Every invertible matrix is diagonally equivalent to a matrix with distinct eigenvalues.pdf

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1、EveryinvertiblematrixisdiagonallyequivalenttoamatrixwithdistincteigenvaluesMan-DuenChoi,ZejunHuangy,Chi-KwongLiz,andNung-SingSzexAbstractWeshowthatforeveryinvertiblenncomplexmatrixAthereisannndiagonalinvertibleDsuchthatADhasdistincteigenvalues.Usingthisresult,wearmaconjecture

2、ofFeng,Li,andHuangthatannnmatrixisnotdiagonallyequivalenttoamatrixwithdistincteigenvaluesifandonlyifitissingularandallitsprincipalminorsofsizen1arezero.AMSSubjectClassication.15A18.Keywords.Invertiblematrices,diagonalmatrices,distincteigenvalues.1IntroductionDenotebyMnthesetof

3、nncomplexmatrices.In[1],theauthorspointedoutthatmatriceswithdistincteigenvalueshavemanyniceproperties.TheythenraisedthequestionwhethereveryinvertiblematrixinMnisdiagonallyequivalenttoamatrixwithdistincteigenvalues,andconjec-turedthatamatrixinMnisnotdiagonallyequivalenttoamatrixw

4、ithdistincteigenvaluesifandonlyifitissingularandeveryprincipalminorofsizen1iszero.TheyprovidedaproofformatricesinMnwithn3,anddemonstratedthecomplexityoftheproblemformatricesinM4usingtheirapproach.Inthisnote,wearmtheirconjecturebyprovingthefollowingtheorem.Theorem1.1SupposeA2Mn

5、isinvertible.ThereisaninvertiblediagonalD2MnsuchthatADhasdistincteigenvalues.Oncethisresultisproved,wehavethefollowingcorollary.DepartmentofMathematics,UniversityofToronto,Toronto,Ontario,CanadaM5S2E4.(choi@math.toronto.edu)yDepartmentofAppliedMathematics,TheHongKongPolytechnicU

6、niversity,HungHom,Kowloon,HongKong.(huangzejun@yahoo.cn)zDepartmentofMathematics,CollegeofWilliam&Mary,Williamsburg,Virginia23187-8795,USA.ThisresearchwasdonewhilehewasvisitingTheHongKongUniversityofScience&Technologyin2011underthesupportofaFulbrightFellowship.Liisanhonoraryprofe

7、ssorofTheUniversityofHongKong,TaiyuanUniversityofTechnology,andShanghaiUniversity.(ckli@math.wm.edu)xDepartmentofAppliedMathematics,TheHongKongPolytechnicUniversity,HungHom,Kowloon,HongKong.(raymond.sze@inet.polyu.edu.hk)1Corollary1.2LetA2Mn.Thefollowingareequivalent.(a)Aisnotdia

8、gonallyequivalenttoamatrixwithdistinctei