236 V Hadamard’s Determinant Problem
Putμ=sup
A∈Sdet(AtA).Since det(AtA)is a continuous function of themnvariablesαjkandSmay be re-
garded as a compact set inRmn,μis finite and there exists a matrixA∈Sfor which
det(AtA)=μ. By repeatedly applying the argument of the preceding paragraph to
thisAwe may replace it by one for which every entry in the first column is eitherαor
βandforwhichalsodet(AtA)=μ. These operations do not affect the submatrixB
formed by the lastm−1 columns ofA. By interchanging thek-th column ofAwith
the first, which does not alter the value of det(AtA), we may apply the same argument
to every other column ofA.
The proof of Proposition 8 actually shows that ifCis a compact subset ofRnand
ifSis the set of alln×mmatricesAwhose columns are inC, then there exists an
n×mmatrixMwhose columns are extreme points ofCsuch that
det(MtM)=sup
A∈Sdet(AtA).Heree∈Cis said to be anextreme pointofCif there do not exist distinctv 1 ,v 2 ∈C
andθ∈( 0 , 1 )such thate=θv 1 +( 1 −θ)v 2.
The preceding discussion concerns weighings by a chemical balance. If instead
we use a spring balance, then we are similarly led to the problem of maximizing
det(BtB)among alln×mmatricesB=(βjk)withβjk=1 or 0 according as thek-th
object is or is not involved in thej-th weighing. Moreover other types of measurement
lead to the same problem. A spectrometer sorts electromagnetic radiation into bundles
of rays, each bundle having a characteristic wavelength. Instead of measuring the
intensity of each bundle separately, we canmeasure the intensity of various combi-
nations of bundles by using masks with open or closed slots.
It will now be shown that in the casem =nthe chemical and spring balance
problems are essentially equivalent.
Lemma 9If B is an(n− 1 )×(n− 1 )matrix of 0 ’s and 1 ’s, and if Jnis the n×n
matrix whose entries are all 1 ,then
A=Jn−[
OO
O 2 B
]
,
is an n×nmatrixof 1 ’s and− 1 ’s, whose first row and column contain only 1 ’s, such
that
detA=(− 2 )n−^1 detB.Moreover, every n×nmatrixof 1 ’s and− 1 ’s, whose first row and column contain only
1 ’s, is obtained in this way.