Page 277 - Advanced Linear Algebra

Metric Vector Spaces: The Theory of Bilinear Forms 261

next theorem tells us that we do not need to consider skew-symmetric forms per
se, since skew-symmetry is always equivalent to either symmetry or
alternateness.

Theorem 11.1 Let be a vector space over a field .
-
=
)
1 If char²-³ ~ , then
alternate ¬ symmetric ¯ skew-symmetric
)
2 If char²-³ £ , then
alternate ¯ skew-symmetric

Also, the only form that is both alternate and symmetric is the zero form:
º%Á&» ~ for all %Á&  = .
Proof. First note that for an alternating form over any base field,

~ º% b &Á% b &» ~ º%Á&» b º&Á%»
and so

º%Á &» ~ cº&Á %»
which shows that the form is skew-symmetric. Thus, alternate always implies
skew-symmetric.

If char²-³ ~ , then c ~ and so the definitions of symmetric and skew-
)
symmetric are equivalent, which proves 1 . If char²-³ £ and the form is
skew-symmetric, then for any %  = , we have º%Á %» ~ cº%Á %» or º%Á %» ~ ,
which implies that º%Á %» ~ . Hence, the form is alternate. Finally, if the form
is alternate and symmetric, then it is also skew-symmetric and so
º"Á#» ~ cº"Á#» for all "Á#  = , that is, º"Á#» ~ for all "Á#  = .

Example 11.2 The standard inner product on =² Á ³ , defined by

²% ÁÃÁ% ³ h ²& ÁÃÁ& ³ ~ % & b Ä b % &
is symmetric, but not alternate, since
² Á ÁÃÁ ³ h ² Á ÁÃÁ ³ ~ £
The Matrix of a Bilinear Form
If 8 ~² Á Ã Á ³ is an ordered basis for a metric vector space = , then a

bilinear form is completely determined by the d matrix of values
4 ~ ² ³ ~ ²º Á »³

Á
8

This is referred to as the matrix of the form (or the matrix of ) with respect to
=
-
8
the ordered basis . Moreover, any d matrix over is the matrix of some
bilinear form on .
=