Graphics Reference
In-Depth Information
1.9.1. Example.
The dot product on
R
n
is a bilinear map. More generally, an inner
product is a bilinear map.
1.9.2. Example.
The determinant function on
R
2
(where we think of either the rows
or columns of the matrix as vectors in
R
2
) is a bilinear map.
Let A be an n ¥ n matrix. The map f :
R
n
¥
R
n
1.9.3. Example.
Æ
R
defined by
(
)
=
T
f
v, w
v
A
w
is a bilinear map.
Let T be a linear transformation on
R
n
. The map f :
R
n
¥
R
n
1.9.4. Example.
Æ
R
defined by
(
)
=∑
()
f
v, w
v
T
w
is a bilinear map.
Definition.
Let f be a bilinear map on a vector space
V
. Let B = (
v
1
,
v
2
,...,
v
n
) be an
ordered basis for
V
and let a
ij
= f(
v
i
,
v
j
). The matrix A = (a
ij
) is called the
matrix for f
with respect to the basis B. The determinant of A is called the
discriminant
of f with
respect to the basis B.
The matrix for a bilinear map clearly depends on the chosen basis. However, the
following is true:
1.9.5. Proposition.
If B¢=(
v
1
¢,
v
2
¢,...,
v
n
¢) is another ordered basis for
V
and if A¢
is the matrix of the bilinear map f with respect to B¢, then
CAC
T
A
¢=
,
where C = (c
ij
) is the matrix relating the basis B to the basis B¢, that is,
n
Â
1
¢
=
v
c
v
.
i
ij
j
j
=
Proof.
This can be checked by a straightforward computation.
Definition.
A real n ¥ n matrix A is said to be
congruent
to a real n ¥ n matrix B if
there exists a nonsingular matrix C such that A = CBC
T
.
It is easy to show that the congruence relation is an equivalence relation on the
set of all n ¥ n real matrices. We can rephrase Proposition 1.9.5.
1.9.6. Corollary.
The matrix of a bilinear map is unique up to congruence, so that
the study of bilinear maps is equivalent to the study of congruence classes of matrices.