Number Theory: An Introduction to Mathematics

226 V Hadamard’s Determinant Problem Let A= [ αβ γδ ] , where at least one ofα,β,γ,δis nonzero. By multiplyingAon the left, or o ...

1 What is a Determinant? 227 SAT=diag[1r− 1 ,δ, (^0) n−r] for some nonzeroδ∈F. The matrixAissingularifr<nandnonsingularifr=n. ...

228 V Hadamard’s Determinant Problem (ii)if the matrix B is obtained from the matrix A by multiplying all entries in one row byλ ...

2 Hadamard Matrices 229 detA=det(SA)=detDδ and det(AB)=det(SAB)=det(DδB). But detDδ =δ,by(vi),anddet(DδB)=δdetB, by (ii). Theref ...

230 V Hadamard’s Determinant Problem Proof We are going to construct inductively mutually orthogonal vectorsw 1 ,...,wm such tha ...

2 Hadamard Matrices 231 [ 11 1 − 1 ] . There is one rather simple procedure for constructingH-matrices. IfA=(αjk)is ann×mmatrix ...

232 V Hadamard’s Determinant Problem Lemma 6Let C be an n×n matrix, with 0 ’s on the main diagonal and all other entries 1 or− 1 ...

3 The Art of Weighing 233 QJq= 0 , QtQ=qIq−Jq. Furthermore, sinceχ(− 1 )=(− 1 )(q−^1 )/^2 ,Qis symmetric ifq≡1 mod 4 and skew- s ...

234 V Hadamard’s Determinant Problem by measuring each separately. Suppose, for definiteness, that we havemobjects whose weights ...

3 The Art of Weighing 235 where+and−stand for 1 and−1 respectively, thenAtA= 4 I 3. With this experi- mental design the individu ...

236 V Hadamard’s Determinant Problem Put μ=sup A∈S det(AtA). Since det(AtA)is a continuous function of themnvariablesαjkandSmay ...

4 Some Matrix Theory 237 Proof Since A= [ 1 O ent− 1 I ][ 1 en− 1 O − 2 B ] , whereemdenotes a row ofm1’s, the matrixAhas determ ...

238 V Hadamard’s Determinant Problem The preceding argument applies equallywell to a hyperbola, since it is also described by an ...

4 Some Matrix Theory 239 It should be noted that, if U is any orthogonal matrix such thatUtHU = diag[λ 1 ,...,λn] then, sinceUUt ...

240 V Hadamard’s Determinant Problem such that TtG 1 T=diag[γ 1 ,...,γp], TtH 1 T=diag[δ 1 ,...,δp]. SinceG 3 −^1 is positive de ...

4 Some Matrix Theory 241 Lemma 15If C=αIm+βJmfor some realα,β,then detC=αm−^1 (α+mβ). Moreover, ifdetC= 0 ,thenC−^1 =γIm+δJm,wh ...

242 V Hadamard’s Determinant Problem with equality only ifγjk=γ 1 jγ 1 k/βfor 2≤j<k≤m. Henceη≤αm−^1 β, with equality only ifγ ...

5 Application to Hadamard’s Determinant Problem 243 H= [ LN Nt M ] , whereL,Mare square matrices of ordersr′,s′respectively. By ...

244 V Hadamard’s Determinant Problem Proof We may assume det(AtA)=0 and thusm≤n.ThenAtA=G=(γjk)is a positive definite real symm ...

5 Application to Hadamard’s Determinant Problem 245 ReplacingAbyDA, we obtainAAt = AtA.ThenAcommutes withAtAand hence also withJ ...