<!--
  morphisms.xml	    Forms package documentation
                                                                   John Bamberg
                                                               and Jan De Beule
                                                           
  Copyright (C) 2008, Ghent University

This is the chapter of the documentation describing all of the functions of forms.
-->

<Chapter Label="morphisms">
<Heading>Morphisms of forms</Heading>

In this chapter we give a very brief overview on morphisms
of sesquilinear and quadratic forms. The reader can find more in the
texts: Cameron <Cite Key="Cameron"/>, Taylor <Cite Key="Taylor"/>,
Aschbacher <Cite Key="Aschbacher"/>, or Kleidman and Liebeck <Cite Key="KleidmanLiebeck"/>.
<P/>
In this chapter we consider an <M>n</M>-dimensional vector space <M>V</M> over a finite field. Suppose that <M>f</M> is a
sesquilinear form or a quadratic form on <M>V</M>, then we call the pair
<M>(V,f)</M> a <E>formed vector space</E>.

<Section Label="morphisms:sesquilinear">
<Heading>Morphisms of sesquilinear forms</Heading>
Consider two formed vector spaces <M>(V,f)</M> and <M>(W,g)</M> over the same
field <M>F</M>, where both <M>f</M> and <M>g</M> are sesquilinear forms. Suppose
that <Alt Not="HTML"><M>\phi</M></Alt><Alt Only="HTML">&#966;</Alt> is a linear
map from <M>V</M> to <M>W</M>. 

The map <Alt Not="HTML"><M>\phi</M></Alt><Alt Only="HTML">&#966;</Alt> is an <E>isometry</E><Index>Isometry</Index> from the formed
space <M>(V,f)</M> to the formed space 
<M>(W,g)</M> if for all <M>v,w</M> in <M>V</M> we have
<Display>
<Alt Not="HTML">f(v,w) = f'(\phi(v), \phi(w)).</Alt>
<Alt Only="HTML">f(v,w) = f'(&#966;(v),&#966;(w))</Alt></Display>
The map <Alt Not="HTML"><M>\phi</M></Alt><Alt Only="HTML">&#966;</Alt> is a <E>similarity</E><Index>Similarity</Index> from the
formed space <M>(V,f)</M> to the formed space <M>(W,g)</M> if for all <M>v,w</M>
in <M>V</M> we have 
<Display>
<Alt Not="HTML">f(v,w) = \lambda f'(\phi(v), \phi(w)).</Alt>
<Alt Only="HTML">f(v,w) = &#955; (f'(&#966;(v),&#966;(w)))</Alt>
</Display>
for some non-zero <Alt Only="HTML">&#955; in <M>F</M></Alt><Alt
Not="HTML"><M>\lambda \in F</M></Alt>. Finally, the map <Alt Only="HTML">&#966;</Alt><Alt
Not="HTML"><M>\phi</M></Alt>. is a
<E>semi-similarity</E><Index>Semi-similarity</Index> from the formed space  
<M>(V,f)</M> to the formed space <M>(W,g)</M> if for all <M>v,w</M> in <M>V</M>
we have
<Alt Not="HTML"><Display>f(v,w) = \lambda f'(\phi(v), \phi(w))^\alpha</Display></Alt>
<Alt Only="HTML"><Display>f(v,w) = &#955; f'(&#966;(v),&#966;(w))&lt;SUP>&#945;&lt;/SUP></Display></Alt>
for some non-zero <Alt Only="HTML">&#955; in <M>F</M></Alt><Alt
Not="HTML"><M>\lambda \in F</M></Alt> and a field automorphism <Alt
Only="HTML">&#945;</Alt><Alt Not="HTML"><M>\alpha</M></Alt> of <M>F</M>.
<P/>
One of the objectives of studying maps between formed vector spaces is the
classification of sesquilinear forms on a vector space <M>V</M>, where it
is sufficient to classify non-degenerate forms. The following results
are well known.
<P/>
It can be proved that (see for example Section 6.3 of <Cite Key="Cameron"/>):
<List>
<Item>all non-degenerate alternating forms of a given vector space over a given finite field
are similar,</Item>
<Item>all non-degenerate hermitian forms of a
given vector space over a given finite field are similar, and,</Item>
<Item>the non-degenerate symmetric
bilinear forms on a vector space over a field with odd characteristic come in
three flavours, two of which occur when the dimension of the vector space is even,
one of which occurs when the dimension of the vector space is odd.</Item>
</List>
In principle, within each similarity class, different isometry classes can
occur, but we will see that in most cases, each similarity class contains
exactly one isometry class.
<P/>
Given a sesquilinear form <M>f</M> over a vector space <M>V</M>,
<Package>Forms</Package> provides functionality to compute the linear map <Alt
Not="HTML"><M>\phi</M></Alt><Alt Only="HTML">&#966;</Alt>
from <M>V</M> to itself (or, equivalently, a matrix describing a change of
basis), such that <M>f</M> is mapped to its canonical representative in
its isometry class. In the next sections, we describe the representative(s) of the
similarity class(es) used in <Package>Forms</Package>, and, when necessary, the
different isometry classes, for each of the three reflexive sesquilinear forms.
The easiest cases are the hermitian and alternating cases.

