Quelle morphisms.xml Sprache: XML

<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 <M>\phi</M> is a linear map from <M>V</M> to <M>W</M>.

The map <M>\phi</M> 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 \in V</M> we have
<Display>
<Alt Only="Text">f(v,w) = f'(\phi(v), \phi(w)).
<Alt Only="LaTeX">f(v,w) = f'(\phi(v), \phi(w)).
<Alt Only="HTML MathJax">f(v,w) = f'(\phi(v), \phi(w)).
<Alt Only="HTML noMathJax">f(v,w) = f'(φ(v),φ(w))
The map <M>\phi</M> 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 \in V</M>
we have
<Display>
<Alt Only="Text">f(v,w) = \lambda f'(\phi(v), \phi(w)).
<Alt Only="LaTeX">f(v,w) = \lambda f'(\phi(v), \phi(w)).
<Alt Only="HTML MathJax">f(v,w) = \lambda f'(\phi(v), \phi(w)).
<Alt Only="HTML noMathJax">f(v,w) = λ (f'(φ(v),φ(w)))
</Display>
for some non-zero <M>\lambda \in F</M>. Finally, the map <M>\phi</M> 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 Only="Text">f(v,w) = \lambda f'(\phi(v), \phi(w))^\alpha
<Alt Only="LaTeX">f(v,w) = \lambda f'(\phi(v), \phi(w))^\alpha
<Alt Only="HTML MathJax">f(v,w) = \lambda f'(\phi(v), \phi(w))^\alpha
<Alt Only="HTML noMathJax">f(v,w) = λ f'(φ(v),φ(w))<sup>α</sup>
for some non-zero <Alt Only="HTML"><M>λ</M> in <M>F</M></Alt><Alt
Not="HTML"><M>\lambda \in F</M></Alt> and a field automorphism <M>\alpha</M>
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">φ</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 <M>\alpha</M>. 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 Only="Text"><M>f(e_i,e_i)=1</M></Alt><Alt Only="LaTeX"><M>f(e_i,e_i)=1</M></Alt>
<Alt Only="HTML MathJax"><M>f(e_i,e_i)=1</M></Alt>
<Alt Only="HTML noMathJax">f(e<SUB>i</SUB>,e<SUB>j</SUB>)=1</Alt> for
each basisvector <Alt Only="Text"><M>e_i</M></Alt><Alt Only="LaTeX"><M>e_i</M></Alt>
<Alt Only="HTML MathJax"><M>e_i</M></Alt><Alt Only="HTML noMathJax">e<SUB>i</SUB></Alt>.
Hence <M>(V,f)</M> is isometric with <M>(V,f')
with <M>f' the non-degenerate hermitian form with the identity matrix over
<M>F</M>. The Witt index of <M>f</M> equals <Alt Only="Text"><M>n/2</M></Alt><Alt Only="LaTeX"><M>\frac{n}{2}</M></Alt>
<Alt Only="HTML MathJax"><M>\frac{n}{2}</M></Alt><Alt Only="HTML noMathJax"><M>n/2</M></Alt> when <M>n</M> is even and
<Alt Only="Text"><M>(n-1)/2</M></Alt><Alt Only="LaTeX"><M>\frac{n-1}{2}</M></Alt>
<Alt Only="HTML MathJax"><M>\frac{n-1}{2}</M></Alt><Alt Only="HTML noMathJax"><M>(n-1)/2</M></Alt>
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="Text"><Display>\left( \begin{array}{cc}0 & r \\ -r & 0 \end{array}
\right)</Display></Alt>
<Alt Only="LaTeX"><Display>\left( \begin{array}{cc}0 & r \\ -r & 0 \end{array}
\right)</Display></Alt>
<Alt Only="HTML MathJax"><Display>\left( \begin{array}{cc}0 & r \\ -r & 0 \end{array}
\right)</Display></Alt>
<Alt Only="HTML noMathJax">
<table style="color:#000"><tr>
<td><table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td><td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
<tr>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>r</td>
</tr>
<tr>
<td align="center" valign="center" width=30>-r</td>
<td align="center" valign="center" width=30>0</td>
</tr>
</table></td><td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td></tr></table></td>
</tr></table>
</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="Text"><Display>\left( \begin{array}{ccccccc}
0 & 1 & 0 & 0 & \ldots & 0 & 0 \\
-1 & 0 & 0 & 0 & \ldots & 0 & 0 \\
0 & 0 & 0 & 1 & \ldots & 0 & 0 \\
0 & 0 & -1 & 0 & \ldots & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & 0 & \ldots & 0 & 1 \\
0 & 0 & 0 & 0 & \ldots & -1 & 0 \\
\end{array} \right).</Display>
</Alt>
<Alt Only="LaTeX"><Display>\left( \begin{array}{ccccccc}
0 & 1 & 0 & 0 & \ldots & 0 & 0 \\
-1 & 0 & 0 & 0 & \ldots & 0 & 0 \\
0 & 0 & 0 & 1 & \ldots & 0 & 0 \\
0 & 0 & -1 & 0 & \ldots & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & 0 & \ldots & 0 & 1 \\
0 & 0 & 0 & 0 & \ldots & -1 & 0 \\
\end{array} \right).</Display>
</Alt>
<Alt Only="HTML MathJax"><Display>\left( \begin{array}{ccccccc}
0 & 1 & 0 & 0 & \ldots & 0 & 0 \\
-1 & 0 & 0 & 0 & \ldots & 0 & 0 \\
0 & 0 & 0 & 1 & \ldots & 0 & 0 \\
0 & 0 & -1 & 0 & \ldots & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & 0 & \ldots & 0 & 1 \\
0 & 0 & 0 & 0 & \ldots & -1 & 0 \\
\end{array} \right).</Display>
</Alt>
<Alt Only="HTML noMathJax">

