On the perturbation theory for unitary eigenvalue

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2020-01-24

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ONTHEPERTURBATIONTHEORYFORUNITARYEIGENVALUEPROBLEMSB.BOHNHORST,A.BUNSE-GERSTNERy,ANDH.FABENDERzAbstract.Someaspectsoftheperturbationtheoryforeigenvaluesofunitarymatricesareconsidered.MakinguseofthecloserelationbetweenunitaryandHermitianeigenvalueproblemsaCourant-Fischer-typetheoremforunitarymatricesisderivedandaninclusiontheoremanaloguetotheKahantheoremforHermitianmatricesispresented.ImplicationsforthespecialcaseofunitaryHessenbergmatricesarediscussed.Keywords.unitaryeigenvalueproblem,perturbationtheoryAMS(MOS)subjectclassications.15A18,65F991.Introduction.Newnumericalmethodstocomputeeigenvaluesofunitarymatriceshavebeendevelopedduringthelasttenyears.UnitaryQR-typemethods[19,9],adivide-and-conquermethod[20,21],abisectionmethod[10],andsomespe-cialmethodsfortherealorthogonaleigenvalueproblem[1,2]havebeenpresented.Interestinthistaskarosefromproblemsinsignalprocessing[11,29,33],inGaussianquadratureontheunitcircle[18],andintrigonometricapproximations[31,16]whichcanbestatedaseigenvalueproblemsforunitarymatrices,ofteninHessenbergform.Asthosenumericalmethodsexploittherichmathematicalstructureofunitaryma-trices,whichiscloselyanalogoustothestructureofHermitianmatrices,themethodsareecientanddeliververygoodapproximationstothedesiredeigenvalues.Thereexist,however,onlyafewperturbationresultsfortheunitaryeigenvalueproblem,whichcanbeusedtoderiveerrorboundsforthecomputedeigenvalueap-proximations.Athoroughandcompletetreatmentoftheperturbationaspectsasso-ciatedwiththenumericalmethodsforunitaryeigenvalueproblemsisstillmissing.Thefollowingperturbationresultshavebeenobtainedsofar.IfUandeUareunitarymatriceswithspectra(U)=fjg,and(eU)=fejg,respectively,wecanarrangetheeigenvaluesindiagonalmatricesande,respectively,andconsiderasameasureforthedistanceofthespectrad((U);(eU)):=minPjjPTePjj;=2;F(1.1)wheretheminimumistakenoverallpermutationmatricesPandthenormiseitherthespectralortheFrobeniusnorm.BytheHoman-Wielandttheorem(see,e.g.,[34])wegetdF((U);(eU))jjUeUjjF:BhatiaandDavis[5]provedthecorrespondingresultforthespectralnormd2((U);(eU))jjUeUjj2:Schmidt,Vogel&PartnerConsult,GesellschaftfurOrganisationundManagementberatungmbH,Gadderbaumerstr.19,33602BielefeldyUniversitatBremen,Fachbereich3-MathematikundInformatik,28334Bremen,Germany,email:angelika@math.uni-bremen.dezUniversitatBremen,Fachbereich3-MathematikundInformatik,28334Bremen,Germany,email:heike@math.uni-bremen.de12OntheperturbationtheoryfortheunitaryeigenvalueproblemElsnerandHeconsiderarelativeerrorin[15].Theyusethemeasureed((U);(eU)):=minPjj(+PTeP)1(PTeP)jj;(1.2)whereagain=2or=F.Theyprovethated((U);(eU))jjC(UHeU)jjwhereC(U)=i(I+U)1(IU)istheCayleytransformationofU(assumingherethat162(U)).ToeacheigenvalueofU,where162(U),wecanassociateananglebydening=arctan[p1(1+)1(1)]with=2=2.Itistheangleformedbythelinefrom-1throughandtherealaxis(seealsoSection2).WithrespecttotheiranglestheeigenvaluesofUandeUhaveanaturalorderingontheunitcircle.ElsnerandHegivesine-andtangent-interpretationsoftheaboveinequalityintermsoftheseangles.FurthermoretheyshowthatwithrespecttoacertaincuttingpointontheunitcircletheeigenvaluesofUandeUhaveanaturalorderingfj()gandfej()gontheunitcirclesuchthatmaxjjj()ej()jjjUeUjj2:Aninterlacingtheoremforunitarymatricesisalsopresentedin[15],showingthattheeigenvaluesofsuitablymodiedprincipalsubmatricesofaunitarymatrixinterlacethoseofthecompletematrixontheunitcircle(seeSection2).InthispaperweconsiderfurtheraspectsoftheperturbationproblemfortheeigenvaluesofaunitarymatrixU.InSection2weshowhowtheanglesarerelatedtotheeigenvaluesoftheCayleytransformofU.Withtheaidofthisrelationwecangiveamin-max-characterizationfortheanglesofU’seigenvaluesinanalogytotheCourant-FischertheoremforHermitianmatrices.WealsoshowthattangentsoftheseanglescanbecharacterizedbyusualRayleighquotientscorrespondingtothegeneralizedeigenvalueproblemp1(I+UH)(IU)x=(I+UH)(I+U)x:FurthermoreweproveaKahan-likeinclusiontheoremshowingthattheeigenvaluesofacertainmodiedleadingprincipalsubmatrixofUdeterminearcsontheunitcirclesuchthateacharccontainsaneigenvalueofU.InapplicationsunitarymatricesareoftenofHessenbergform.InSection3werecallthataunitaryunreducedHessenbergmatrixHhasauniqueparameterizationH=H(1;:::;n),wherethereectionparameters1;:::;n2C,withjij1fori=1;:::;n1andjnj=1,determineHcompletely.WeshowtheimplicationsoftheresultsinSection2forthespecialcaseofunitaryHessenbergmatrices.InparticularitwillbeseenthatthemodiedkthleadingprincipalsubmatrixinthisspecialcaseisjustH(1;:::;k1;)wherejj=1.Wediscussthedependenceoftheeigenvaluesonthislastreectionparameter.FinallySection4willgivenumericalexampleswhichelucidatethestatementsprovedinSection3.Ontheperturbationtheoryfortheunitaryeigenvalueproblem32.PerturbationResultsforunitaryMatrices.UnitarymatriceshavearichmathematicalstructurethatiscloselyanalogoustothatofHermitianmatrices.InthissectionwerstdiscusstheintimaterelationshipbetweenunitaryandHermitianmatriceswhichindicatesthatonecanhopetondu