<Subsection Label="morphisms:hermitian">
<Heading>Hermitian forms</Heading>
We suppose that <M>f</M> is a non-degenerate hermitian form on a
vector space <M>V</M> over the finite field <M>F</M>, with involutory field
automorphism <Alt Not="HTML"><M>\alpha</M></Alt><Alt Only="HTML">&#945;</Alt>.
It can be proved (see <Cite Key="KleidmanLiebeck"/>) 
that any vector space equipped with a non-degenerate hermitian
form <M>f</M> contains an orthogonal basis such that <Alt
Not="HTML"><M>f(e_i,e_i)=1</M></Alt><Alt
Only="HTML">f(e&lt;SUB>i&lt;/SUB>,e&lt;SUB>j&lt;/SUB>)=1</Alt> for
each basisvector <Alt Not="HTML"><M>e_i</M></Alt><Alt
Only="HTML">e&lt;SUB>i&lt;/SUB></Alt>. Hence <M>(V,f)</M> is isometric with <M>(V,f')</M>
with <M>f'</M> the non-degenerate hermitian form with the identity matrix over
<M>F</M>. The Witt index of <M>f</M> equals <M>n/2</M> when <M>n</M> is even and
<M>(n-1)/2</M> when <M>n</M> is odd.
</Subsection>