<table style="color:#000"><tr>
<td><table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td><td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
<tr>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>1</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>...</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
</tr>
<tr>
<td align="center" valign="center" width=30>-1</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>...</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
</tr>
<tr>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>1</td>
<td align="center" valign="center" width=30>...</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
</tr>
<tr>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>-1</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>...</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
</tr>
<tr>
<td align="center" valign="center" width=30>:</td>
<td align="center" valign="center" width=30>:</td>
<td align="center" valign="center" width=30>:</td>
<td align="center" valign="center" width=30>:</td>
<td align="center" valign="center" width=30><sup><big>·</big></sup>·<sub><big>·</big></sub></td>
<td align="center" valign="center" width=30>:</td>
<td align="center" valign="center" width=30>:</td>
</tr>
<tr>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>...</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>1</td>
</tr>
<tr>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>...</td>
<td align="center" valign="center" width=30>-1</td>
<td align="center" valign="center" width=30>0</td>
</tr>
</table></td><td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td></tr></table></td>
</tr></table>

</Alt>
The Witt index of <M>f</M> equals <Alt Only="Text"><M>n/2</M></Alt>
<Alt Only="LaTeX"><M>\frac{n}{2}</M></Alt><Alt Only="HTML MathJax"><M>\frac{n}{2}</M></Alt>
<Alt Only="HTML noMathJax"><M>n/2</M></Alt>.
</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 <M>v</M> 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 noMathJax">
<table style="color:#000"><tr>
<td><table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td><td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
<tr>
<td align="center" valign="center" width=30>0</td>
<td align="center" valign="center" width=30>1</td>
</tr>
<tr>
<td align="center" valign="center" width=30>1</td>
<td align="center" valign="center" width=30>0</td>
</tr>
</table></td><td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td></tr></table></td>
</tr></table>
</Alt>
<Alt Only="Text"><Display>\left( \begin{array}{cc}0 & 1 \\ 1 & 0 \end{array}
\right).</Display></Alt>
<Alt Only="LaTeX"><Display>\left( \begin{array}{cc}0 & 1 \\ 1 & 0 \end{array}
\right).</Display></Alt>
<Alt Only="HTML MathJax"><Display>\left( \begin{array}{cc}0 & 1 \\ 1 & 0 \end{array}
\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'), where the Gram matrix of f'</M> consists of blocks as
described above. The chosen canonical form has polynomial
<Alt Only="Text"><Display>x_1 x_2 + \ldots + x_{n-1}x_n</Display></Alt>
<Alt Only="LaTeX"><Display>x_1 x_2 + \ldots + x_{n-1}x_n</Display></Alt>
<Alt Only="HTML MathJax"><Display>x_1 x_2 + \ldots + x_{n-1}x_n</Display></Alt>
<Alt Only="HTML noMathJax"><center>x<sub>1</sub> x<sub>2</sub> + ... + x<sub>n<font face="symbol">-</font
>1</sub>x<sub>n</sub></center></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 <Alt Only="Text"><M>n/2</M></Alt>
