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1、LinearAlgebraandLinearModels,SecondEditionR.B.BapatSpringerPrefaceThemainpurposeofthepresentmonographistoprovidearigorousintroductiontothebasicaspectsofthetheoryoflinearestimationandhypothesistesting.Thenecessaryprerequisitesinmatrices,multivariatenormaldistr
2、ibution,anddistribu-tionofquadraticformsaredevelopedalongtheway.Themonographisprimarilyaimedatadvancedundergraduateand?rst-yearmaster’sstudentstakingcoursesinlinearalgebra,linearmodels,multivariateanalysis,anddesignofexperiments.Itshouldalsobeofusetoresearchw
3、orkersasasourceofseveralstandardresultsandproblems.Somefeaturesinwhichwedeviatefromthestandardtextbooksonthesubjectareasfollows.Wedealexclusivelywithrealmatrices,andthisleadstosomenonconventionalproofs.Oneexampleistheproofofthefactthatasymmetricmatrixhasreale
4、igenvalues.Werelyonranksanddeterminantsabitmorethanisdoneusually.Thedevelopmentinthe?rsttwochaptersissomewhatdifferentfromthatinmosttexts.Itisnottheintentiontogiveanextensiveintroductiontomatrixtheory.Thus,severalstandardtopicssuchasvariouscanonicalformsandsi
5、milarityarenotfoundhere.Weoftenderiveonlythoseresultsthatareexplicitlyusedlater.Thelistoffactsinmatrixtheorythatareelementary,elegant,butnotcoveredhereisalmostendless.Weputagreatdealofemphasisonthegeneralizedinverseanditsapplications.Thisamountstoavoidingthe“
6、geometric”orthe“projections”approachthatisfavoredbysomeauthorsandtakingrecoursetoamorealgebraicapproach.Partlyasapersonalbias,IfeelthatthegeometricapproachworkswellinprovidinganviPrefaceunderstandingofwhyaresultshouldbetruebuthaslimitationswhenitcomestoprovin
7、gtheresultrigorously.The?rstthreechaptersaredevotedtomatrixtheory,linearestimation,andtestsoflinearhypotheses,respectively.Chapter4collectsseveralresultsoneigenval-uesandsingularvaluesthatarefrequentlyrequiredinstatisticsbutusuallyarenotprovedinstatisticstext
8、s.Thischapteralsoincludessectionsonprincipalcompo-nentsandcanonicalcorrelations.Chapter5preparesthebackgroundforacourseindesigns,establishingthelinearmodelastheunderlyingmathematicalframe
