Page 307 - Advanced Linear Algebra

Metric Vector Spaces: The Theory of Bilinear Forms 291

Now consider the nonsingular orthogonal geometry =~ span ²" Á " . ³

.
According to Theorem 11.28, the form is universal when restricted to =
for which º#Á #» ~ .
Hence, there exists a #  =

Now, #~ " b " for Á  - not both , and we may swap " and " if

necessary to ensure that £ . Hence,
~ ²# Á " Á ÃÁ" Á' ÁÃÁ' ³
8
is an ordered basis for for which the matrix 4 is diagonal and has a in the

=
8
upper left entry. We can repeat the process with the subspace =~ span ²# Á # . ³

Continuing in this way, we can find an ordered basis
9 ~ ²#Á #Á Ã Á # Á 'Á Ã Á ' ³

-

for which 4~ ? ² ³ for some nonzero  - . Now, if is a square in ,

9
:
then we can replace by # ² ° j ³ # to get a basis for which 4 ~ ? : ² ³ . If
- is not a square in , then ~ for some  - # and so replacing by
gives a basis for which
:
² ° ³# : 4 ~ ? ² ³.
be the set of all d symmetric matrices over a finite
Theorem 11.30 Let I
field . If char ² - ³ ~ , we restrict I to the set of all symmetric matrices with
-
at least one nonzero entry on the main diagonal.
)
1 If char²-³ ~ , then any matrix in I is congruent to a unique matrix of the
form ? ² ³ and the matrices ¸? ² ³ “ ~ Á à Á ¹ form a set of

under congruence. Also, the rank is a complete
canonical forms for I
invariant.
)

-
2 If char²-³ £ , let be a fixed nonsquare in . Then any matrix I is
congruent to a unique matrix of the form ?² ³ or ?² ³ . The set

¸? ² ³Á ? ² ³ “ ~ Á à Á ¹ is a set of canonical forms for congruence

on I (
. Thus, there are exactly two congruence classes for each rank .)
The Orthogonal Group
Having “settled” the classification question for orthogonal geometries over
certain types of fields, let us turn to a discussion of the structure-preserving
maps, that is, the isometries.
Rotations and Reflections
We begin by examining the matrix of an orthogonal transformation. If is an
8
=
ordered basis for , then for any Á % &  = ,
!
8
º%Á &» ~ ´%µ 4 ´&µ 8
8
B

and so if ²= ³ , then
!
!
!

8
º%Á &» ~ ´%µ 4 ´&µ ~ ´%µ ²´µ 4´µ ³´&µ 8 8
8
8
8 8 8
Hence, is an orthogonal transformation if and only if