<Alt Only="LaTeX"><M>\frac{n}{2}</M></Alt><Alt Only="HTML MathJax"><M>\frac{n}{2}</M></Alt>
<Alt Only="HTML noMathJax"><M>n/2</M></Alt>.
</Item>
<Item>If the dimension of <M>U</M> is one, then necessarily the polynomial of <M>f</M>
equals
<Alt Only="Text"><Display>\mu x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n</Display></Alt>
<Alt Only="LaTeX"><Display>\mu x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n</Display></Alt>
<Alt Only="HTML MathJax"><Display>\mu x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n</Display></Alt>
<Alt Only="HTML noMathJax"><center>μ x<sub>1</sub><sup>2</sup> + x<sub>2</sub>
x<sub>3</sub> + ... + x<sub>n<font face="symbol">-</font
>1</sub>x<sub>n</sub></center></Alt>
for some <M>\mu \in F</M>, 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 <M>\mu</M>
is a square in <M>F</M>, then rescaling the first basis vector yields a polynomial
<Alt Only="Text"><Display>x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n</Display></Alt>
<Alt Only="LaTeX"><Display>x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n</Display></Alt>
<Alt Only="HTML MathJax"><Display>x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n</Display></Alt>
<Alt Only="HTML noMathJax"><center>x<sub>1</sub><sup>2</sup> + x<sub>2</sub>
x<sub>3</sub> + ... + x<sub>n<font face="symbol">-</font
>1</sub>x<sub>n</sub></center></Alt>
which we choose as the canonical form for this similarity class. If
<M>\mu</M> is a non-square, a rescaling of
<Alt Only="Text"><M>x_2,x_4,\ldots,x_{n-1}</M></Alt>
<Alt Only="LaTeX"><M>x_2,x_4,\ldots,x_{n-1}</M></Alt>
<Alt Only="HTML MathJax"><M>x_2,x_4,\ldots,x_{n-1}</M></Alt>
<Alt Only="HTML noMathJax">x<sub>2</sub>,x<sub>4</sub>,...,x<sub>n<font face="symbol">-</font
>1</sub></Alt> yields a polynomial
<Alt Only="Text"><Display>\mu (x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n)</Display></Alt>
<Alt Only="LaTeX"><Display>\mu (x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n)</Display></Alt>
<Alt Only="HTML MathJax"><Display>\mu (x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n)</Display></Alt>
<Alt Only="HTML noMathJax"><center>μ(x<sub>1</sub><sup>2</sup> + x<sub>2</sub>
x<sub>3</sub> + ... + x<sub>n<font face="symbol">-</font
>1</sub>x<sub>n</sub>)</center></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
<Alt Only="Text"><M>(n-1)/2</M></Alt>
<Alt Only="LaTeX"><M>\frac{n-1}{2}</M></Alt><Alt Only="HTML MathJax"><M>\frac{n-1}{2}</M></Alt>
<Alt Only="HTML noMathJax"><M>(n-1)/2</M></Alt>.
</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 Only="Text"><Display>\mu x_1^2 + x_2^2 + x_3 x_4 + \ldots + x_{n-1}x_n</Display></Alt>
<Alt Only="LaTeX"><Display>\mu x_1^2 + x_2^2 + x_3 x_4 + \ldots + x_{n-1}x_n</Display></Alt>
<Alt Only="HTML MathJax"><Display>\mu x_1^2 + x_2^2 + x_3 x_4 + \ldots + x_{n-1}x_n</Display></Alt>
<Alt Only="HTML noMathJax"><center>μx<sub>1</sub><sup>2</sup> +
x<sub>2</sub><sup>2</sup> + x<sub>3</sub> x<sub>4</sub> + ... + x<sub>n<font
face="symbol">-</font>1</sub>x<sub>n</sub></center></Alt> for a
non-square <M>\mu \in F</M>, 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
<Alt Only="Text"><M>(n-2)/2</M></Alt>
<Alt Only="LaTeX"><M>\frac{n-2}{2}</M></Alt><Alt Only="HTML MathJax"><M>\frac{n-2}{2}</M></Alt>
<Alt Only="HTML noMathJax"><M>(n-2)/2</M></Alt>.
</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 &radf; 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 &radf;, 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 noMathJax">

