Page 301 - Advanced Linear Algebra

Metric Vector Spaces: The Theory of Bilinear Forms 285

Let us summarize.

Theorem 11.21 Let be an orthogonal geometry.
=
)
1 If is also symplectic, then has an orthogonal basis if and only if it is
=
=
totally degenerate. When char²-³ £ , all orthogonal symplectic
geometries have an orthogonal basis, but this is not the case when
char²-³ ~ .
2 If = ) is not symplectic, then = has an ordered orthogonal basis
8 ~ ²" Á ÃÁ" Á' ÁÃÁ' ³ for which º" Á" » ~ £ and º' Á' » ~ .

has the diagonal form
Hence, 4 8
v y
x Æ {
x {
x {
4~ x 8 {
x {
x {
Æ
w z
with ~ rk ²4 ³ nonzero entries on the diagonal.
8
As a corollary, we get a nice theorem about symmetric matrices.
Corollary 11.22 Let 4 be a symmetric matrix and assume that 4 is not
alternate if char²-³ ~ . Then 4 is congruent to a diagonal matrix.
The Classification of Orthogonal Geometries: Canonical
Forms
We now want to consider the question of improving upon Theorem 11.21. The
diagonal matrices of this theorem do not form a set of canonical forms for
are nonzero scalars, then the matrix of with
congruence. In fact, if Á Ã Á =
respect to the basis 9 ~ ² " ÁÃÁ " Á ' Á ÃÁ' ³ is

v y
x Æ {
x {

4~ x 9 x { { (11.2 )
x {
x {
Æ
w z
Hence, 4 and 4 8 9 are congruent diagonal matrices. Thus, by a simple change
of basis, we can multiply any diagonal entry by a nonzero square in .
-
(
The determination of a set of canonical forms for symmetric nonalternate when
char²-³ ~ ) matrices under congruence depends on the properties of the base
field. Our plan is to consider three types of base fields: algebraically closed