<Subsection Label="morphisms:alternating">
<Heading>Alternating forms</Heading>
We suppose that <M>f</M> is a non-degenerate alternating bilinear form on a
vector space <M>V</M> over a finite field <M>F</M>. As already mentioned in Section
<Ref Sect="theory:sesquilinearforms"/>, non-degenerate alternating forms only
exist on even dimensional vector spaces. Restricting to a two dimensional
vector space, it is clear immediately that the Gram matrix of <M>f</M> must be
<Alt Only="HTML">&lt;!-- matrix expression begin -->
&lt;table style="color:#000">&lt;tr>
&lt;td>&lt;table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000">&lt;tr>
&lt;td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;&lt;/td>&lt;td>&lt;table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
&lt;tr>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>r&lt;/td>
&lt;/tr>
&lt;tr>
&lt;td align="center" valign="center" width=30>-r&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;/tr>
&lt;/table>&lt;/td>&lt;td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;&lt;/td>&lt;/tr>&lt;/table>&lt;/td>
&lt;/tr>&lt;/table>
&lt;!-- matrix expression end --></Alt>
<Alt Only="LaTeX"><M>\left( \begin{array}{cc}0 &amp; r \\ -r &amp; 0 \end{arry}
\right)</M></Alt>
for some non-zero <M>r \in F</M>. If we rescale one of the basisvectors, which
induces an isometry, then we see that there always exists a basis such that
<M>r=1</M>. We call a two dimensional vector space equipped with a non-degenerate
alternating form a <E>symplectic hyperbolic line</E>, and it is proved (see Theorem 6.7 of <Cite Key="Cameron"/>) that the
formed space <M>(V,f)</M> can be written as an orthogonal direct sum of symplectic hyperbolic
planes. Hence, up to isometry, there is only one non-degenerate alternating form
of an even dimensional vector space, and we choose as canonical representative the
alternating form with Gram matrix
<Alt Only="LaTeX"><Display>\left( \begin{array}{ccccccc}
0 &amp; 1 &amp; 0 &amp; 0 &amp; \ldots &amp; 0 &amp; 0 \\ 
-1 &amp; 0 &amp; 0 &amp; 0 &amp; \ldots &amp; 0 &amp; 0 \\
0 &amp; 0 &amp; 0 &amp; 1 &amp; \ldots &amp; 0 &amp; 0 \\
0 &amp; 0 &amp; -1 &amp; 0 &amp; \ldots &amp; 0 &amp; 0 \\
\vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots &amp; \vdots \\
0 &amp; 0 &amp; 0 &amp; 0 &amp; \ldots &amp; 0 &amp; 1 \\
0 &amp; 0 &amp; 0 &amp; 0 &amp; \ldots &amp; -1 &amp; 0 \\
\end{array} \right).</Display>
</Alt>
<Alt Only="HTML">
&lt;!-- matrix expression begin -->
&lt;table style="color:#000">&lt;tr>
&lt;td>&lt;table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000">&lt;tr>
&lt;td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;&lt;/td>&lt;td>&lt;table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
&lt;tr>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>1&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>...&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;/tr>
&lt;tr>
&lt;td align="center" valign="center" width=30>-1&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>...&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;/tr>
&lt;tr>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>1&lt;/td>
&lt;td align="center" valign="center" width=30>...&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;/tr>
&lt;tr>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>-1&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>...&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;/tr>
&lt;tr>
&lt;td align="center" valign="center" width=30>:&lt;/td>
&lt;td align="center" valign="center" width=30>:&lt;/td>
&lt;td align="center" valign="center" width=30>:&lt;/td>
&lt;td align="center" valign="center" width=30>:&lt;/td>
&lt;td align="center" valign="center" width=30>&lt;sup>&lt;big>&#183;&lt;/big>&lt;/sup>&#183;&lt;sub>&lt;big>&#183;&lt;/big>&lt;/sub>&lt;/td>
&lt;td align="center" valign="center" width=30>:&lt;/td>
&lt;td align="center" valign="center" width=30>:&lt;/td>
&lt;/tr>
&lt;tr>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>...&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>1&lt;/td>
&lt;/tr>
&lt;tr>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>...&lt;/td>
&lt;td align="center" valign="center" width=30>-1&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;/tr>
&lt;/table>&lt;/td>&lt;td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;&lt;/td>&lt;/tr>&lt;/table>&lt;/td>
&lt;/tr>&lt;/table>
<!-- matrix expression end -->
</Alt>
The Witt index of <M>f</M> equals <M>n/2</M>.
</Subsection>

