Elementary symmetric functions of two solvents of

arXiv:0804.2978v1[math-ph]18Apr2008ElementarysymmetricfunctionsoftwosolventsofaquadraticmatrixequationMAJivulescu1,2,ANapoli1,AMessina11MIUR,CNISMandDipartimentodiScienzeFisicheedAstronomiche,Universit`adiPalermo,viaArchirafi36,90123Palermo,Italy2DepartmentofMathematics,”Politehnica”UniversityofTimi¸soara,P-taVictorieiNr.2,300006Timi¸soara,RomaniaEmailAddress:1,[e-mail]:1messina@fisica.unipa.itApril18,2008AbstractQuadraticmatrixequationsoccurinavarietyofapplications.Inthispaperweintroducenewpermutationallyinvariantfunctionsoftwosolventsofthen×nquadraticmatrixequationX2−L1X−L0=0,playingtheroleofthetwoelementarysymmetricfunctionsofthetworootsofaquadraticscalarequation.OurresultsrelyontheconnectionexistingbetweentheQMEandthetheoryoflinearsecondorderdifferenceequationswithnoncommutativecoefficients.Anapplicationofourresultstoasimplephysicalproblemisbrieflydiscussed.Keywords:quadraticmatrixequation;solvent;differenceequation;symmet-ricfunctions.1INTRODUCTIONMatrixlanguagebelongstoallScienceswheremulticomponentvariablesandnoncommutativityentersceneinthedescriptionofthesystemunderscrutiny.Insuchsituationsmatrixmethodsallowindeedcompactformu-lationsandelegantresolutionsoflinearandsometimesnonlinearproblems.Asanexample,considerthetimedevelopmentofaquantumsysteminvesti-gatedintheHeisenbergpicture.Oneisingeneralfacedwithamatrix,oftennonlinear,systemsofdifferentialequationstypicallyverydifficulttohandleduetothenoncommutativityoftheinvolvedobservables.Evenwhentheproblemunderinvestigationenablesthedecouplingofthissystemachieving1lineardifferentialHeisenbergequationsinan-dimensionalHilbertspace,theapplicationofthecharacteristicequationmethod,sousefultowritedownthegeneralintegralinthescalarcase,originatesan-dimensionalmatrixnonlinearalgebraicequationwhoseresolutioncannotunfortunatelyrelyongeneralpropositions.Differentlyfromthecasen=1(thatisthescalarcase),thefundamentaltheoremofalgebradoesnotindeedhold,sothattheroots(usuallycalledsolvents)ofanalgebraicmatrixequationmayexistornotandevenstipulatingtheirexistence,theirnumbercannotbesimplyrelatedtothedegreeoftheequationandcouldinparticularbeinfinite.Itisthusnotsurprisingthatthetheoryofalgebraicmatrixequationsbothforitswideapplicabilityandasresearchsubjectaimingextendingthebeauti-fulchapterofthescalaralgebraicequations,hasrecentlyreceivedagreatdealofattentioninthemathematicalliterature(Gantmacher1998,Gohberg2006,Horn1999).Herewerecalltheanalysisofthespectralpropertiesoftheassociatedmatrixpencilviathatofthematrixequationitself(Gohbergetal.1978,Kreinetal.1978).Manyresearchpapersonthepropertiesofaquadraticmatrixequation,especiallynumericalapproaches,haveappearedoverthelastyears(Baietal.2005,Butleretall1985,Dennisetall1978,Highham1987,2000,2001,Shurbetetal1974).Ourpaperinvestigatestheseconddegreen-dimensionalmatrixalgebraicequationspossessingthefollowingcanonicalformX2−L1X−L0=0(1)wheretheunknownXandthetwo,generallynotcommuting,coefficientsL0,L1belongtoMn(C),thealgebraofallcomplexsquarematricesofordern.Wecallthisequationarightquadraticmatrixequation(RQME).Itsresolutionisfarfrombeingtrivialessentiallybecausethesimpleresolutiveformulaholdingforthequadraticscalarequation(QSE)isnotingeneralapplicableduetothenoncommutativecharacteroftheproblem.Hererightmeansthatthelineartermineq.(1)hastheformL1XinsteadofXL1orL1X+XL′1definingaleft(LQME)orabilateralquadraticmatrixequation(BQME)respectively.Right,leftandbilateralmatrixequationsareparticularcasesoftheRiccatialgebraicmatrixequation(Horeetal.1999)whosecanonicalformisXAX+BX+XC+D=0(2)InthenextsectionwewillshowthattheresolutionofthisequationisalwaystraceablebacktothatofarelatedRQME.Thus,inthispaperweconcentrateonthepropertiespossessedbyeq.(1)andexplorewhetheroratwhichextenttheymaybethoughtofasgeneralizationsofwell-knownpropertiesofelementaryQSE.Inthisspiritweseekanalogiesanddifferencesbetweenthetwoquadraticequations,matrixandscalar,and,inconnectionwitheq.(1),weformulatethefollowingquestions:2Q1)Tofindthenumberofitssolvents;Q2)Toexpressitssolvents,ifany,intermsofitscoefficientsandvicev-ersa;Q3)Todefinesymmetricfunctionsofapairofsolvents.Itiswellknownthatallthreequestionsmaybesatisfactorilycopedwithwhenthequadraticequationisthescalarone.WeshallshowthatthematrixnatureoftheunknownXofeq.(1)togetherornotwiththenoncommu-tativitybetweenthetwocoefficientsaswellasbetweenasolventandthesameL1andL0,determinepropertiesofeq.(1)withnocounterpartamongthosepossessedbythescalarequation.Ourinvestigationwillbringtolightinterestingrelationsbetweenthesolventsofeq.(1)anditscoefficientsL1andL0interpretableasgeneralizationsoftheclassicalGirard-NewtonandWaringformulas(Sansone1952).Inthesection3,devotedtothequestionsQ1)andQ2),weintroducethenotation,giveashortsketchoftheexistingliteratureandreportad-hocbuiltexamplessupportingsometheoreticalstatements.Globallyspeaking,developstepbystepconvincingargumentsenablingustointroduce,onaheuristicbasis,ournewdefinitionofelementaryfunctionsoftwosolventsofaquadraticmatrixequation.Themainandnovelresultsofthispaperareconstructedinconne