Page 293 - Advanced Linear Algebra
P. 293

Metric Vector Spaces: The Theory of Bilinear Forms  277



            To see why, suppose that       is an isometry and  8¢= ¦ >  ~ ²# Á ÃÁ# ³  is an


                          =
            ordered basis for  . Then  ~  ²  #  Á 9  Ã  Á       #     ³     is an ordered basis for  >   and


                          4 ²= ³ ~²º# Á # »³ ~²º # Á # »³ ~4 ²> ³

                           8

                                                          9


            Thus,  the  congruence  class of matrices representing  =   is identical to the
            congruence class of matrices representing >  .
            Conversely, suppose that  =   and  >   are metric vector  spaces  with  the  same
            congruence  class of representing matrices. Then if  8 ~²# Á Ã Á # ³  is an


                                                9
                          =
            ordered basis for  , there is an ordered basis  ~ ²$ ÁÃÁ$ ³  for >  for which


                          ²º# Á # »³ ~ 4 ²= ³ ~ 4 ²> ³ ~ ²º$ Á $ »³


                                               9


                                      8
            Hence, the map         defined by  # ~ $       is an isometry from   to >  .
                                            ¢= ¦ >
                                                                   =
            We have shown that two metric vector spaces are isometric if and only if they
            have the same congruence class of representing matrices.  Thus,  we  can
            determine whether any two metric vector spaces are isometric by representing
            each space with a matrix and determining whether these matrices are congruent,
            using a set of canonical forms or a set of complete invariants.
            Symplectic Geometry
            We now turn to a study of the structure of orthogonal and symplectic geometries
                                                                          )
                                                       (
            and their isometries. Since the study of the structure  and the structure itself  of
            symplectic geometries is simpler than that of orthogonal geometries, we begin
            with the symplectic case. The reader who is interested only in the orthogonal
            case may omit this section.
            Throughout this section, let   be a nonsingular symplectic geometry.
                                  =
            The Classification of Symplectic Geometries
            Among  the  simplest  types  of metric vector spaces are those that possess an
            orthogonal basis. However, it is easy to see that a symplectic geometry   has an
                                                                     =
            orthogonal basis if and only if it is totally degenerate and so no “interesting”
            symplectic geometries have orthogonal bases.
            Thus, in searching for an orthogonal decomposition of  =  ,  we  turn  to  two-
            dimensional subspaces and this puts us in mind of hyperbolic spaces. Let   be
                                                                        <
            the family of all hyperbolic subspaces of  , which is nonempty since the zero
                                              =
                                                                         =
            subspace ¸ ¹  is singular and so has a nonzero hyperbolic extension. Since   is
            finite-dimensional,  <   has a maximal member  >  . Since  >   is  nonsingular,  if
            > £= , then
                                       =~ >  p >  ž
            where  >  ž   . But then if #   > £ ¸ ¹  ž   is nonzero, there is a hyperbolic extension
   288   289   290   291   292   293   294   295   296   297   298