<Subsection Label="morphisms:bilinear">
<Heading>Bilinear forms</Heading>
We suppose that <M>f</M> is a non-degenerate symmetric bilinear form on a
vector space <M>V</M> over a finite field <M>F</M> with odd characteristic. We
call a two dimensional vector space a <E>hyperbolic line</E> if it contains a
non-zero vector such that <M>f(v,v) = 0</M>. It is proved (see Proposition 6.9 of 
<Cite Key="Cameron"/>) that any two
hyperbolic lines are isometric, and we choose as canonical representative the
orthogonal form with Gram matrix
<Alt Only="HTML">&lt;!-- matrix expression begin -->
&lt;table style="color:#000">&lt;tr>
&lt;td>&lt;table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000">&lt;tr>
&lt;td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;&lt;/td>&lt;td>&lt;table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
&lt;tr>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;td align="center" valign="center" width=30>1&lt;/td>
&lt;/tr>
&lt;tr>
&lt;td align="center" valign="center" width=30>1&lt;/td>
&lt;td align="center" valign="center" width=30>0&lt;/td>
&lt;/tr>
&lt;/table>&lt;/td>&lt;td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;&lt;/td>&lt;/tr>&lt;/table>&lt;/td>
&lt;/tr>&lt;/table>
&lt;!-- matrix expression end --></Alt>
<Alt Only="LaTeX"><Display>\left( \begin{array}{cc}0 &amp; 1 \\ 1 &amp; 0 \end{arry}
\right).</Display></Alt>
<P/>
It can be proved (see Theorem 6.10 of <Cite Key="Cameron"/>)
that the formed space <M>(V,f)</M> can be written as the
orthogonal direct sum of hyperbolic lines and one subspace <M>U</M> of
dimension at most two. The behaviour of <M>f</M> on the subspace <M>U</M>
determines the similarity class of <M>f</M>. We describe the three occurring
cases, to describe the chosen canonical form, we use the polynomial rather than
the Gram matrix.
<P/>
<List><Item>If the dimension of <M>U</M> is zero, then <M>(V,f)</M> is the orthogonal direct
sum of hyperbolic lines, and hence <M>(V,f)</M> is isometric to the formed
space <M>(V,f')</M>, where the Gram matrix of <M>f'</M> consists of blocks as
described above. The chosen canonical form has polynomial
<Alt Not="HTML"><Display>
x_1 x_2 + \ldots + x_{n-1}x_n
</Display></Alt><Alt Only="HTML"><Display>x&lt;sub>1&lt;/sub> x&lt;sub>2&lt;/sub> + ... + x&lt;sub>n&lt;font face="symbol">-&lt;/font
>1&lt;/sub>x&lt;sub>n&lt;/sub></Display></Alt>
Note that the dimension of the vector space <M>V</M> is necessarily even. We call
<M>f</M> <E>hyperbolic</E> (see also Section <Ref Sect="theory:sesquilinearforms"/>). 
It follows also that in this similarity class, there
is only one isometry class. The Witt index of <M>f</M> equals <M>n/2</M>.
</Item>
<Item>If the dimension of <M>U</M> is one, then necessarily the polynomial of <M>f</M>
equals
<Alt Not="HTML"><Display>
\mu x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n
</Display></Alt><Alt Only="HTML"><Display>&#956; x&lt;sub>1&lt;/sub>&lt;sup>2&lt;/sup> + x&lt;sub>2&lt;/sub>
x&lt;sub>3&lt;/sub> + ... + x&lt;sub>n&lt;font face="symbol">-&lt;/font
>1&lt;/sub>x&lt;sub>n&lt;/sub></Display></Alt>
for some <Alt Not="HTML"><M>\mu \in F</M></Alt><Alt Only="HTML">&#956; in
<M>F</M></Alt>, and the dimension of the vector space <M>V</M> is odd. We call
<M>f</M> <E>parabolic</E> (see also Section <Ref Sect="theory:sesquilinearforms"/>). It is clear that if  <Alt
Not="HTML"><M>\mu</M></Alt><Alt Only="HTML">&#956;</Alt> is a square in
<M>F</M>, then rescaling the first basis vector yields a polynomial <Alt Not="HTML"><Display>
x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n
</Display></Alt><Alt Only="HTML"><Display>x&lt;sub>1&lt;/sub>&lt;sup>2&lt;/sup> + x&lt;sub>2&lt;/sub>
x&lt;sub>3&lt;/sub> + ... + x&lt;sub>n&lt;font face="symbol">-&lt;/font
>1&lt;/sub>x&lt;sub>n&lt;/sub></Display></Alt>
which we choose as the canonical form for this similarity class. If <Alt
Not="HTML"><M>\mu</M></Alt><Alt Only="HTML"><M>&#956;</M></Alt>
is a non-square, a rescaling of <Alt
Not="HTML"><M>x_2,x_4,\ldots,x_{n-1}</M></Alt><Alt
Only="HTML">x&lt;sub>2&lt;/sub>,x&lt;sub>4&lt;/sub>,...,x&lt;sub>n&lt;font face="symbol">-&lt;/font
>1&lt;/sub></Alt> yields a polynomial
<Alt Not="HTML"><Display>
\mu (x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n)
</Display></Alt><Alt Only="HTML"><Display>&#956;(x&lt;sub>1&lt;/sub>&lt;sup>2&lt;/sup> + x&lt;sub>2&lt;/sub>
x&lt;sub>3&lt;/sub> + ... + x&lt;sub>n&lt;font face="symbol">-&lt;/font
>1&lt;/sub>x&lt;sub>n&lt;/sub>)</Display></Alt>
which represents now a bilinear form that is <B>similar but not isometric</B> to
the given one. Hence, the parabolic similarity class contains two isometry
classes. The Witt index of <M>f</M> equals <M>(n-1)/2</M>.
</Item>
<Item>
Suppose at last that the dimension of <M>U</M> is two. We may suppose that
<M>U</M> is not a hyperbolic line. It is not too difficult to see that a suitable base change
yields the polynomial <Alt Not="HTML"><Display>
\mu x_1^2 + x_2^2 + x_3 x_4 + \ldots + x_{n-1}x_n
</Display></Alt><Alt Only="HTML"><Display>&#956;x&lt;sub>1&lt;/sub>&lt;sup>2&lt;/sup> +
x&lt;sub>2&lt;/sub>&lt;sup>2&lt;/sup> + x&lt;sub>3&lt;/sub> x&lt;sub>4&lt;/sub> + ... + x&lt;sub>n&lt;font
face="symbol">-&lt;/font>1&lt;/sub>x&lt;sub>n&lt;/sub></Display></Alt> for a
non-square <Alt Not="HTML"><M>\mu \in F</M></Alt><Alt Only="HTML"><M>&#956; in
F</M></Alt>, and the dimension of the vector space <M>V</M> is even. We call
<M>f</M> <E>elliptic</E>. The Witt index of <M>f</M> equals <M>(n-2)/2</M>.
</Item>
</List>
</Subsection>

