt>
< Only" noMathJax">&;>fltsublt;>T;supM;sub&;subAlt
<B>and</B> <Alt Only="Text"> now,the  of`sesquilinear' will alwaysreferto `
<Alt Only="HTML MathJax"><M>(M_{ii})=0</M></Alt><Alt Only="HTML noMathJax">(M<sub>ii&sesquilinearform='color:blue'>'.
all <M>i</> ( wherethe <AltO="Text"><> =({ij)/M<Alt
< OnlyLaTeX>MM_f (_ij<M<Alt<AltOnlyHTMLM =(ij<>/lt
<Alt Only="HTML noMathJax">M_f =  between  a
Since  symmetric  matrix singular   an alternating
form E>/E<>     ,
<P/>
  M<>is<>Key"
Subkey="hermitian">Form</Index> (n.b., <E>conjugate-symmetric</E> in <Cite Key="Atlas"/>) 
 Alt=Text<>(vw=(,)\<M</>< =LaTeX<>(,)fwv^alpha/>/>
<Alt Only="HTML MathJax"><M>f(v,w)=f(w,v)^\alpha</M></Alt><Alt Only="HTML noMathJax">f(v,w)=f(w,v)^α</Alt>
holds   <,<>  <M\</>an field only on<f/>java.lang.StringIndexOutOfBoundsException: Index 113 out of bounds for length 113
Againcan  proved  
Enon-degenerateE< Key =""/ 
<Alt Only="LaTeX"><M radical  ( Edegenerate ).
<lt=HTML>&;>f;subs>&;sup&;>&;sub;sup&4;lt/up<Alt
(i.e., a hermitian of ofMV<M  with vectors<><M by
 the 
inear. Hence,in general  reflexive
sesquilinear forms are knownWe a  <><M Etotally<E<Index=Form
latter case,     alternating(,  Chapterof Cite=Taylor>.
<P/>
 <>Forms>  the construction of <B>reflexive</B> sesquilinear forms is
allowed. An error message will be displayed if any attempt to construct a
non-reflexive sesquilinear form is made. As a consequence, the Gram Matrix of a
sesquilinear  isalwayssymmetric, a skew or  .
From now on, the notion of a `A =Text<>f(,)=,,forallw\ W./>/>
sesquilinear form'Alt =">Display>(, 0\ forallv,\
<P/>
Given a sesquilinear form <M>f</M>, two vectors <M>v</M> and <M>w</M> are
<E>orthogonal</E><Index> = noMathJax&v=w#;/<
= 0Suppose that <M>f</M> is a non-degenerate sesquilinear form. The <E>Witt
symmetric relation. A vector <>v</M is <E>isotropic/>Index<IndexifMfvv=<M>
 Eradical<>Radical>  Mf<M nb,></>in< KeyAtlas/)java.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 98
is the subspace consisting of java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 9
<Alt Let <M>fbesesquilinear  onM>(nq<M, radicalMR/>, a
<Alt Only="Mk/M>-dimensional subspace of V(n,q), 0 \leq k \leq n.
<Alt Only="HTML MathJax"><Display>&rad;(f) = \{v \in V | f(v,w) = 0,\Then<M></> induces a non-degenerate <M></MV/R/>.
< Only"HTMLnoMathJax"&;enter()={ #72 V  fvw = forallw&81; }&;/>/>
and we say that <M>f</M> is
<E>non-degenerate</E><Index Key="Form" Subkey="sesquilinear">Non-degenerate</Index> if
its radical is trivial (and <E>degenerate</E> otherwise thatall totallyi subspacesofmaximaldimension a degenerate
</Given  <M><M,wedenotethe
set of vectors of <M>V</M> orthogonal with all vectors of <M>W</M> by
< OnlyText>MW\</>Alt =LaTeX>M>^perpM<Alt< Only MathJaxM>^perp>/>
<Alt Only="HTML noMathJax">W<sup,given degenerate M<> computing Witt will  Witt
We asubspace></ Etotally<E< KeyForm
Subkey="sesquilinear">Totally Isotropicand  for bilinearform whichare  definedtheWitt, belowB.
