Page 279 - Advanced Linear Algebra

Metric Vector Spaces: The Theory of Bilinear Forms 263

Thus, if two matrices represent the same bilinear form on = , they must be
represents a bilinear form on and
=
congruent. Conversely, if )~ 4 8
!
(~ 7 )7
where is invertible, then there is an ordered basis for for which
9
7
=
7~ 4 98Á
and so
!
(~ 4 98 4 4 9 8 Á 8
Á
represents the same form with respect to .
9
Thus, (~ 4 9
Theorem 11.2 Let 8 ~² Á Ã Á ³ be an ordered basis for an inner product

space , with matrix
=
4~ ²º Á »³

8
)
1 The form can be recovered from the matrix by the formula
!
º%Á &» ~ ´%µ 4 ´&µ 8
8
8
)
2 If 9 ~² Á Ã Á ³ is also an ordered basis for , then
=

4~ 4 ! 4 4 9 8 Á 8 98 Á
9
is the change of basis matrix from to .
8
9
where 4 98Á
3 Two matrices and represent the same bilinear form on a vector space
)
(
)
= if and only if they are congruent, in which case they represent the same
set of bilinear forms on .
=
In view of the fact that congruent matrices have the same rank, we may define
the rank of a bilinear form (or of ) to be the rank of any matrix that represents
=
that form.
The Discriminant of a Form
If and are congruent matrices, then
(
)
!
det²(³ ~ det²7 )7³ ~ det²7³ det²)³
and so det²(³ and det²)³ differ by a square factor. The discriminant of a
"
bilinear form is the set of determinants of all of the matrices that represent the
form. Thus, if is an ordered basis for , then
=
8

" ~- det ²4 ³~¸ det ²4 ³“ £ -¹
8
8