generalized_inverse

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1、Chapter4GeneralizedInversesDefinition4.1:Ageneralizedinverseofanm×nmatrixAisanyn×mmatrixGsuchthatAGA=A.−ThenotationAisoftenusedtoindicateageneralizedinverseofA.Result4.2:IfAisann×nnon-singularmatrix,theuniquegeneralizedinverseofAis−1A.−1−1−1Proof:SinceAA=In,wehaveAAA=A.Clearly,Aisag-

2、inverseofA.SupposeB(n×n)isag-inverseofA.ThenABA=A−1−1−1−1⇔(AA)BAA=AAA−1−1−1i.e.,B=A,(sinceAA=AA=In).Sog-inverseofanon-singularmatrixisuniqueanditistheregularinverse.1010100100Example4.3:LetA=.LetG1=01.ThenAG1=01=010010000010=I2⇒AG1A=I2A=A.Thus,kG1isag-inv

3、erseofA.LetG2=0110101001001.ThenAG2=01=.ThusAG2A=I2A=01001g31g32g31g32A,henceforarbitraryg31andg32,G2isag-inverseofA.Therefore,thereexistinfinitelymanyg-inversesofA.Result4.4:ForanysymmetricmatrixA,thereexistsag-inverseofA.0Proof:thereexistsan⊥matrixP(n×n)soth

4、atA=PΛPwhereΛ=Diag(λ1,...,λn)andλi’saretheeigenvaluesofA.Letλ−1ifλ6=0iiγi=0ifλi=0.13Notethatλ2γ=λforalli=1,...,n.Letiii0G=PDiag(γ1,...,γn)P.Notethat000AGA=PΛPPDiag(γ1,...,γn)PPΛP0=PΛInDiag(γ1,...,γn)InΛP2220=PDiag(λ1γ1,λ2γ2,...,λnγn)P00=PDiag(λ1,λ2,...,λn)P=PΛP=A.ThusGisag-invers

5、eofA.Result4.5:IfG1andG2aregeneralizedinversesofA,thensoisG1AG2.Proof:Exercise.−−Result4.6:ForA(n×n)symmetric,thereexistsAsuchthatAissymmetricand−−−AAA=A.−Proof:LetA=GdefinedinResult4.4above.Notethat−000(A)=[PDiag(γ1,...,γn)P]0000−=(P)Diag(γ1,...,γn)P=PDiag(γ1,...,γn)P=A.Also,−−000AA

6、A=PDiag(γ1,...,γn)PPΛPPDiag(γ1,...,γn)P2200=PDiag(γ1λ1,...,γnλn)P=PDiag(γ1,...,γn)P−=Asinceγ2λ=γforalli.iii−−−−Definition4.7:Ag-inverseAforamatrixAthathasthepropertyAAA=Aissaidtobereflexive.ConsiderthelinearmodelY=Xβ+²where²isarandomvectorwithE(²)=0andVar(²)=σ2I.SupposeXisann×pnmatrix

7、.Leastsquaresestimateofβisobtainedbysolving00XXβ=XY.(8)14Let(X0X)−beag-inverseofX0X.Thenβˆ=(X0X)−X0Yisasolutionto(1).Beforeweshowthisweprovethefollowingresults.Result4.8:(X0X)(X0X)−X0=X0.Proof:Notethat00−0000−000[XX(XX)X−X][XX(XX)X−X]00−000−00=[XX(XX)X−X][X{(XX)}XX−X]00−000−00=[XX(X

8、X)XX−XX][{(XX)}XX−I