<table style="color:#000"><tr>
<td><table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td><td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
<tr>
<td align="center" valign="center" width=30>M</td>
<td align="center" valign="center" width=30>A</td>
</tr>
<tr>
<td align="center" valign="center" width=30>B</td>
<td align="center" valign="center" width=30>C</td>
</tr>
</table></td><td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td></tr></table></td>
</tr></table>

</Alt>
<Alt Only="HTML MathJax">
<Display>\left( \begin{array}{cc} M & A \\ B & C \end{array}
\right)</Display>
</Alt>
<Alt Only="LaTeX">
    <Display>\left( \begin{array}{cc} M & A \\ B & C \end{array}
        \right)</Display>
</Alt>
<Alt Only="Text">
    <Display>\left( \begin{array}{cc} M & A \\ B & C \end{array}
        \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. As explained in Section <Ref
Sect="theory:sesquilinearforms"/>, the form <M>f</M> induces a non-degenerate form <M>g</M>
on the vector space &vmodradf;. The computed matrix <M>M</M> can be taken as Gram matrix
for the form <M>g</M>. As defined in Section <Ref Sect="theory:sesquilinearforms"/>, the
Witt index of the degenerate form <M>f</M> is the Witt index of the non-degenerate inducedform <M>g</M>.
The dimension of the maximal isotropic subspaces with relation to <M>f</M> 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">φ</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">φ</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 Only="Text">f(v) = f'(\phi(v)).
<Alt Only="LaTeX">f(v) = f'(\phi(v)).
<Alt Only="HTML MathJax">f(v) = f'(\phi(v)).
<Alt Only="HTML noMathJax">f(v) = f'(φ(v)).
The map <M>\phi</M> 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 Only="Text">f(v) = \lambda f'(\phi(v)).
<Alt Only="LaTeX">f(v) = \lambda f'(\phi(v)).
<Alt Only="HTML MathJax">f(v) = \lambda f'(\phi(v)).
<Alt Only="HTML noMathJax">f(v) = λ (f'(φ(v)))
</Display>
for some non-zero <M>\lambda \in F</M>. Finally, the map <M>\phi</M> 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 Only="Text">f(v)=\lambda f'(\phi(v))^\alpha
<Alt Only="LaTeX">f(v)=\lambda f'(\phi(v))^\alpha
<Alt Only="HTML MathJax">f(v)=\lambda f'(\phi(v))^\alpha
<Alt Only="HTML noMathJax">f(v) = λ f'(φ(v))^α
</Display>
for some non-zero <M>\lambda \in F</M> and a field automorphism <M>\alpha</M> 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-singular 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 Only="Text"><M>x_1 x_2</M></Alt><Alt Only="LaTeX"><M>x_1 x_2</M></Alt>
<Alt Only="HTML MathJax"><M>x_1 x_2</M></Alt><Alt Only="HTML noMathJax">x<sub>1</sub>x<sub>2</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'), with polynomial
