238 V Hadamard’s Determinant Problem
The preceding argument applies equallywell to a hyperbola, since it is also
described by an equation of the form (∗). We now wish to extend this result to higher
dimensions. Ann-dimensional conic with centre at the origin has the form
Q:=xtAx=const.,wherex∈RnandAis ann×nreal symmetric matrix. The analogue of a rotation is a
linear transformationx=Tywhich preserves Euclidean lengths, i.e.xtx=yty.This
holds for ally∈Rnif and only if
TtT=I.AmatrixTwhich satisfies this condition is said to beorthogonal.ThenTt=T−^1 and
hence alsoTTt=I.
The single most important fact about real symmetric matrices is theprincipal axes
transformation:
Theorem 10If H is an n×n real symmetric matrix, then there exists an n×nreal
orthogonal matrix U such that UtH U is a diagonal matrix:
UtHU=diag[λ 1 ,...,λn].Proof Letf:Rn→Rbe the map defined by
f(x)=xtHx.Sincefis continuous and the unit sphereS={x∈Rn:xtx= 1 }is compact,
λ 1 :=sup
x∈Sf(x)is finite and there exists anx 1 ∈Ssuch thatf(x 1 )=λ 1. We are going to show that, if
x∈Sandxtx 1 =0, then alsoxtHx 1 =0.
For any realε, put
y=(x 1 +εx)/( 1 +ε^2 )^1 /^2.Then alsoy∈S,sincexandx 1 are orthogonal vectors of unit length. Hencef(y)≤
f(x 1 ), by the definition ofx 1 .Butx 1 tHx=xtHx 1 ,sinceHis symmetric, and hence
f(y)={f(x 1 )+ 2 εxtHx 1 +ε^2 f(x)}/( 1 +ε^2 ).For small|ε|it follows that
f(y)=f(x 1 )+ 2 εxtHx 1 +O(ε^2 ).IfxtHx 1 were different from zero, we could chooseεto have the same sign as it and
obtain the contradictionf(y)>f(x 1 ).
On the intersection of the unit sphereSwith the hyperplanextx 1 =0, the function
fattains its maximum valueλ 2 at some pointx 2. Similarly, on the intersection of the
unit sphereSwith the(n− 2 )-dimensional subspace of allxsuch thatxtx 1 =xtx 2 =0,
the functionfattains its maximum valueλ 3 at some pointx 3. Proceeding in this way
we obtainnmutually orthogonal unit vectorsx 1 ,...,xn. MoreoverxtjHxj=λjand,
by the argument of the previous paragraph,xtjHxk=0ifj>k. It follows that the
matrixUwith columnsx 1 ,...,xnsatisfies all the requirements.