3. Background Theory on Forms
  
  In this section, we give a very brief overview on the theory of sesquilinear
  and  quadratic  forms. The reader can find more in the texts: Cameron [C00],
  Taylor [T92], Aschbacher [A00], or Kleidman and Liebeck [KL90].
  
  
  3.1 Sesquilinear forms, dualities, and polarities
  
  A  sesquilinear  form  on  a  vector space V over a field F, is a map f from
  Vtimes V to F which is linear in the first coordinate, but semilinear in the
  second  coordinate;  that  is,  there  is  a  field  automorphism alpha (the
  companion automorphism of f) such that f(v,lambda w)=lambda^alpha f(v,w) for
  all  v,win  V  and lambdain F. If alpha is the identity, then f is bilinear.
  Two  vectors v and w are orthogonal (w.r.t. f) if f(v,w) = 0. The radical of
  f  is  the  subspace  consisting  of  vectors  which are orthogonal to every
  vector,  and  we say that f is non-degenerate if its radical is trivial (and
  degenerate  otherwise). A duality delta of a projective space mathcalP is an
  incidence reversing permutation of the subspaces of mathcalP, and a polarity
  of  mathcalP  is  a  duality  of  order  2. An example of such arises from a
  non-degenerate  sesquilinear  form; given a subspace W, we let W^perp be the
  set  of  points  which are orthogonal with every element of W. We say that a
  subspace  W is totally isotropic with respect to a polarity if W contains or
  is  contained  in W^perp. The Birkhoff-von Neumann Theorem states that every
  duality  of  the  projective  space  PG(n,q)  arises  from  a non-degenerate
  sesquilinear  form  (up  to a scalar). Such a duality is a polarity if it is
  reflexive,  i.e.,  f(v,w)=0  implies  f(w,v)=0. Now a sesquilinear form f is
  hermitian   if  f(v,w)=f(w,v)^alpha  holds  where  alpha  is  the  companion
  automorphism  of  f and alpha has order 2. But if alpha is trivial then f is
  symmetric. If f inturn satisfies f(v,v)=0 (for all v) then f is alternating.
  It is a well-known theorem of polar geometry that a non-degenerate reflexive
  sesquilinear  form  is  either  alternating,  symmetric,  or  similar  to an
  hermitian  form.  The  associated polarity is called symplectic, orthogonal,
  and  unitary  respectively  (though there are some other conventions for the
  characteristic 2 case).
  
  
  3.1-1 Example
  
  Let  M  be an invertible 4-dimensional square matrix over F and consider the
  following  map on pairs of elements of the 4-dimensional vector space V over
  F:
  
  \[
       f(v, w) = v M w^T.
  \]
  
  Then f is a sesquilinear form of V and M is the Gram matrix of f.
  
  
  3.2 Quadratic forms
  
  We  have  seen  that  a  polar space can arise from a reflexive sesquilinear
  form,  but  there  are  other  polar spaces which do not arise this way, but
  instead  have an associated quadratic form. A map Q from a vector space V to
  a field F is a quadratic form if it satisfies Q(lambda v) = lambda^2Q(v) for
  all  vin  V  and lambdain F. We say that a subspace W is totally singular if
  the  restriction  of  Q  to  W  is  trivial. Note that a subspace is totally
  isotropic  (with  respect  to  the  associated  polarity)  if  it is totally
  singular, but the converse is not always true.
  
  
  3.2-1 Example
  
  Let  M  be an invertible 4-dimensional square matrix over F and consider the
  following map on elements of the 4-dimensional vector space V over F:
  
  \[
       f(v) = v M v^T.
  \]
  
  Then f is a quadratic form of V and M is the Gram matrix of f.
  
  Given  a quadratic form Q, there is an associated sesquilinear form f (which
  may not be reflexive) defined as follows
  
  \[
       f(v,w)=Q(v+w)-Q(v)-Q(w).
  \]
  
  For characteristic not 2, the quadratic form and its associated sesquilinear
  form f determine one another, as 2 Q(v)= f(v,v) (for all v).
  
