简介：Itiswell-knownthatifwehaveanapproximateeigenvalueλ-ofanormalmatrixAofordern,agoodapproximationtothecorrespondingeigenvectorucanbecomputedbyoneinverseiterationprovidedtheposition,saykmax,ofthelargestcomponentofuisknown.Inthispaperwegiveadetailedtheoreticalanalysistoshowrelationsbetweentheeigenvecoruandvectorxk,k=1,…,n,obtainedbysimpleinverseiteration,i.e.,thesolutiontothesystem(A-λI)x=ekwithekthekthcolumnoftheidentitymatrixI.Weprovethatundersomeweakconditions,theindexkmaxisofsomeoptimalpropertiesrelatedtothesmallestresidualandsmallestapproximationerrortouinspectralnormandFrobeniusnorm.Wealsoprovethatthenormalizedabsolutevectorv=|u|/||u||∞ofucanbeapproximatedbythenormalizedvectorof(||x1||2,…||xn||2)^T，Wealsogivesomeupperboundsof|u(k)|forthose'optimal'indexedssuchasFernando'sheuristicforkmaxwithoutanyassumptions,Astabledoubleorthogonalfactorizationmethodandasimplerbutmaylessstableapproachareproposedforlocatingthelargestcomponentofu.

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