<Subsection Label="morphisms:degenerate">
<Heading>Degenerate forms</Heading>
Suppose that <M>f</M> is a degenerate sesquilinear form on the vector space
<M>V</M>, then <M>Rad(f)</M> is a non-trivial subspace of the vector space
<M>V</M>. The vector space <M>V</M> can be written as the orthogonal direct sum
of a subspace <M>W</M> and <M>Rad(f)</M>, and the restriction of <M>f</M> to
<M>W</M> is a non-degenerate sesquilinear form on <M>W</M>. Hence, <M>f</M> is
isometric with a sesquilinear form having Gram matrix
<Alt Only="HTML">
&lt;!-- matrix expression begin -->
&lt;table style="color:#000">&lt;tr>
&lt;td>&lt;table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000">&lt;tr>
&lt;td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;&lt;/td>&lt;td>&lt;table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
&lt;tr>
&lt;td align="center" valign="center" width=30>M&lt;/td>
&lt;td align="center" valign="center" width=30>A&lt;/td>
&lt;/tr>
&lt;tr>
&lt;td align="center" valign="center" width=30>B&lt;/td>
&lt;td align="center" valign="center" width=30>C&lt;/td>
&lt;/tr>
&lt;/table>&lt;/td>&lt;td style ="border-top:1px solid #000; border-bottom:1px solid #000;">&nbsp;&lt;/td>&lt;/tr>&lt;/table>&lt;/td>
&lt;/tr>&lt;/table>
&lt;!-- matrix expression end -->
</Alt>
<Alt Only="LaTeX">
<Display>\left( \begin{array}{cc} M &amp; A \\ B &amp; C \end{arry}
\right)</Display>
</Alt>
where <M>M</M> is the Gram matrix of a non-degenerate sesquilinear form and
<M>A,B</M> and <M>C</M> are appropriate zero matrices. The dimension of the
maximal isotropic subspaces is the sum of the Witt index and the dimension of
the radical.
</Subsection>
</Section>

