Combinatorial Matrix Theory and Generalized Inverses of Matrices PDF
Preview Combinatorial Matrix Theory and Generalized Inverses of Matrices
Combinatorial Matrix Theory and Generalized Inverses of Matrices Ravindra B. Bapat (cid:2) Steve J. Kirkland (cid:2) K. Manjunatha Prasad (cid:2) Simo Puntanen Editors Combinatorial Matrix Theory and Generalized Inverses of Matrices Editors RavindraB.Bapat K.ManjunathaPrasad IndianStatisticalInstitute ManipalUniversity NewDelhi,India Manipal,Karnataka,India SteveJ.Kirkland NationalUniversityofIrelandMaynooth SimoPuntanen MaynoothCo.Kildare UniversityofTampere Ireland Tampere,Finland ISBN978-81-322-1052-8 ISBN978-81-322-1053-5(eBook) DOI10.1007/978-81-322-1053-5 SpringerNewDelhiHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013932346 ©SpringerIndia2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub- lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect tothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Foreword The success of a conference has to be evaluated by the quality of the scientific presentationsandtheabilityoftheorganizerstopublishtheproceedings.Ifthisis theyardstickused,IreckontheInternationalConferenceonCombinatorialMatrix TheoryandGeneralizedInversesofMatricesmakesthegradeeasily. IcongratulatetheDepartmentofStatistics,ManipalUniversity,onthesuccessful conductoftheconferenceandthepublicationoftheproceedings. Manipal,India Dr.H.VinodBhat v Preface TheInternationalWorkshopandConferenceonCombinatorialMatrixTheoryand GeneralizedInversesofMatrices-2012wasorganizedbytheDepartmentofStatis- tics,ManipalUniversity,Manipal,India.Thereweremorethan100registeredpar- ticipants for both of the workshop and conference and 22 invited speakers repre- senting12nationsacrosstheglobe.Two-dayconferenceconsistedoffourplenary sessions.Invitedpapersandcontributedpaperswerepresentedinthreeparallelses- sions dedicated, respectively, to the branches of “Combinatorial Matrix Theory,” “GeneralizedInverseofMatrices,”and“MatrixMethodsforStatistics.”Theinter- national workshop and conference on CMTGIM provideda delightful and fruitful opportunityforallIndianandforeignmathematiciansandstatisticiansatalllevels ofexperiencetointeractinpersonandexchangeideas. “Combinatorial Matrix Theory and Generalized Inverses of Matrices” is being publishedinanattempttodocumenttherecentdevelopmentsintheareasdiscussed attheconference.Thisbookcontainsabout18researchandexpositoryarticlesfrom differentspeakersandtheircollaborators.Onlytheselectedarticles,aftertherefer- eeingprocess,wereretainedfortheinclusioninthisvolume.Areportonthework- shop and conference is presented as an appendix, covering narrative photographs andmessagesfromtheProVice-Chancellor,JointSecretary,CSIR,andtheDirec- torGeneral,CSO. NewDelhi,India RavindraB.Bapat Maynooth,Ireland SteveKirkland Manipal,India K.ManjunathaPrasad Tampere,Finland SimoPuntanen vii Acknowledgement The editors are thankful to all contributors and the referees who have helped in reviewing the articles for this volume. The following is the list of referees who kindlyagreedtoreviewthearticlesreceived. • BibhasAdhikari,IndianInstituteofTechnology,Rajasthan,India. • AdelchiAzzalini,DepartmentofStatisticalSciences,UniversityofPadua,Italy. • OskarM.Baksalary,AdamMickiewiczUniversity,Poland. • AmitavaBhattacharya,TataInstituteofFundamentalResearch,Mumbai,India. • AlokDey,IndianStatisticalInstitute,Delhi,India. • N.Eagambaram,CentralStatisticsOffice,MOSPI,NewDelhi,India. • MiguelFonseca,NovaUniversityofLisbon,Portugal. • AliS.Hadi,TheAmericanUniversityinCairo,Egypt. • SteveKirkland,NationalUniversityofIrelandMaynooth,Ireland. • S.H. Kulkarni,IndianInstituteofTechnologyMadras,Chennai,India. • SarojMalik,HinduCollege,UniversityofDelhi,Delhi,India. • SimoPuntanen,UniversityofTampere,Finland. • T.S.S.R.K.Rao,IndianStatisticalInstitute,Bangalore,India. • TarkeshwarSingh,BirlaInstituteofTechnology,Goa,India. • K.C.Sivakumar,IndianInstituteofTechnologyMadras,Chennai,India. • G.Sudhakara,MIT,ManipalUniversity,Manipal,India. • ShaunS.Wulff,UniversityofWyoming,USA. • YiminWie,FudanUniversity,China. We are grateful to all patrons of the conference and the Chairman of organizing committee Dr. H. Vinod Bhat, Pro Vice-Chancellor, MU, for their consistent sup- port.Also,wethankDr.N.S.K.Nair,theHeadoftheDepartment,andotherfaculty