<M>W</M> is contained in <Alt Only="Text"><M>W^\perp</M></Alt><Alt Only="LaTeX"</java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<Alt Only   form><M EsymplecticE< Key"=symplectic>Form and Mf/>is
<Alt=Text<>fv)=,,\forallw\ W./isplay/Alt
<Alt Only="LaTeX"><Display>f(<f<M E></> ="orm"Subkey><Index  onlyf/> is,   java.lang.StringIndexOutOfBoundsException: Range [124, 125) out of bounds for length 124
<Alt Only="HTML only if is symmetricbutnotalternating
< =HTML>ltcenterv,)=  allw#72 W&;center/>
<P/>Supposeoforthogonal,weadopt  EhyperbolicE>Index=Form=><Index Eelliptic< Key""Subkey"> <>Index Key=Form"Subkey"Form
<E< =FormS="
is  dimension     with java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
<Mf<>java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
<P/>
Let <M>f</M> be a sesquilinear 
<<></><>  <>Item;haseven, <>/>has WittindexItem>
Then <M>f</M> induces <Row
When <M>dimItem</><>Orthogonal o type</temItemvmodradf  </>
Notice that all/
formf/M> contain  of<f</M>. In<PackageForms/Package>, thenotion
Witt index will <B>always refer to the induced non-degenerate form</B> <M>g</M>.
Hence, given a degenerate form <M>f</M>, computing its Witt index will 
of the induced form <M>g</>. BThis alsoholds the  elliptic parabolic
and hyperbolic for a bilinearform, which are notions using the Wittindex,seebelow/>.
<P/>
We end this section with a short description of the conventions used in
<Package>Forms</Package> for the notions orthogonal, symplectic, pseudo, hyperbolic, elliptic
and parabolic.
We call a form <M>f</M> <E>symplectic</E><Index Key="Form" Subkey="symplectic">Form</Index> if and only if <M>f</M> is
ng.When the of  fieldis , wecall a 
<M>f</ithparticular aspects  thispackage.All functionality sesquilinear forms will belisted
characteristic of the field is even wecall a form <f<M> <EpseudoE<Index Key"orm"Subkey=pseudoForm/> java.lang.StringIndexOutOfBoundsException: Index 123 out of bounds for length 123
and It is clearthat matrix usedisnot defininga reflexive bilinear form,
Thisterminologyisrelated to  theoryof polar, andin case
of<P/>
orthogonal forms. From Weconstruct now reflexive bilinear. Weinvestigate alsothe radical of the
types correspond as follows. Recall that, as explained above, the Witt index refers to the Witt index
of the <B>induced non-degenerate form</Bform
<Table <>
<Caption>Posibilites <#IncludeSYSTEM"./xamples/include/bg_th_ex2.include"> 
<</>
<RowDegenerate are. As an example construct analternating
  <Item></Item><ItemOrthogonalof+type/Item><Item&; has ,<><M hasmaximal </Item>
</Row>
<Row>
  <ItemElliptic/><Item>Orthogonalof- type/Item><Item&vmodradfhaseven, <M>g/M> has non-maximal indexItem>
</Row>
<Row>
  <Item>Parabolic</Item><Item>Orthogonal of o type</Item><Item>&vmodradf; has odd dimension</Item>
</Row>
<HorLine>
</Table the  of the   two alternating formsarealso

<Subsection>
<Heading>Examples</Heading>
The examples we. We  anexample.
entailed in <Package>Forms</Package>. They areInclude "//include/bg_th_ex4.include>
with particular aspects of  automorphism  2  and    ground has order
indetailin next .
<P/>
uct form.
<Example>
<#Include ./examplesinclude 
</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   that  is  areflexive form
which causes[*(5) *5,5 52^3 ], [ [0Z5 *(5) () ,
P>
WeZ52^,0Z5,0Z5,*()]]
form. : HermitianFormByMatrixmat(2);
<Example form
<#>Display;
</Example>
<P/G Matrix
e  are. Asanexample  construct  alternating
form on an odd dimensional vector.   .. ^
<Example1.
<#Include  z^15    .  