<Alt Not="HTML noMathJax"><Display>x_1 x_2 + \ldots + x_{n-1}x_n,</Display></Alt>
<Alt Only="HTML noMathJax"><center>x<sub>1</sub> x<sub>2</sub> + ... + x<sub>n<font face="symbol">-</font
>1</sub>x<sub>n</sub>,</center></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 <Alt Only="Text"><M>n/2</M></Alt><Alt Only="LaTeX"><M>\frac{n}{2}</M></Alt>
<Alt Only="HTML MathJax"><M>\frac{n}{2}</M></Alt><Alt Only="HTML noMathJax"><M>n/2</M></Alt>.
</Item>
<Item>If the dimension of <M>U</M> is one, then necessarily the polynomial of <M>f</M>
equals
<Alt Not="HTML noMathJax"><Display>
\mu x_1^2 + x_2 x_3 + \ldots + x_{n-1}x_n
</Display></Alt>
<Alt Only="HTML noMathJax"><center>μ x<sub>1</sub><sup>2</sup> + x<sub>2</sub>
x<sub>3</sub> + ... + x<sub>n<font face="symbol">-</font
>1</sub>x<sub>n</sub></center></Alt>
for some <M>\mu \in F</M>, 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
<M>\mu=1</M> The Witt index of <M>f</M> equals <Alt Only="Text"><M>(n-1)/2</M></Alt><Alt Only="LaTeX"><M>\frac{n-1}{2}</M></Alt>
<Alt Only="HTML MathJax"><M>\frac{n-1}{2}</M></Alt><Alt Only="HTML noMathJax"><M>(n-1)/2</M></Alt>.
</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 noMathJax"><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 noMathJax"><center>d
x<sub>1</sub><sup>2</sup> + x<sub>1</sub>x<sub>2</sub> +
x<sub>2</sub><sup>2</sup> + x<sub>3</sub> x<sub>4</sub> + ... + x<sub>n<font
face="symbol">-</font>1</sub>x<sub>n</sub></center></Alt> for an
element of category 1, this is, an element <M>d</M> such that <Alt Only="Text"><M>Tr(d)=1</M></Alt>
<Alt Only="LaTeX"><M>\mathrm{Tr}(d)=1</M></Alt><Alt Only="HTML MathJax"><M>\mathrm{Tr}(d)=1</M></Alt>
<Alt Only="HTML noMathJax"><M>Tr(d)=1</M></Alt> with
<Alt Only="Text"><M>Tr</M></Alt><Alt Only="LaTeX"><M>\mathrm{Tr}</M></Alt><Alt Only="HTML MathJax"><M>\mathrm{Tr}</M></Alt>
<Alt Only="HTML noMathJax"><M>Tr(d)</M></Alt> the trace map from <M>F</M> to
<Alt Only="Text"><M>GF(2)</M></Alt><Alt Only="LaTeX"><M>\mathrm{GF}(2)</M></Alt>
<Alt Only="HTML MathJax"><M>\mathrm{GF}(2)</M></Alt><Alt Only="HTML noMathJax"><M>GF(2)</M></Alt>.
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 <Alt Only="Text"><M>(n-2)/2</M></Alt><Alt Only="LaTeX"><M>\frac{n-2}{2}</M></Alt>
<Alt Only="HTML MathJax"><M>\frac{n-2}{2}</M></Alt><Alt Only="HTML noMathJax"><M>(n-2)/2</M></Alt>.
</Item>
</List>
Hence, non-singular 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 singular quadratic form on the <M>n</M>-dimensional
vector space <M>V</M>, then &radf; 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 &radf;, and the restriction of
<M>f</M> to <M>W</M> is a non-singular 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.

