1 Quadratic Spaces 293(ei,ek)=∑nj,h= 1τjiβjhτhk,whereβjh=(e′j,e′h), the matrixB=(βjh)is symmetric and
A=TtBT. (2)Two symmetric matricesA,Bwith elements fromFare said to becongruentif (2)
holds for some invertible matrixTwith elements fromF. Thus congruence of sym-
metric matrices corresponds to a change of basis in the quadratic space. Evidently
congruence is an equivalence relation, i.e. it is reflexive, symmetric and transitive. Two
quadratic forms are said to beequivalent over Fif their coefficient matrices are con-
gruent. Equivalence overFof the quadratic formsfandgwill be denoted byf∼Fg
or simplyf∼g.
It follows from (2) that
detA=(detT)^2 detB.Thus, although detAis not uniquely determined by the quadratic space, if it is nonzero,
itssquare classis uniquely determined. By abuse of language, we will call any repre-
sentative of this square class thedeterminantof the quadratic spaceVand denote it by
detV.
Although quadratic spaces are better adapted for proving theorems, quadratic
forms and symmetric matrices are useful for computational purposes. Thus a famil-
iarity with both languages is desirable. However, we do not feel obliged to give two
versions of each definition or result, and a version in one language may later be used
in the other without explicit comment.
A vectorvis said to beorthogonalto a vectoruif(u,v)=0. Then alsouis
orthogonal tov.Theorthogonal complement U⊥of a subspaceUofVis defined to
be the set of allv∈Vsuch that(u,v)=0foreveryu∈U. EvidentlyU⊥is again a
subspace. A subspaceUwill be said to benon-singularifU∩U⊥={ 0 }.
The whole spaceV is itself non-singular if and only ifV⊥ ={ 0 }. ThusV is
non-singular if and only if some, and hence every, symmetric matrix describing it is
non-singular, i.e. if and only if detV=0.
We say that a quadratic spaceVis theorthogonal sumof two subspacesV 1 and
V 2 , and we writeV=V 1 ⊥V 2 ,ifV=V 1 +V 2 ,V 1 ∩V 2 ={ 0 }and(v 1 ,v 2 )=0forall
v 1 ∈V 1 ,v 2 ∈V 2.
IfA 1 is a coefficient matrix forV 1 andA 2 a coefficient matrix forV 2 ,then
A=
(
A 1 0
0 A 2
)
is a coefficient matrix forV=V 1 ⊥V 2. Thus detV=(detV 1 )(detV 2 ). EvidentlyVis
non-singular if and only if bothV 1 andV 2 are non-singular.
IfW is any subspace supplementary to the orthogonal complementV⊥of the
whole spaceV,thenV=V⊥⊥WandWis non-singular. Many problems for arbitrary
quadratic spaces may be reduced in this way to non-singular quadratic spaces.