Graphics Reference
In-Depth Information
Definition.
The
rank
of a bilinear map f is the rank of any matrix for f. A bilinear
map on an n-dimensional vector space is said to be
degenerate
or
nondegenerate
if its
rank is less than n or equal to n, respectively.
It follows from Proposition 1.9.5 that the rank of a bilinear map is well
defined.
1.9.7. Proposition.
The matrix associated to a symmetric bilinear map with respect
to any basis is a symmetric matrix.
Proof.
Exercise.
Definition.
A
quadratic map
on a vector space
V
over a field k is any map q :
V
Æ k
that can be defined in the form q(
v
) = f(
v
,
v
), where f is some bilinear map on
V
.
In that case, q is also called the
quadratic map associated to f
. The quadratic map
q is said to be
degenerate
or
nondegenerate
if f is. A
discriminant
of q is defined to
be discriminant of f with respect to some basis for
V
. The quadratic map q and the
bilinear map f are said to be
positive definite
if q(
v
) = f(
v
,
v
) > 0 for all
v
π
0
.
Let f be a bilinear map on
R
2
and assume that
1.9.8. Example.
(
)
=
f
vw
,
a
v w
+
a
v w
+
a
v w
+
a
v w
.
11
1
1
12
1
2
21
2
1
22
2
2
The quadratic map q associated to f is then given by
()
=
2
(
)
2
.
qa
v
+
a
+
a
v v
+
a
v
11
12
21
1
2
22
We see that q is just a homogeneous polynomial of degree 2 in v
1
and v
2
.
If the field k does not have characteristic 2 and if f is a symmetric bilinear map
with associated quadratic map q, then
1
2
(
)
=
[
(
)
-
()
-
()
]
f
vw
,
q
v w
+
q
v
q
w
.
In other words, knowledge of q alone allows one to reconstruct f, so that the concepts
“symmetric bilinear map” and “quadratic map” are really just two ways of looking at
the same thing.
The form of the quadratic map in Example 1.9.8 and others like it motivates
what is basically nothing but some alternate terminology for talking about quadratic
maps.
Definition.
A
d-ic form
over a field k is a homogeneous polynomial over k of degree
d in an appropriate number of variables. A
linear
or
quadratic form
is a d-ic form
where d is 1 or 2, respectively.
For example, 2x + 3y is a linear form in variables x and y and
2
2
2
xyz y z
+-++
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