  
  3.3 Morphisms of forms
  
  An  isometry  from  a  formed  space  (V,f)  to  a  formed space (W,f') is a
  bijection phi such that for all v,w in V we have
  
  \[
       f(v,w) = f'(\phi(v), \phi(w)).
  \]
  
  The  weaker  notions of similarity and semi-similarity are also important in
  polar  geometry.  If there exists a scalar lambda such that for all v,w in V
  we have
  
  \[
       f(v,w) = \lambda f'(\phi(v), \phi(w))
  \]
  
  then  we  say  that  phi  is  a  similarity.  If  we also have a fixed field
  automorphism alpha such that
  
  \[
       f(v,w)=\lambda f'(\phi(v), \phi(w))^\alpha,
  \]
  
  then  phi  is  a  semi-similarity.  Naturally, we say that the formed spaces
  (V,f)  and  (W,f') are isometric (resp. similar) if there exists an isometry
  (resp. similarity) between them. Every non-degenerate reflexive sesquilinear
  form is alternating, symmetric, or similar to an hermitian form. Thus, up to
  similarity,   the   non-degenerate  polar  spaces  come  in  five  flavours:
  symplectic,   unitary,   orthogonal-elliptic,   orthogonal-hyperbolic,   and
  orthogonal-parabolic.  In  the  case  of  the  orthogonal  spaces,  they are
  distinguished  by  their  Witt  Index (the common dimension of their maximal
  totally singular/isotropic subspaces).
  
  
  3.4 An important convention
  
  In  Forms, we have stipulated a convention on the creation of forms so as to
  cause  as  little  confusion as possible. The hermitian forms will simply be
  those  with  the  Frobenius  Automorphism  is the companion automorphism. We
  should  also  caution  the user on what information is "enough" to specify a
  form as problems can arise in even characteristic.
  
  
  3.4-1 Example
  
  Let  F be a finite field of square order and let M be the following 4times 4
  matrix over F:
  
     -------------------
      | 0   1   0   0 | 
      | 1   0   0   0 | 
      | 0   0   0   1 | 
      | 0   0   1   0 | 
     -------------------
  
  Let alpha be the unique automorphism of F of order 2. Then the form
  
  \[
       (u, v) := uM(v^T)^\alpha
  \]
  
  defines   a  non-degenerate  hermitian  sesquilinear  form.  If  F  has  odd
  characteristic, then the form
  
  \[
       (u, v) := uMv^T
  \]
  
  defines  a non-degenerate orthogonal form, but if F has even characteristic,
  then this form is both:
  
  (1)   a symplectic bilinear form, and
  
  (2)   the associated bilinear form arising from a quadratic form.
  
  In  the  latter,  case  we  see  that  the bilinear form does not define the
  quadratic  form, but rather that the quadratic form is necessary in order to
  define the polar geometry.
  
  
  3.5 Canonical forms
  
  Every nondegenerate polar space has a direct decomposition into a sum
  
  \[
       L_1\perp L_2\perp\cdots L_n\perp U
  \]
  
  where  each of the L_i are hyperbolic lines and U is an anisotropic subspace
  of  dimension  at  most  2.  Thus  if  the given polar space is defined by a
  sesquilinear  form  f,  then  there is an isometric polar space defined by a
  Gram Matrix of the form
  
     ------------------------------------------------------------------------
      | U |              |              |       |              |     |     | 
     ------------------------------------------------------------------------
      |   |    0       1 |              |       |              |     |     | 
      |   | \epsilon   0 |              |       |              |     |     | 
     ------------------------------------------------------------------------
      |   |              |    0       1 |       |              |     |     | 
      |   |              | \epsilon   0 |       |              |     |     | 
     ------------------------------------------------------------------------
      |   |              |              | *     |              |     |     | 
      |   |              |              |     * |              |     |     | 
     ------------------------------------------------------------------------
      |   |              |              |       |    0       1 |     |     | 
      |   |              |              |       | \epsilon   0 |     |     | 
     ------------------------------------------------------------------------
  
  were the top left hand corner represents the anisotropic part, and there are
  zeros  everywhere  else.  The  value  of  epsilon  is  -1  if  the  form  is
  alternating, otherwise it is 1.