Number Theory: An Introduction to Mathematics

1 Quadratic Spaces 301

Proposition 18Let V be a quadratic space with two orthogonal sum representations

V=U⊥W=U′⊥W′.

If there exists an isometryφ:U→U′, then there exists an isometryψ:V→Vsuch
thatψu=φu for all u∈U andψW=W′. Thus if U is isometric to U′,thenW is
isometric to W′.

Proof Letu 1 ,...,um andum+ 1 ,...,unbe bases forU and W respectively. If
u′j=φuj( 1 ≤j≤m),thenu′ 1 ,...,u′mis a basis forU′.Letu′m+ 1 ,...,u′nbe a basis
forW′. The symmetric matrices associated with the basesu 1 ,...,unandu′ 1 ,...,u′n
ofVhave the form
(
A 0
0 B

)

,

(

A 0

0 C

)

,

which we will write asA⊕B,A⊕C. Thus the two matricesA⊕B,A⊕Care
congruent. It is enough to show that this implies thatBandCare congruent. For
supposeC=StBSfor some invertible matrixS=(σij).Ifwedefineu′′m+ 1 ,...,u′′nby

u′i=

∑n

j=m+ 1

σjiu′′j (m+ 1 ≤i≤n),

then (u′′j,u′′k)=(uj,uk)(m+ 1 ≤j,k≤n)and the linear mapψ:V→Vdefined by

ψuj=u′j( 1 ≤j≤m), ψuj=u′′j(m+ 1 ≤j≤n),

is the required isometry.
By taking the bases forU,W,W′to be orthogonal bases we are reduced to the
case in whichA,B,Care diagonal matrices. We may choose the notation so that
A=diag[a 1 ,...,am], whereaj=0forj≤randaj=0forj>r.Ifa 1 =0, i.e.
ifr>0, and if we writeA′=diag[a 2 ,...,am], then it follows from Propositions 1
and 16 that the matricesA′⊕BandA′⊕Care congruent. Proceeding in this way, we
are reduced to the caseA=O.
Thus we now supposeA=O. We may assumeB =O,C=O, since other-
wise the result is obvious. We may choose the notation also so thatB=Os⊕B′and
C=Os⊕C′,whereB′is non-singular and 0≤s<n−m.IfTt(Om+s⊕C′)T=
Om+s⊕B′,where

T=

(

T 1 T 2

T 3 T 4

)

,

thenT 4 tC′T 4 = B′.SinceB′is non-singular, so also isT 4 and thusB′andC′are
congruent. It follows thatBandCare also congruent.

Corollary 19If a non-singular subspace U of a quadratic space V is isometric to
another subspace U′,thenU⊥is isometric to U′⊥.