Linear Algebra and Linear Models， 2ed (R.B. Bapat).pdf

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1、LinearAlgebraandLinearModels,SecondEditionR.B.BapatSpringerPrefaceThemainpurposeofthepresentmonographistoprovidearigorousintroductiontothebasicaspectsofthetheoryoflinearestimationandhypothesistesting.Thenecessaryprerequisitesinmatrices,multivariatenormaldistr

2、ibution,anddistribu-tionofquadraticformsaredevelopedalongtheway.Themonographisprimarilyaimedatadvancedundergraduateandfirst-yearmaster’sstudentstakingcoursesinlinearalgebra,linearmodels,multivariateanalysis,anddesignofexperiments.Itshouldalsobeofusetoresearchw

3、orkersasasourceofseveralstandardresultsandproblems.Somefeaturesinwhichwedeviatefromthestandardtextbooksonthesubjectareasfollows.Wedealexclusivelywithrealmatrices,andthisleadstosomenonconventionalproofs.Oneexampleistheproofofthefactthatasymmetricmatrixhasreale

4、igenvalues.Werelyonranksanddeterminantsabitmorethanisdoneusually.Thedevelopmentinthefirsttwochaptersissomewhatdifferentfromthatinmosttexts.Itisnottheintentiontogiveanextensiveintroductiontomatrixtheory.Thus,severalstandardtopicssuchasvariouscanonicalformsandsi

5、milarityarenotfoundhere.Weoftenderiveonlythoseresultsthatareexplicitlyusedlater.Thelistoffactsinmatrixtheorythatareelementary,elegant,butnotcoveredhereisalmostendless.Weputagreatdealofemphasisonthegeneralizedinverseanditsapplications.Thisamountstoavoidingthe“

6、geometric”orthe“projections”approachthatisfavoredbysomeauthorsandtakingrecoursetoamorealgebraicapproach.Partlyasapersonalbias,IfeelthatthegeometricapproachworkswellinprovidinganviPrefaceunderstandingofwhyaresultshouldbetruebuthaslimitationswhenitcomestoprovin

7、gtheresultrigorously.Thefirstthreechaptersaredevotedtomatrixtheory,linearestimation,andtestsoflinearhypotheses,respectively.Chapter4collectsseveralresultsoneigenval-uesandsingularvaluesthatarefrequentlyrequiredinstatisticsbutusuallyarenotprovedinstatisticstext

8、s.Thischapteralsoincludessectionsonprincipalcompo-nentsandcanonicalcorrelations.Chapter5preparesthebackgroundforacourseindesigns,establishingthelinearmodelastheunderlyingmathematicalframe