<Subsection Label="morphisms:singular">
<Heading>Singular forms</Heading>
Suppose that <M>f</M> is a singular quadratic form on the vector space
<M>V</M>, then &radf; 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 &radf;, and the restriction of <M>f</M> to
<M>W</M> is a non-singular quadratic form on <M>W</M>. Hence, <M>f</M> is
isometric with a quadratic form having Gram matrix
<Alt Only="HTML noMathJax">

<table style="color:#000"><tr>
<td><table border=0 cellpadding=0 cellspacing=0px style="border-left:1px solid #000; border-right:1px solid #000; color:#000"><tr>
<td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td><td><table border=0 cellpadding=0 cellspacing=0 style="color:#000;">
<tr>
<td align="center" valign="center" width=30>M</td>
<td align="center" valign="center" width=30>A</td>
</tr>
<tr>
<td align="center" valign="center" width=30>B</td>
<td align="center" valign="center" width=30>C</td>
</tr>
</table></td><td style ="border-top:1px solid #000; border-bottom:1px solid #000;"> </td></tr></table></td>
</tr></table>

</Alt>
<Alt Only="HTML MathJax">
<Display>\left( \begin{array}{cc} M & A \\ B & C \end{array}
\right)</Display>
</Alt>
<Alt Only="LaTeX">
    <Display>\left( \begin{array}{cc} M & A \\ B & C \end{array}
        \right)</Display>
</Alt>
<Alt Only="Text">
    <Display>\left( \begin{array}{cc} M & A \\ B & C \end{array}
        \right)</Display>
</Alt>
where <M>M</M> is the Gram matrix of a non-singular quadratic form and
<M>A,B</M> and <M>C</M> are appropriate zero matrices. As explained in Section <Ref
Sect="quadforms"/>, the form <M>f</M> induces a non-singular form <M>g</M>
on the vector space &vmodradf;. The computed matrix <M>M</M> can be taken as Gram matrix
for the form <M>g</M>. As defined in Section <Ref Sect="quadforms"/>, the
Witt index of the singular form <M>f</M> is the Witt index of the non-singular induced form <M>g</M>.
The dimension of the maximal isotropic subspaces with relation to <M>f</M> is the sum of
the Witt index and the dimension of the radical.
</Subsection>
</Section>

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

<ManSection>
<Attr Name="BaseChangeToCanonical" Arg="f"/>
<Returns>a transition matrix <M>b</M> 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 <M>M</M> is
the Gram matrix of the form <A>f</A>, then this method returns an invertible
matrix <M>b</M> such that <M>b M</M> <C>TransposedMat</C><M>(b)</M>
(or <M>b M</M> <C>TransposedFrobeniusMat</C><M>(b, q)</M> for suitable <M>q</M>
if <A>f</A> is a hermitian form) is
the Gram matrix of the canonical representative. That is, <M>b</M> 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 &gldq; 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 &gldq; 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 noMathJax"><Display>f(u^g,v^g) = c f(u,v).</Display></Alt>
<Alt Only="HTML noMathJax"><Display>
f(u<sup>g</sup>,v<sup>g</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 on the vector space <M>V</M>.
When <A>f</A> is degenerate, respectively singular, its Witt index is defined as the Witt index of the
induced non-degenerate, respectively non-singular form on the vector space &vmodradf;, see Sections
<Ref Sect="theory:sesquilinearforms"/> and <Ref Sect="quadforms"/>.
<Example>
<#Include SYSTEM "../examples/include/wittindex.include">
</Example>
</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 on the vector space <M>V</M>. 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). If <A>f</A> is degenerate, respectively singular,
then this operation refers to the inuced non-degenerate, respectively non-singular form induced on the
vector space &vmodradf;.
</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 on the vector space <M>V</M>. 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). If <A>f</A> is degenerate, respectively singular,
then this operation refers to the inuced non-degenerate, respectively non-singular form induced on the
vector space &vmodradf;.

</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 on the vector space <M>V</M>. 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). If <A>f</A> is degenerate, respectively singular,
then this operation refers to the inuced non-degenerate, respectively non-singular form induced on the
vector space &vmodradf;.
</Description>
</ManSection>

<ManSection>
<Oper  Name="TypeOfForm" Arg="f"/>
<Returns>a number.
</Returns>
<Description>
The argument <A>f</A> is a sesquilinear or quadratic form on the vector space <M>V</M>
with radical <M>R</M>, a <M>k</M>-dimensional space.
Then <A>f</A> induces a non-degenerate/non-singular
form <M>g</M> on <M>V/R</M>. When <M>R</M> is the trivial vector space,
the form <M>g</M> is just the given form <A>f</A>. This operation returns
<List>
<Item>0 when <M>g</M> is symplecitc or parabolic;</Item>
<Item>+1 when <M>g</M> is hyperbolic;</Item>
<Item>-1 when <M>g</M> is elliptic;</Item>
<Item>-1/2 when <M>g</M> is hermitian in odd dimension;</Item>
<Item>+1/2 when <M>g</M> is hermitian in even dimension;</Item>
<Item>an error message when <A>f</A> is a pseudo form.</Item>
</List>
Note that no method is installed for the trivial form. The methods
for this operation rely on <F>IsParabolicForm</F>, <F>IsHyperbolicForm</F> and
<F>IsEllipticForm</F> for orthogonal bilinear forms and quadratic forms.
<Example>
<#Include SYSTEM "../examples/include/typeofform.include">
</Example>

</Description>
</ManSection>

</Section>

</Chapter>

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