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1、[406]AGENERALIZEDINVERSEFORMATRICESBYR.PENROSECommunicatedbyJ.A.TODDReceived26July1954Thispaperdescribesageneralizationoftheinverseofanon-singularmatrix,astheuniquesolutionofacertainsetofequations.Thisgeneralizedinverseexistsforany(possiblyrectangular)matrixwh
2、atsoeverwithcomplexelementsJ.Itisusedhereforsolvinglinearmatrixequations,andamongotherapplicationsforfindinganexpressionfortheprincipalidempotentelementsofamatrix.Alsoanewtypeofspectraldecom-positionisgiven.Inanotherpaperitsapplicationtosubstitutionalequations
3、andthevalueofhermitianidempotentswillbediscussed.Notation.Capitallettersalwaysdenotematrices(notnecessarilysquare)withcom-plexelements.TheconjugatetransposeofthematrixAiswrittenA*.Smalllettersareusedforcolumnvectors(withanasteriskforrowvectors)andsmallGreeklet
4、tersforcomplexnumbers.Thefollowingpropertiesoftheconjugatetransposewillbeused:A**=A,(A+B)*=A*+B*,(BA)*=A*B*,AA*=0implies^4=0.ThelastofthesefollowsfromthefactthatthetraceofAA*isthesumofthesquaresofthemodulioftheelementsofA.EromthelasttwoweobtaintheruleBAA*=CAA*
5、impliesBA=GA,(1)since(BAA*-CAA*)(B-C)*=(BA-CA)(BA-GA)*.SimilarlyBA*A=CA*AimpliesBA*=GA*.(2)THEOREM.ThefourequationsAXA—AC\XAX=X,(4){AX)*=AX,(5){XA)*=XA,(6)haveauniquesolutionforanyA.%Matricesovermoregeneralringswillbeconsideredinalaterpaper.Ageneralizedinver
6、seformatrices407Proof.Ifirstshowthatequations(4)and(5)areequivalenttothesingleequationXX*A*=X.(7)Equation(7)followsfrom(4)and(5),sinceitismerely(5)substitutedin(4).Con-versely,(7)impliesAXX*A*=AX,theleft-handsideofwhichishermitian.Thus(5)follows,andsubstitutin
7、g(5)in(7)weget(4).Similarly,(3)and(6)canbereplacedbytheequationXAA*=A*.(8)ThusitissufficienttofindanXsatisfying(7)and(8).SuchanXwillexistifaBcanbefoundsatisfyingM,,3BBA*AA*=A*.ForthenX=BA*satisfies(8).Also,wehaveseenthat(8)impliesA*X*A*=A*andthereforeBA*X*A*=B
8、A*.ThusXalsosatisfies(7).23NowtheexpressionsA*A,(A*A),(A*A),...cannotallbelinearlyindependent,i.e.thereexistsarelationkA1A*A+2(A*A)*+...+Ak(A*A)=0>(9)whereA,,...,Aftarenotallzero.