<Section Label="morphisms:quadratic">
<Heading>Morphisms of quadratic forms</Heading>
Consider two formed vector spaces <M>(V,f)</M> and <M>(W,g)</M> over the same
field <M>F</M>, where both <M>f</M> and <M>g</M> are quadratic forms. Suppose
that <Alt Not="HTML"><M>\phi</M></Alt><Alt Only="HTML">&#966;</Alt> is a linear
map from <M>V</M> to <M>W</M>. The map <Alt
Not="HTML"><M>\phi</M></Alt><Alt Only="HTML">&#966;</Alt> is an
<E>isometry</E><Index>Isometry</Index> from the formed space <M>(V,f)</M> to the
formed space <M>(W,g)</M> if for all <M>v,w</M> in <M>V</M> we have
<Display>
<Alt Not="HTML">f(v) = f'(\phi(v)).</Alt>
<Alt Only="HTML">f(v) = f'(&#966;(v)).</Alt></Display>
The map <Alt Not="HTML"><M>\phi</M></Alt><Alt Only="HTML">&#966;</Alt> is a
<E>similarity</E><Index>Similarity</Index> from the formed space <M>(V,f)</M> to
a formed space <M>(W,g)</M> if for all <M>v,w</M> 
in <M>V</M> we have 
<Display>
<Alt Not="HTML">f(v) = \lambda f'(\phi(v)).</Alt>
<Alt Only="HTML">f(v) = &#955; (f'(&#966;(v)))</Alt>
</Display>
for some non-zero <Alt Only="HTML">&#955; in <M>F</M></Alt><Alt
Not="HTML"><M>\lambda \in F</M></Alt>. Finally, the map <Alt Only="HTML">&#966;</Alt><Alt
Not="HTML"><M>\phi</M></Alt>. is a
<E>semi-similarity</E><Index>Semi-similarity</Index> from the formed space  
<M>(V,f)</M> to the formed space <M>(W,g)</M> if for all <M>v,w</M> in <M>V</M>
we have
<Display><Alt Not="HTML">
f(v)=\lambda f'(\phi(v))^\alpha</Alt>
<Alt Only="HTML">f(v) = &#955; f'(&#966;(v))&lt;SUP>&#945;&lt;/SUP></Alt>
</Display>
for some non-zero <Alt Only="HTML">&#955; in <M>F</M></Alt><Alt
Not="HTML"><M>\lambda \in F</M></Alt> and a field automorphism <Alt
Only="HTML">&#945;</Alt><Alt Not="HTML"><M>\alpha</M></Alt> of <M>F</M>.
<P/>
Also in this case, one of the objectives of studying maps between formed vector spaces is the
classification of quadratic forms of the same vector space <M>V</M>, where it
is sufficient to classify non-degenerate forms. 
<P/>Since there is a one-to-one relationship between quadratic forms in odd
characteristic and orthogonal bilinear forms in odd characteristic, we suppose
in this section that <M>f</M> is a quadratic form in even characteristic. We
call a two dimensional vector space a <E>hyperbolic line</E> if it contains a
non-zero vector such that <M>f(v) = 0</M>. It is proved (see Proposition 6.9 of 
<Cite Key="Cameron"/>) that any two
hyperbolic lines are isometric, and we choose as canonical representative the
quadratic form with polynomial <Alt Not="HTML"><M>
x_1 x_2</M></Alt><Alt Only="HTML">x&lt;sub>1&lt;/sub>x&lt;sub>2&lt;/sub></Alt>.
As in the case of the orthogonal bilinear forms, it can be proved (see Theorem 6.10 of
 <Cite Key="Cameron"/>) that <M>(V,f)</M> can be written as the orthogonal
direct sum of hyperbolic lines
and one subspace <M>U</M> of dimension at most two. The behaviour of <M>f</M> on
the subspace <M>U</M> determines the similarity class of <M>f</M>. The
classification of quadratic forms in even characteristic is analogous to
the one in odd characteristic.
<List><Item>If the dimension of <M>U</M> is zero, then <M>(V,f)</M> is the orthogonal direct
sum of hyperbolic lines, and hence <M>(V,f)</M> is isometric to the formed
space <M>(V,f')</M>, with polynomial 
<Alt Not="HTML"><Display>
x_1 x_2 + \ldots + x_{n-1}x_n,
</Display></Alt><Alt Only="HTML"><Display>x&lt;sub>1&lt;/sub> x&lt;sub>2&lt;/sub> + ... + x&lt;sub>n&lt;font face="symbol">-&lt;/font
>1&lt;/sub>x&lt;sub>n&lt;/sub>,</Display></Alt>
which is chosen as the canonical form. Note that the dimension of the
vector space <M>V</M> is necessarily even. We call <M>f</M> <E>hyperbolic</E> (see also Section <Ref Sect="theory:sesquilinearforms"/>). It
follows also that in this similarity class, there is only one isometry class.
The Witt index of <M>f</M> equals <M>n/2</M>.
</Item>
<Item>If the dimension of <M>U</M> is one, then necessarily the polynomial of <M>f</M>
equals
<Alt Not="HTML"><Display>
\mu x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n
</Display></Alt><Alt Only="HTML"><Display>&#956; x&lt;sub>1&lt;/sub>&lt;sup>2&lt;/sup> + x&lt;sub>2&lt;/sub>
x&lt;sub>3&lt;/sub> + ... + x&lt;sub>n&lt;font face="symbol">-&lt;/font
>1&lt;/sub>x&lt;sub>n&lt;/sub></Display></Alt>
for some <Alt Not="HTML"><M>\mu \in F</M></Alt><Alt Only="HTML">&#956; in
<M>F</M></Alt>, and the dimension of the vector space <M>V</M> is odd. We call
<M>f</M> <E>parabolic</E> (see also Section <Ref Sect="theory:sesquilinearforms"/>). Since every element is a square in even
characteristic, rescaling the first basis vector yields <Alt
Not="HTML"><M>\mu=1</M></Alt><Alt Only="HTML">&#956;=1</Alt>. The Witt index of
<M>f</M> equals <M>(n-1)/2</M>.
</Item>
<Item>
Suppose at last that the dimension of <M>U</M> is two. We may suppose that
<M>U</M> is not a hyperbolic line. It is not difficult to see that a suitable base change
yields the polynomial <Alt Not="HTML"><Display>
d x_1^2 + x_1x_2 + x_2^2 + x_3 x_4 + \ldots + x_{n-1}x_n
</Display></Alt><Alt Only="HTML"><Display>d
x&lt;sub>1&lt;/sub>&lt;sup>2&lt;/sup> + x&lt;sub>1&lt;/sub>x&lt;sub>2&lt;/sub> +
x&lt;sub>2&lt;/sub>&lt;sup>2&lt;/sup> + x&lt;sub>3&lt;/sub> x&lt;sub>4&lt;/sub> + ... + x&lt;sub>n&lt;font
face="symbol">-&lt;/font>1&lt;/sub>x&lt;sub>n&lt;/sub></Display></Alt> for an
element of category 1, this is, an element <M>d</M> such that <M>T(d)=1</M> with
<M>T</M> the trace map from <M>F</M> to <M>GF(2)</M>. Furthermore, an easy
argument shows that an appropriate base change allows to choose any element of
category 1 for <M>d</M>. It follows also that the dimension of the vector space
<M>V</M> is even. We call <M>f</M> <E>elliptic</E> (see also Section <Ref Sect="theory:sesquilinearforms"/>). 
The Witt index of <M>f</M> equals <M>(n-2)/2</M>.
</Item>
</List>
Hence, non-degenerate quadratic forms in even characteristic come in three
similarity classes, which is analogous to the odd characteristic case, and
each similarity class contains only one isometry class, which is different than
in the odd characteristic case
<P/>
Suppose that <M>f</M> is a degenerate quadratic form on the <M>n</M>-dimensional
vector space <M>V</M>, then <M>Rad(f)</M> is a non-trivial subspace of the
vector space <M>V</M>. The vector space <M>V</M> can be written as the orthogonal
direct sum of a subspace <M>W</M> and <M>Rad(f)</M>, and the restriction of
<M>f</M> to <M>W</M> is a non-degenerate quadratic form on <M>W</M>. Hence,
<M>f</M> is isometric with a quadratic form with one of the three above
polynomials. The dimension of the maximal isotropic subspaces is the sum of the
Witt index and the dimension of the radical.

