Page 304 - Advanced Linear Algebra
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288 Advanced Linear Algebra
v y
x Æ {
x {
x {
x {
x x c { {
4~ Z Á Á ~ x Æ {
8
x {
x c {
x {
x {
x {
Æ
w z
with ones, negative ones and zeros on the diagonal.
Here is the matrix version of Theorem 11.25.
Theorem 11.26 Let I be the set of all d symmetric matrices over the real
field .
s
) for
1 Any matrix in I is congruent to a unique matrix of the form Z Á Á Á
some and ~ c c .
Á
) is a set of
2 The set of all matrices of the form Z Á Á for b b ~
.
canonical forms for congruence on I
) is the rank of
3 Let 4 I and let 4 be congruent to A Á Á . Then b
4 and c is the signature 4 of and the triple ² Á Á ³ is the inertia
of 4 . The pair ² Á ³ , or equivalently the pair ² b Á c ³ , is a
.
complete invariant under congruence on I
)
Proof. We need only prove the uniqueness statement in part 1 . Let
8 ~ ²" Á ÃÁ" Á# ÁÃÁ# Á' ÁÃÁ' ³
and
Z
Z
Z
Z
Z
Z
9 ~ ²" Á ÃÁ" Á# ÁÃÁ# ' ÁÃÁ' ³
Z
Z
Z
be ordered bases for which the matrices 4 and 4 8 9 have the form shown in
Theorem 11.25. Since the rank of these matrices must be equal, we have
b ~ b and so ~ .
Z
Z
Z
If % span ²" ÁÃÁ" ³ and % £ , then
º%Á %» ~ L " Á " M ~ º" Á "» ~ ~
Á
Á Á
Z
On the other hand, if & span ²# ÁÃÁ# ³ and & £ , then
Z
Z
Z
Z
º&Á &» ~ L #Á Z # M Z ~ º# Á # » ~ c ~ c
Á
Á Á
Z
Hence, if & span ²# ÁÃÁ# Á' ÁÃÁ' ³ then º&Á &» . It follows that
Z
Z
Z
Z
Z
