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Metric Vector Spaces: The Theory of Bilinear Forms 279
invariants under congruence and that the set of all matrices of the form ? Á c
is a set of canonical forms.
Witt's Extension and Cancellation Theorems
We now prove the Witt theorems for symplectic geometries.
(
Theorem 11.15 Witt's extension theorem) Let = and = Z be isometric
nonsingular symplectic geometries over a field . Then any isometry
-
¢: ¦ : = Z
on a subspace of can be extended to an isometry from to = Z .
:
=
=
Proof. According to Theorem 11.12, we can extend to a nonsingular
completion of , so we may simply assume that and : : are nonsingular.
:
Hence,
=~ : p :
and
Z
=~ : p ² :³
To complete the extension of to , we need only choose a hyperbolic basis
=
² Á ÁÃÁ Á ³
for : and a hyperbolic basis
Z
Z
Z
Z
² Á ÁÃÁ Á ³
for ²:³ and define the extension by setting ~ Z and ~ Z .
As a corollary to Witt's extension theorem, we have Witt's cancellation theorem.
(
Theorem 11.16 Witt's cancellation theorem) Let = and = Z be isometric
nonsingular symplectic geometries over a field . If
-
Z
=~ : p : and = ~ ; p ;
then
: ; ¬ : ;
The Structure of the Symplectic Group: Symplectic Transvections
Let us examine the nature of symplectic transformations (isometries) on a
nonsingular symplectic geometry = . Recall that for a real vector space, an
isometric isomorphism, which corresponds to an isometry in the present context,
is the same as an orthogonal map and orthogonal maps are products of
is defined as an
reflections (Theorem 10.17). Recall also that a reflection / #
operator for which