</Section>

<Section Label="morphisms_functions">
<Heading>Operations based on morphisms of forms</Heading>

<ManSection>
<Attr Name="BaseChangeToCanonical" Arg="f"/>
<Returns>a transition matrix <A>b</A> from one basis to another</Returns>
<Description>
The argument <A>f</A> is a sesquilinear or quadratic form.
For every isometry class of forms, there is a canonical
representative, as described in Section <Ref Sect="morphisms:sesquilinear"/>. If <A>M</A> is
the Gram matrix of the form <A>f</A>, then this method returns an invertible
matrix <A>b</A> such that <A>b * M * TransposedMat(b)</A> is
the Gram matrix of the canonical representative. That is, <A>b</A> is the <E>transition matrix</E>
from a basis of the underlying vector space of <A>f</A> to another basis.
<Example>
<#Include SYSTEM "../examples/include/basechangetocanonical.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Oper Name="BaseChangeHomomorphism" Arg="b, gf"/>
<Returns>the inner automorphism of GL(d,q) associated
to the transition matrix <A>b</A>.</Returns>
<Description>
The argument <A>b</A> must be an invertible matrix of size <M>d</M> over
the finite field <A>gf</A> of order <M>q</M>. This method returns the inner automorphism
of <M>GL(d,q)</M> induces by conjugation by <M>b</M>.
<Example>
<#Include SYSTEM "../examples/include/basechangehom.include"> 
</Example>
</Description>
</ManSection>


