Page 280 - Advanced Linear Algebra

264 Advanced Linear Algebra

Quadratic Forms

There is a close link between symmetric bilinear forms on = and quadratic
forms on .
=
Definition A quadratic form on a vector space is a map ¢ 8 = ¦ - with the
=
following properties:
)
1 For all  -Á #  = ,

8² #³ ~ 8²#³
)
2 The map
º"Á #» ~ 8²" b #³ c 8²"³ c 8²#³
8
)
(
is a symmetric bilinear form.
=
8
Thus, every quadratic form on defines a symmetric bilinear form " º Á # » 8
on . Conversely, if char ² - ³ £ and if Á º » is a symmetric bilinear form on ,
=
=
then the function

8²%³ ~ º%Á %»

is a quadratic form . Moreover, the bilinear form associated with is the
8
8
original bilinear form:
º"Á #» ~ 8²" b #³ c 8²"³ c 8²#³
8

~ º" b #Á " b #» c º"Á "» c º#Á #»

~ º"Á #» b º#Á "» ~ º"Á #»

Thus, the maps ºÁ » ¦ 8 and 8 ¦ ºÁ » 8 are inverses and so there is a one-to-one
correspondence between symmetric bilinear forms on and quadratic forms on
=
= . Put another way, knowing the quadratic form is equivalent to knowing the
corresponding bilinear form.

Again assuming that char²-³ £ , if ~ ²# Á Ã Á # ³ is an ordered basis for an
8

orthogonal geometry = and if the matrix of the symmetric form on = is
4~ ² ³, then for % ~ % # ,'
8
Á

!
8²%³~ º%Á %»~ ´%µ 4 ´%µ ~ % %
Á
8
8
8
Á
%
and so 8²%³ is a homogeneous polynomial of degree 2 in the coordinates .
(The term “form” means homogeneous polynomial —hence the term quadratic
form.)