Then det Equality holds if and only if X is a Hadamard matrix This is a nice example of a theorem which seems to lack any reasonable ap proach we are asked to optimise a highly nonlinear function over a multidimen sional region yet when looked at ID: 25599
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az }| {++++++az }| {++++��az }| {++��++az }| {++����Figure1:ThreerowsofaHadamardmatrixNow,sinceeveryotherrowisorthogonaltotherst,weseethateachfurtherrowhasmentries+andmentries�,wheren=2m.Moreover,ifn�2,therstthreerowsarenowasinFigure1,withn=4a.ItisconjecturedthatthereisaHadamardmatrixofeveryorderdivisibleby4(inotherwords,theabovenecessaryconditionissufcient).Thesmallestmultipleof4forwhichnomatrixhasbeenconstructediscurrently428.Wewillseeafewconstructionsinthefollowingsections.See[4]formore.(Note:Thisisnowoutofdate;aHadamardmatrixoforder428wasconstructedbyHadiKharaghaniandBehruzTayfeh-Rezaie[3]attheIPMinTehran,Iranin2004.ThematrixcanbedownloadedfromtheIPMwebsite[2],orfromNeilSloane'slibraryofHadamardmatrices[5].Thecurrentsmallestunknownorderis668.)2SomeconstructionsTheconstructionsgivenherebynomeansexhaustthoseknown,butsufcetogiveaHadamardmatrixofeachadmissibleorderlessthan52.2.1SylvestermatricesThesimplestconstructionofnewHadamardmatricesfromoldistheKronecker(ortensor)product.Ingeneral,ifA=(aij)andB=(bkl)arematricesofsizemnandpqrespectively,theKroneckerproductA Bisthempnqmatrixmadeupofpqblocks,wherethe(i;j)blockisaijB.Thenwehavethefollowingresult:Theorem3TheKroneckerproductofHadamardmatricesisaHadamardmatrix.TheSylvestermatrixS(k)oforder2kistheiteratedKroneckerproductofkcopiesoftheHadamardmatrix+++�oforder2.Bytheprecedingtheorem,TheEncyclopaediaofDesignTheoryHadamardmatrices/2 ConstructionfromLatinsquaresLetL=(lij)beaLatinsquareoforder6:thatis,a66arraywithentries1;:::;6suchthateachentryoccursexactlyonceineachroworcolumnofthearray.NowletHbeamatrixwithrowsandcolumnsindexedbythe36cellsofthearray:itsentryinthepositioncorrespondingtoapair(c;c0)ofdistinctcellsisdenedtobe+1ifcandc0lieinthesamerow,orinthesamecolumn,orhavethesameentry;allotherentries(includingthediagonalones)are�1.ThenHisaHadamardmatrix.SteinertriplesystemsLetSbeaSteinertriplesystemoforder15:thatis,Sisasetof“triples”or3-elementsubsetsoff1;:::;15gsuchthatanytwodistinctelementsofthissetarecontainedinauniquetriple.Thereare35triples,andtwodistincttripleshaveatmostonepointincommon.NowletAbeamatrixwithrowsandcolumnsindexedbythetriples,withentryinposition(t;t0)being�1iftandt0meetinasinglepoint;allotherentries(includingthediagonalones)are+1.NowletHbeobtainedbyborderingAbyarowandcolumnof+1s.ThenHisaHadamardmatrix.3EquivalenceofHadamardmatricesTheSylvesterandPaleymatricesoforders4and8areequivalent–indeedthereisessentiallyauniquematrixofeachoftheseorders.Foralllargerordersforwhichbothtypesexist(thatis,n=p+1,wherepisaMersenneprime),theyarenotequivalent.Weproceedtomakethesenseof“equivalence”ofHadamardmatricesprecise.ThereareseveraloperationsonHadamardmatriceswhichpreservetheHadamardproperty:(a)permutingrows,andchangingthesignofsomerows;(b)permutingcolumns,andchangingthesignofsomecolumns;(c)transposition.WecalltwoHadamardmatricesH1andH2equivalentifonecanbeobtainedfromtheotherbyoperationsoftypes(a)and(b);thatis,ifH2=P�1H1Q,wherePandQaremonomialmatrices(havingjustonenon-zeroelementineachroworcolumn)withnon-zeroentries1.Accordingly,theautomorphismgroupofaHadamardmatrixHisthegroupconsistingofallpairs(P;Q)ofmonomialmatriceswithnon-zeroentries1TheEncyclopaediaofDesignTheoryHadamardmatrices/4 ThisisthePaleydesign.Yetanotherdesigncanbeobtainedasfollows.LetHbeaHadamardmatrixoforder4a.ThepointsofthedesignarethecolumnsofH;foreachpairofrowsofH,therearetwoblocksofsize2a,thesetofcolumnswheretheentriesintherowsagree,andthesetwheretheydisagree.Thisisa3-(4a;2a;2a(a�1))design.Equivalentmatricesgivethesamedesign.Remarkablyitturnsoutthatthedesignisa4-designifandonlyifa=3,inwhichcaseitisevena5-design(specicallythe5-(12;6;1)Steinersystem,whoseautomorphismgroupistheMathieugroupM12whichwemetabove).5SymmetricmatriceswithconstantrowsumIfaHadamardmatrixHissymmetricwithconstantrowsum,thenitsorderisasquare,say4m2,andtherowsumiseither2mor�2m.Ifwereplacetheentries�1inthematrixby0,weobtaintheincidencematrixofasquare2-(4m2;2m2m;m2m)design.AnySylvestermatrixofsquareorderisequivalenttoasymmetricmatrixwithconstantrowsum,andthusgivesrisetosuchdesigns;thesecanbeconstructedusingquadraticformsonavectorspaceoverGF(2).TheHadamardmatricesoforder36constructedabovefromLatinsquaresarealsoofthisform.References[1]M.Hall,Jr.,NoteontheMathieugroupM12,Arch.Math.13(1962),334–340.[2]InstituteforStudiesinTheoreticalPhysicsandMathematics,IPM,Iran;http://math.ipm.ac.ir.[3]H.KharaghaniandB.Tayfeh-Rezaie,AHadamardmatrixoforder428,J.CombinatorialDesigns13(2005),435–440.[4]J.SeberryandM.Yamada,Hadamardmatrices,sequencesandblockdesigns,pp.431–560inContemporaryDesignTheory:ACollectionofSurveys(ed.J.H.DinitzandD.R.Stinson),Wiley,NewYork,1992.[5]NeilSloane,AlibraryofHadamardmatrices;http://www.research.att.com/˜njas/hadamard/TheEncyclopaediaofDesignTheoryHadamardmatrices/6