<ManSection>
<Attr  Name="IsometricCanonicalForm" Arg="f"/>
<Returns>the canonical form isometric to the
sesquilinear or quadratic form <A>f</A>.</Returns>
<Description>
The argument <A>f</A> is a sesquilinear or quadratic form.
For every isometry class of forms, there is a canonical
representative, as described in Section <Ref Sect="morphisms:sesquilinear"/>,
which is the returned form.
<Example>
<#Include SYSTEM "../examples/include/isometriccanonicalform.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Oper Name="ScalarOfSimilarity" Arg="M, form"/>
<Returns>a finite field element
</Returns>
<Description>
Recall that a similarity of a form <M>f</M> on a vector space <M>V</M>, is a
linear transformation <M>g</M> of <M>V</M> where there exists some nonzero 
scalar <M>c</M> such that for all <M>v,w</M> in <M>V</M>,
<Alt Not="HTML"><M>f(u^g,v^g) = c f(u,v).</M></Alt><Alt
Only="HTML"><Display>
f(u&lt;sup>g&lt;/sup>,v&lt;sup>g&lt;/sup>) = c f(u,v)</Display></Alt>
This operation finds for a particular matrix <A>M</A>, giving rise to
a similarity of the sesquilinear form <A>form</A>, the said scalar <M>c</M>.
<Example>
<#Include SYSTEM "../examples/include/scalarfromsim.include"> 
</Example>
</Description>
</ManSection>

<ManSection>
<Attr  Name="WittIndex" Arg="f"/>
<Returns>the Witt index of the form <A>f</A>.
</Returns>
<Description>
The argument <A>f</A> is a sesquilinear or quadratic form.
When <A>f</A> is degenerate, its Witt index is defined as the Witt index of its
non-degenerate part, see Sections <Ref Sect="theory:sesquilinearforms"/> 
and <Ref Sect="quadforms"/>.
</Description>
</ManSection>

<ManSection>
<Prop  Name="IsEllipticForm" Arg="f"/>
<Returns>true or false.
</Returns>
<Description>
The argument <A>f</A> is a sesquilinear or quadratic form. This operation
returns <A>true</A> is and only if <A>f</A> is elliptic; that is,
it is orthogonal of minus type, or in other words, has even dimension
and non-maximal Witt index (see Section <Ref
Sect="morphisms:bilinear"/> for sesquilinear forms and Section <Ref
Sect="morphisms:quadratic"/> for quadratic forms).
</Description>
</ManSection>

<ManSection>
<Prop  Name="IsParabolicForm" Arg="f"/>
<Returns>true or false.
</Returns>
<Description>
The argument <A>f</A> is a sesquilinear or quadratic form. This operation
returns <A>true</A> is and only if <A>f</A> is parabolic; that is,
it is orthogonal of neutral type, or in other words, it has odd dimension (see Section <Ref
Sect="morphisms:bilinear"/> for sesquilinear forms and Section <Ref
Sect="morphisms:quadratic"/> for quadratic forms).
</Description>
</ManSection>

<ManSection>
<Attr  Name="IsHyperbolicForm" Arg="f"/>
<Returns>true or false.
</Returns>
<Description>
The argument <A>f</A> is a sesquilinear or quadratic form. This operation
returns <A>true</A> is and only if <A>f</A> is hyperbolic; that is,
it is orthogonal of plus type, or in other words, has even dimension
and maximal Witt index (see Section <Ref
Sect="morphisms:bilinear"/> for sesquilinear forms and Section <Ref
Sect="morphisms:quadratic"/> for quadratic forms).
</Description>
</ManSection>

</Section>

</Chapter>
