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 : and = ~ ; p ;
then
: ; ¬ : ;
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 : and = ~ ; p ;
and : ; . Then any isometry ¢ :¦ ; can be extended to an isometry from
= = to Z . According to Theorem 11.9, we have ² : ³ ~ ; and so : ; .
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 :
Since is an isometry, : is also nonsingular and we can write
Z
=~ : p ² :³
Since : : , Witt's cancellation theorem implies that : ² :³ . If
Z
¢: ¦ ² :³ is an isometry, then the map ¢= ¦ = defined by
²" b #³ ~ " b #
for ": and # : 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.