membersoftheDepartmentofStatisticsfortheiractiveinvolvementintheconduct ofconference. Our sincere thanks are also due to the National Board of Higher Mathematics (NBHM),theCentralStatisticsOffice(CSO)MinistryofStatisticsandProgramme Implementation,theCouncilofScientificandIndustrialResearch(CSIR),andthe InternationalCenterforTheoreticalPhysics(ICTP)fortheirfinancialsupport. ix x Acknowledgement Assistanceof K.S.Mohana,SanthiSheelaandM. Vinayincompilingthisvol- umewasincredible,andweproudlyapplaudtheireffort. Finally,wethanktheteamatSpringerfortheircooperationatallstagesofpub- lishingthisvolume. September2012 RavindraB.Bapat SteveKirkland K.ManjunathaPrasad SimoPuntanen Contents SkewSpectrumoftheCartesianProductofanOrientedGraphwithan OrientedHypercube . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 A.AnuradhaandR.Balakrishnan NotesonExplicitBlockDiagonalization . . . . . . . . . . . . . . . . . . . 13 MuraliK.Srinivasan TheThirdImmanantofq-LaplacianMatricesofTreesandLaplacians ofRegularGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 R.B.BapatandSivaramakrishnanSivasubramanian MatrixProductofGraphs. . . . . . . . . . . . . . . . . . . . . . . . . . . 41 K.ManjunathaPrasad,G.Sudhakara,H.S.Sujatha,andM.Vinay DeterminantoftheLaplacianMatrixofaWeightedDirectedGraph . . . 57 DebajitKalita FromMultivariateSkewedDistributionstoCopulas . . . . . . . . . . . . 63 TõnuKollo,AnneSelart,andHelleVisk RevisitingtheBLUEinaLinearModelviaProperEigenvectors . . . . . 73 JanHauke,AugustynMarkiewicz,andSimoPuntanen InferenceinErrorOrthogonalModels . . . . . . . . . . . . . . . . . . . . 85 FranciscoCarvalhoandJoãoTiagoMexia OntheEntriesofOrthogonalProjectionMatrices . . . . . . . . . . . . . 101 OskarMariaBaksalaryandGötzTrenkler Moore–PenroseInverseofPerturbedOperatorsonHilbertSpaces . . . . 119 ShaniJoseandK.C.Sivakumar TheReverseOrderLawinIndefiniteInnerProductSpaces . . . . . . . . 133 SachindranathJayaraman xi xii Contents GeneralizedInversesandApproximationNumbers. . . . . . . . . . . . . 143 K.P.Deepesh,S.H.Kulkarni,andM.T.Nair OntheLevel-2ConditionNumberforMoore–PenroseInverseinHilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 HuaianDiaoandYiminWei ProductsandSumsofIdempotentMatricesoverPrincipalIdealDomains 171 K.P.S.BhaskaraRao PerfectSemiringofNonnegativeMatrices . . . . . . . . . . . . . . . . . . 177 AdelAlahmedi,YousefAlkhamees,andS.K.Jain RegularMatricesoveranIncline . . . . . . . . . . . . . . . . . . . . . . . 183 AR.Meenakshi MatrixPartialOrdersAssociatedwithSpacePreorder . . . . . . . . . . 195 K.ManjunathaPrasad,K.S.Mohana,andY.SanthiSheela AnIllustratedIntroductiontoSomeOldMagicSquaresfromIndia . . . 227 GeorgeP.H.StyanandKaLokChu AReportonCMTGIM2012,Manipal . . . . . . . . . . . . . . . . . . . . 253 R.B.BapatandK.ManjunathaPrasad Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Editors RavindraB.BapatisaProfessorattheIndianStatisticalInstitute,DelhiCentre.He isaFellowoftheIndianAcademyofSciences,IndianNationalScienceAcademy andaJ.C.BoseFellow.HeservedasthePresidentoftheIndianMathematicalSoci- etyduring2007–2008.HehasbeenontheeditorialboardsofLinearandMultilinear Algebra,ElectronicJournalofLinearAlgebra,IndianJournalofPureandApplied MathematicsandKeralaMathematicalAssociationBulletin. Steve J. Kirkland is a Stokes Professor at the National University of Ireland Maynooth.HeiscurrentlytheEditor-in-ChiefoftheJournalLinearandMultilinear Algebra,andservesontheeditorialboardsofLinearAlgebraanditsApplications, theElectronicJournalofLinearAlgebra,andSIAMReview.HeisalsothePresident oftheInternationalLinearAlgebraSociety. K.ManjunathaPrasadisaProfessorofMathematicsattheDepartmentofStatis- tics, Manipal University, Manipal. He has several publications in the area of his research interest. His research interests are in matrix theory, generalized inverses, ringtheory,projectivemodulesandmatricesingraphtheory. SimoPuntanenearnedhisPh.D.inStatisticsfromtheUniversityofTampere(Fin- land)in1987,whereheispresentlyaDocentandLecturerofStatistics.Hewasa Senior Fellow at the Academy of Finland (autumn 1988 and 1993–1995), an As- sociateProfessoratTampereUniversityduring1997–1998andworkedasaSenior Scientistof the Academyin 1998–1999.Heis currentlythe bookrevieweditor of theInternationalStatisticalReview. xiii
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