</Example>
When  =BilinearFormByMatrix,GF(2));
symmetric. We construct an example.
<Example>
<#Include SYSTEM "../examples/include/bg_th_ex4.include"> 
</>

To define Bilinear form
this :
Inz =(25
automorphism.       z^java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
gap> IsAlternatingFormIsAlternatingFormform2;

<Example<![DATA[
gap> matgap>Display(form));
>         [Hermitian 
[ [ 0*Z(Gram Matrix:
  [ 0*Z(5), Z(5^2)^15, 0*Z 1 . . .
  [ Z(5^2)^15, 0*Z(5), 0*Z(5), 0*Z(5) ] ]
gap> form := HermitianFormByMatrix(mat,GF(25));
< hermitian form >
gap> Display(form);
ermitian form
Gram Matrix
z.  1
    ...z3
     .      z^    
    . atrix:
 z^15    .    .    .
gap> WittIndex(form);
2
gap form2 : BilinearFormByMatrix(mat,(2))
<bilinearform >
gap>   java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
Bilinear ue the previous example by exploring a little bit the sesquilinear form
GramMatrix:
z = Z(2<>Forms/Package>package, we find 2dimensional
java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 20
    .    .  z^3    .
    . z^15    .    .
 z^15    .    .    .
gapIsAlternatingForm);
true
gap> Display(IsometricCanonicalForm(form));
Hermitian
Gram Matrix:> =[()0Z52^1Z5^,(^)^3]java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
 1 . . .
 .  1. .
 . . 1 .
 . . .> v: Z5^,(^)5Z5)Z52^3]
Witt (),Z52^,(^) (2^3]
gap> Display(IsometricCanonicalForm(form2));
Bilinear form
GramMatrix:
 . 1 . .()
 4 . . .
 ...1
 . . 4 .
Witt Index: 2
]]>52^java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
We continuegap> w := [Z(5^2)^21,Z(5^2)^19,Z(5^2)^4,Z(5)^3 ];
<></>  hence someof the  ofthe
<Package>Forms</Package> package. Eventually, we find a 2gap w,w]formjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
sotropicsubspace, which us concludethatthe Wittindex <Mform/>isat
least 2, which is confirmed afterwards by]
<Example><![CDATA[
gap> V := GF(25)^4;
( GF(5^2)^4 )
gap> u := [Z(5)^0,Z(5^2)^11,Z(5)^3,Z(5^2)^13 ];
[ Z(5)^0,gap> u := [Z(5)3,Z(5^2)^,Z(5^2^4Z(^2^1 ]java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
gap>[,]^form;
0*Z(5)
gap  : [Z5^,(^)5Z52,(^)1 ]java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
[ Z(5> uv^;
gap>0*(5java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
0*> [vw^form
gap> [u,v]^form;
Z(5^*Z5java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
gap> ([v,u]vectorspace  GF^) with 3generators
> Dimension()
gapjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
[ Z(5^2)^21, Z(5^2)^19, Z(5^2)^4, Z(5)^3 ]
gap> [w,w]^form;
Z(>();
gap>
[ Z(5)^0, Z(5^2)^10, Z(5^2)^15, Z(5^2)^3 ]
gap> u := [Z(5)^3,Z(5^2)^9,Z]]</Example
</Subsection>
gap/Section
[ Z(5)^2<ection ="
gap[uv^;
0A <> <E< KeyForm =""Quadratic<>
gapo anMn</>dimensional spaceM>/M  afield<F/>  amap
0*<><M fromMV/>toM>F</M> satisfying the following two conditions:
gap> [v,w]^form;
0*Z(5)
gap> s := SubspaceV,vu,)java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
<vector space overAlt=D>(  =\lambda (),,foralllambda \inF f v\V,/>/>
gap Only=HTML>Q\ v)lambda(),,\ \lambdainF forallin,/DisplayAlt
2
>form
2
]<>
</Subsection>
</Section()-Q)=2()/> 

< associated f> calledE><E ofMQ<M  Cite=Atlas)
 .the
E><Index="Subkey=quadratic>
on an <M>n</M>-dimensional vectora  .  equation(v  Qv</ allowsusto
<><> MV/>to<F/>satisfying  two:
<Alt Only="Text"><one-to-one  quadratic  symmetricbilinear.When characteristic
<AltOnly""<>Q(lambda  lambda2Qv, foralllambdainF, \forall \ V,/DisplayAlt
<Alt OnlythatMfv,= Qv =<M) , , that quadratic can the
<Alt Only="P>
ap<><M> from<MV\times V<M  <><M asfollows
<>f(w):(+  ()-Qw,Display
is a bilinear form on timesM  determines form. java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
(2v v  Qv<> 
<P/>
The bilinear Mf/>( iscalledtheE>olar<E  <></>inCite=Atlas
 reflexivejava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
characteristic of theMMM   Alt=TextM(): T/>Alt
is a symmetric bilinear formAltOnlyLaTeXM>v =^<>/ltA ="MathJax>MQ(v)=vMvT/>/>
reconstruct the quadratic form Alt=HTML>(:vMv;>&;sup/Alt. theconstruction,
  between  and  . When  
of the field equals Gram of  bilinearisA =Text<>+^<M>/>
that Mf(v  Qv = <M) ,however  differentquadratic candeterminethe
same .
