Page 278 - Advanced Linear Algebra

262 Advanced Linear Algebra

Note that if %~ then
'
v º Á %» y

4´%µ ~ Å
8
8
w º Á %» z

and
!
´%µ 4 ~ º%Á » Ä º%Á »
8
8
It follows that if &~ , then

v y
!
´%µ 4 ´&µ ~ º%Á » Ä º%Á » Å ~ º%Á &»
8 8 8
w z

!
8
and this uniquely defines the matrix 4 8 , that is, if ´%µ (´&µ ~ º%Á &» for all
8
%Á &  = , then ( ~ 4 .
8
A matrix is alternate if it is skew-symmetric and has 's on the main diagonal.

Thus, we can say that a form is symmetric (skew-symmetric, alternate) if and
is symmetric (skew-symmetric, alternate).
only if the matrix 4 8
Now let us see how the matrix of a form behaves with respect to a change of
basis. Let 9 ~² Á Ã Á ³ be an ordered basis for . Recall from Chapter 2 that
=

the change of basis matrix 4 , whose th column is ´ µ 8 , satisfies
98 Á
´#µ ~ 4 9 8 ´#µ 9 Á
8
Hence,
!
º%Á &» ~ ´%µ 4 ´&µ 8
8
8
!
8
~ ²´%µ 4 9 ! Á 8 ³4 ²4 9 8 ´&µ ³
9 Á
9
!
~´%µ ²4 ! 8
9 9 Á 8 4 4 9 Á 8 ³´&µ 9
and so
4~ 4 ! 4 4 9 8 Á 8 98 Á
9
This prompts the following definition.
Definition Two matrices (Á )  C ²-³ are congruent if there exists an
invertible matrix for which
7
!
(~ 7 )7
The equivalence classes under congruence are called congruence classes .