Linear Algebra Topics - Computer Graphics and Geometric Modeling - page 47

is a quadratic form in variables x, y, and z. Note that the quadratic form can be rewrit-
ten with symmetric cross-terms as

52 3
2

3
2

1
2

1
2

2 2 2

xyz y x z y +-+ + + +.

It follows that one can associate the symmetric matrix

3
2

Ê ˆ

1 0

Á
Á
Á
Á
Á

˜
˜
˜
˜
˜

3
2

1
2

5

1
2

0 -

Ë ¯

with this form. More generally, if the field k does not have characteristic 2, such as,
for example, R or C , we can make the cross-terms symmetric with the trick shown in
the example above. It follows that in this case every quadratic form in n variables is
simply an expression of the type

n n

 Â

axx
ij

,

i j

i = 1 j = 1

where A = (a ij ) is a symmetric matrix. This means that every quadratic form defines
a unique quadratic map

q : V Æ k

on a vector space V in the following way: Choose an ordered basis B = ( v 1 , v 2 ,..., v n )
for V . Let v ΠV and suppose that

n

 x ii
i

v = v .

= 1

Then

n n

 Â

T

() =

q vxx A = a x x ,

ij i j

i = 1 j = 1

where x = (x 1 ,x 2 ,...,x n ). In this case also, the matrix for the associated bilinear map
is just the matrix A. Of course, for all this to make sense we are treating the x i as
values rather than variables, but we can see that from a theoretical point of view there
is no difference between the theory of quadratic forms and quadratic maps. This
explains why in the literature the terms “quadratic map” and “quadratic form” are