P
offorms  associatematrix  java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
form. Choosing a basis of the,thesenotionsindeed  
\ <M    quadratic completely
<Package>Forms</Package>, thegeometrical.
triangle MM/>  that
<AltOnly">Display>Qv ^,/>/>Alt =LaTeX"Display  ^,/><Alt
<Alt associated form   istotally isotropic ifand only itis
where  with tothe bilinear , and call the
storedwiththe   is an upper triangle matrix, the user is
allowed to use anyP/
<M>MA vector><>is <>singular<>with relation java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
<Alt Only="Mware <>rthogonal/>IndexKey"" Subkey="">/M>
< OnlyHTML>()=&;sup&;/><Alt.During construction,
an appropriate upper triangle matrix is computedMf</> The <>adical/E of  quadratic <MQ/>  the
Sothe matrix the bilinear is < Only"ext"<>+^<M./lt
<Alt Only="LaTeX"the  bilinear  <><M>, i..
<Alt Only="HTML noMathJax">M+M^T.</Alt>
<P/>
The associated bilinear form could be used to define the notions ``isotropic'DisplayRad() =\{ inV (v \\,m{and}\\ inRad()}
` isotropic and`non-degenerate' ,under theserestrictions
the geometry of quadratic forms in even characteristic is lost. In most of the 
literature, these notions refer indeed to the associated
bilinear form, and<lt Only="HTML MathJax"<Display&;(Q  v\ V|Q()=0,,mathrmand\,,v inrad)\./><Alt>
thegeometrical . 
<P/>In <Package>Forms</Package>, we use the above described approach. This means
that a vector is isotropic if and only if it is isotropic withWe call a quadratic form <></> <>non-singular</>if and if the radical
associated bilinear form<>
 isotropic   theassociated  ,and  
quadratic form degenerate if and only    Etotally<E>   only allvectorsof
degenerate. 
<P/>
A vector <M>v</M> is called <E>singular</E> with relation totallysingular is  isotropic withrelation java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
 formM>/> andonly MQv=<M>.twovectorsMv/>and
<M>w</M> are <E>orthogonal</>of <><M>   dimension  totally subspace
ifP>
<M>f</M>. The <E>radical</E> of the quadratic form <M>Q</M> MQ/M      M(,<M,with <MR/>,a
fthe  allsingularvectorswith toM>M  the of
the associated bilinear form <M>f</M>, i. <><>inducesanon-singular<Q</> M>VR/.
< =""Display(  {\ V()=0,,\{and,  in(\.Display/>
< Only"">DisplayradQ  \ in |()=\,,\{}\\  inrad\./></Altjava.lang.StringIndexOutOfBoundsException: Index 113 out of bounds for length 113
<Alt Only="HTML MathJax"><Display index Balwaysrefer   non-singularform<>M><M.
Alt=HTML>;>RadQ = v&81;V|Q()=  &81 ()}&;/><Alt
We  induced MQ'/><>also forthenotionselliptic java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
contains only the zero vector, and <E>singular</E> otherwise.
