简介：LetAbearealsquarematrixandVTAV=GbeanupperHessenbergmatrixwithpositivesubdiagonalentries,whereVisanorthogonalmatrix.ThentheimplicitQ-theoremstatesthatoncethefirstcolumnofVisgiventhenVandGareuniquelydetermined.Inthispaper,threeresultsareestablished.First,itholdsareverseorderimplicitQ-theorem:oncethelastcolumnofVisgiven,thenVandGareuniquelydeterminedtoo.Second,itisprovedthatforaKrylovsubspacetwoformulationsoftheArnoldiprocessareequivalentandinonetoonecorrespondence.Finally,bytheequivalencerelationandthereverseorderimplicitQ-theorem,itisprovedthatfortheKrylovsubspace,ifthelastvectorofvectorsequencegeneratedbytheArnoldiprocessisgiven,thenthevectorsequenceandresultingHessenbergmatrixareuniquelydetermined.