Page 292 - Advanced Linear Algebra
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276    Advanced Linear Algebra




            The Witt extension theorem  says that if   is a subspace of  , then any isometry
                                                           =
                                            :
                                             ¢: ¦ : ‹ =  Z
                                           =
            can be extended to an isometry from   to =  Z . The Witt cancellation theorem
            says that if
                                                 Z
                              =~ : p : ž  and  = ~ ; p ; ž
            then
                                              ž
                                    :š ; ¬ : š ;   ž
            We will prove these theorems in both the orthogonal and symplectic cases a bit
            later in the chapter. For now, we simply want to show that it is easy to prove
            one Witt theorem using the other.

            Suppose that the Witt extension theorem holds and assume that
                                                 Z
                              =~ : p : ž  and  = ~ ; p ; ž
            and :š ; . Then any isometry  ¢ :¦ ;   can be extended to an isometry   from


            =   =  to   Z  . According to Theorem 11.9, we have   ²  :  ž  ³  ~  ;  ž   and so  :  ž  š  ;  ž  .
            Hence, the Witt cancellation theorem holds.
            Conversely, suppose that the Witt cancellation  theorem  holds  and  let
                         Z
                   ¢: ¦ : ‹ =   be an isometry. Since   can be extended to a nonsingular

            completion of  , we may assume that   is nonsingular. Then
                                          :
                       :
                                       =~ : p : ž
            Since   is an isometry,  :  is also nonsingular and we can write


                                      Z


                                     =~ : p ² :³  ž
                                                                ž

            Since  :š : , Witt's cancellation theorem implies  that  : š ² :³ ž  .  If

                                                         Z
                        ž
               ž     ¢: ¦ ² :³  is an isometry, then the map     ¢= ¦ =  defined by

                                               ²" b #³ ~ " b #

            for  ":  and  # : ž  is an isometry that extends  .  Hence  Witt's  extension
            theorem holds.
            The Classification Problem for Metric Vector Spaces
                                                                  (
            The  classification problem  for a class of metric vector spaces  such as the
                                      )
            orthogonal or symplectic spaces  is the problem of determining when two metric
            vector spaces in the class are isometric. The classification problem is considered
            “solved,” at least in a theoretical sense, by finding a set of canonical forms or a
            complete set of invariants for matrices under congruence.
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