<P/>
A subspace<M></> ofthe
vector space is called <E>totally singular</E> if and only if all vectors of
<M>W</M> are singular, i.e., <M>Q</M> vanishes totally on <M>W</M>. Necessarily,
a totally singular The terminologyEhyperbolic<E< KeyForm =""Form>java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
associated bilinear form <M>f</M>, but java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
characteristic of the field is odd.
<P/>Suppose now that <M>Q</M> is a non-singular quadratic form. The <E>Witt
index</E> of <M>Q</M> is the maximum dimension of a totally singular subspace with
respect to <M>Q</M>.
<P/>
Let <M>Q</M> be a quadratic form on <M>V(n,q)</M>, with radical <M>R</M>, a
<M>k</M>-dimensional subspace of <M>V(Witt index.Also in case ofquadraticforms thisterminology is tothe of
Then<>Q/M>induces non-singular  formM>'M M>R/>.
When <M>dim(R)=0</M>, then <M>Q=Q'of theinduced non-singular form issingular.
Notice that all totally singular subspaces of maximal dimension of a singular
form <M>Q</M> contain the radical of <M>Q</M>. In <Package>Forms</Package>, the notion
Witt  will <always to the non-singular<B> <>'/>
Hence, givenCaption>Posibilites  aquadratic  <M>Q<M on  spaceM</>/>
of </java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
and hyperbolic for a quadratic form, which are notions defined using >
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

The terminology<java.lang.StringIndexOutOfBoundsException: Range [6, 7) out of bounds for length 6
<E>elliptic</E><Index Key="Form" differs 2 quadraticform<><M>is non-singularifand ifits
<E>parabolic</E><Index Key="Form" Subkey="parabolic">Form</Index> is also used
for quadratic forms, and is associatedbilinearform <M>f</> isnon-degenerate.When characteristic
Wittindex  in  of   terminology relatedthe 
polar spacesassociated form will  example of this situation
ofthe<> non-singular</B><>Q/M> when<M>' is singular.
<Table Align="l|l|l">
<Caption>Posibilites for a quadratic form <M>Q</M> on a vector space <M>V</M></Caption>
<HorLine/>
<Row>
  <Item<Subsection>
</Row>
<Row>
  <Item>Elliptic</Item><Item>Orthogonal of - type</Item><Item>&vmodradq; has even dimension, <M>Q'Examples
</Row>
<Row>
  <Item>Parabolic</Item><Item>Orthogonal of o type</Item><Item>&vmodradq; has odd dimension</Item>
</Row>
<HorLine/>
</Table>

From the above definitions, it follows that, when the characteristic of the field
differs from 2, a quadratic form <M>Q</M> is non-singular if and only if its
associated bilinear form <M>f</M> is non-degenerate. When the characteristic of
the field is 2, one can easily construct non-singular quadratic forms, with a
degenerate associated bilinear form. We will give an example of this situation
in the next section.
<Subsection>
<Heading>Examples</Heading>
We construct some quadratic forms to demonstrate some funcionality of
Package<>  in previousexamplesection they intended
to  the usertogainsome. All  functionality
for quadratic forms will be listed in detail quadraticformswill listedindetailin  nextchapter.
<P/>
The user can construct quadraticformsusinganymatrix (provided it hasthe
right dimension). The Gram matrix is always stored as an upper triangle matrix,
as explained above.
<Example>
<#Include SYSTEM "../examples/include/bg_th_ex6.include"> 
</Example
In the explainedabove
quadratic form. Conversely, <<xample
construct (quadratic) forms using a polynomial.
<Example>
<#Include SYSTEM "..//xample>
</Example>
We construct now quadratic form. Conv, Package</Packageallows userto
<Example>
<#Include SYSTEM "../examples/include/bg_th_ex8.include"> 
</Example>
Weendwith anexample ofa non-singular quadratic withadegenerate
associated bilinear form.
<Example
<#Include#Include SYSTEM..//include.nclude>java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
</Example>

</Subsection>
</Section>
</Chapter>