Page 303 - Advanced Linear Algebra

Metric Vector Spaces: The Theory of Bilinear Forms 287

Theorem 11.23 Let be an orthogonal geometry over an algebraically closed
=
field . Provided that is not symplectic as well when char ² - ³ ~ , then =
-
=
has an ordered orthogonal basis 8 ~ ²" Á ÃÁ" Á' ÁÃÁ' ³ for which

º" Á " » ~ and º'Á '» ~ . Hence, 4 has the diagonal form

8

v y
x Æ {
x x { {
4~ A Á ~ x {
8
x {
x {
Æ
w z
with ones and zeros on the diagonal. In particular, if is nonsingular,
=

then has an orthonormal basis.
=
The matrix version of Theorem 11.23 follows.
Theorem 11.24 Let I be the set of all d symmetric matrices over an
algebraically closed field . If ² - ³ ~char , we restrict I to the set of all
-
symmetric matrices with at least one nonzero entry on the main diagonal.
)
1 Any matrix 4 in I is congruent to a unique matrix of the form Z Á , in
fact, ~ rk ²4³ and ~ c rk ²4 . ³
) for b ~ is a set of canonical
2 The set of all matrices of the form Z Á
.
forms for congruence on I
) .
3 The rank of a matrix is a complete invariant for congruence on I
The Real Field s
If -~ s , we can choose ~ ° j (( , so that all nonzero diagonal elements in

(11.2 ) will be either , or c .

(
Theorem 11.25 Sylvester's law of inertia) Any real orthogonal geometry =
has an ordered orthogonal basis
8 ~ ²" Á ÃÁ" Á# ÁÃÁ# Á' ÁÃÁ' ³

for which º" Á " » ~ , º#Á #» ~ c and º'Á '» ~ . Hence, the matrix 4 8 has

the diagonal form