Number Theory: An Introduction to Mathematics

300 VII The Arithmetic of Quadratic Forms

The concept of isometry is only another way of looking at equivalence. For if
φ:V →V′is an isometry, thenVandV′have the same dimension. Ifu 1 ,...,un
is a basis forVandu′ 1 ,...,u′nabasisforV′then, since(uj,uk)=(φuj,φuk),the
isometry is completely determined by the change of basis inV′fromφu 1 ,...,φunto
u′ 1 ,...,u′n.
A particularly simple type of isometry is defined in the following way. LetVbe a
quadratic space andwa vector such that(w,w)=0. The mapτ:V→Vdefined by

τv=v−{ 2 (v,w)/(w,w)}w

is obviously linear. IfWis the non-singular one-dimensional subspace spanned byw,
thenV=W⊥W⊥.Sinceτv=vifv∈W⊥andτv=−vifv∈W, it follows thatτ
is bijective. Writingα=− 2 (v,w)/(w,w),wehave

(τv,τv)=(v,v)+ 2 α(v,w)+α^2 (w,w)=(v,v).

Thusτis an isometry. Geometrically,τis areflectionin the hyperplane orthogonal
tow. We will refer toτ =τwas the reflection corresponding to the non-isotropic
vectorw.

Proposition 16If u,u′ are vectors of a quadratic space V such that(u,u) =
(u′,u′)= 0 , then there exists an isometryφ:V→V such thatφu=u′.

Proof Since

(u+u′,u+u′)+(u−u′,u−u′)= 2 (u,u)+ 2 (u′,u′)= 4 (u,u),

at least one of the vectorsu+u′,u−u′is not isotropic. Ifu−u′is not isotropic,
the reflectionτ corresponding tow = u−u′has the propertyτu = u′,since
(u−u′,u−u′)= 2 (u,u−u′). Ifu+u′is not isotropic, the reflectionτcorresponding
tow=u+u′has the propertyτu=−u′.Sinceu′is not isotropic, the corresponding
reflectionσmapsu′onto−u′, and hence the isometryστmapsuontou′.

The proof of Proposition 16 has the following interesting consequence:

Proposition 17Any isometryφ:V→V of a non-singular quadratic space V is a
product of reflections.

Proof Letu 1 ,...,unbe an orthogonal basis forV. By Proposition 16 and its proof,
there exists an isometryψ, which is either a reflection or a product of two reflections,
such thatψu 1 =φu 1 .IfUis the subspace with basisu 1 andWthe subspace with
basisu 2 ,...,un,thenV=U⊥WandW=U⊥is non-singular. Since the isometry
φ 1 =ψ−^1 φfixesu 1 ,wehavealsoφ 1 W =W.Butifσ :W →Wis a reflection,
the extensionτ:V→Vdefined byτu=uifu∈U,τw=σwifw∈W,isalso
a reflection. By using induction on the dimensionn, it follows thatφ 1 is a product of
reflections, and hence so also isφ=ψφ 1.

By a more elaborate argument E. Cartan (1938) showed that any isometry of an
n-dimensional non-singular quadratic space is a product of at mostnreflections.