幂零矩阵的m次根

幂零矩阵的m次根 第21卷第6期 2005年12月 大学数学 COLLEGEMATHEMATICS Vo1.21,?.6 Dec.2005 m——RootsofNilpotentMatrices ZHANGXiao—wang,XUChang—qing (Dept.ofMath.,AnhuiUniv.,Hefei230039,China) Abstract:Thispaperisderivedtothestudyofthe with2?m?N.Wecanderivesomegoodrelationship equation=A. equation一AwhereAisa×nilpotentmatrix betweenthecharactersofAandunsolvabilityofthe Keywords:nilpotentmatrix;unsolvability;m—root;nilpotencyindex CLChumber:O151.21Documentcode:AArticleID:1672—1454(2005)06—0042—03 1Introduction Definition1.1(Algebraicrootoforder)XiscalledanalgebraicrootofAoforderm.ifX satisfyingX一A. Definition1.2(Nilpotentmatrices)Aiscalledanilpotentmatrixifforsomem(m?N)A一0. Definition1.3(Nilpotentindex)WithagivennilpotentmatrixA.ApositiveintegerPiscalled thenilpotencyindexofAifPsatisfiesA一OandA一?0. 2MainResults IfA一0,theequationX一Amustbesolvable nilpotencyindexP,wherePisadivisorofmmustbe sinceallmatriceswithnilpoltencyindexmor m-rootsofA.Inthefollowingweonlydiscuss nonzeronilpotentmatrices,i.e.nilpotencyindex>1.Therearesomelemmasandmycone1 USions providesomesufficientconditionsofunsolvabilityofX一A,whereAisnilpotent. Lemma2.1LetAft(C)andAbeanonzeronilpotentmatrixofnilpotencyindexP.Thenin (c),—Ahasnos.luti.n,providedthat>. Wecanslightlyimprovetheresult. Theorem2?2LetA?(C)andAbeanonzeronilpotentmatrixofnilpotencyindexP.Then _An0?0lvabln)f0rany. ProofAssumebycontradictionthatsuchanxexists,i.e.xm—A,for>.ThenO=A P一1 一X,SOXisnilpotent.LetkbeindexofnilpotencyofX.FromX=Ap—OandXp一)m=Ap一?0. Weclearlyobtain(p--1)<志?and志?,hence(一1)+1?志?,inthat,?三{. Reiceiveddate:2003—11—26 Fundationitem:安徽大学创新团队项目基金 第6期ZHANGXiao—wangetal:m—RootsofNilpotentMatrices43 Thisc0ntradsthefactthat>. Corollary2.3IetA?M((e)andnilpotencyindexofAben.ThentheequationX一Aisnot solvableinM(,C),foranym>1. Proofobservethat>一1. Lemma2.4LetA?M((e)wheren?{2,3,…)andAbeanilpotentmatrixofrankn一1.Then X一Ahasnom-rootinM(C),foranym>1. Lemma2.5(Solvabilityof=A)Letm?2andA?M((c).ThentheequationX一Ahas solutionsifandonlyifthesingularJordanblocksof】, canbeplacedingroupsofm,withorders differingbynomorethanoneineachgroup.althoughthesingular1一by一1blocksneednotbegrouped. Theorem2.6IetA?M((e),wheren?{2,3,…)andAbeanilpotentmatrixofrankn— (where14s<n一1).Then—AhasnorootinM((e),foranym>s. ProofSinceAisnilpotent.i.e.rank(A)>rank(A),itfollowsthatAhasanontrivialsingular block.0bservethatthenumberofsingularJordanblocksisnotmorethans(where<).Sincet he numberofsingularblocksislessthanm.theconditionsoflemma2.5arenotmet.SoX一 Ahasno m-rootinM((e),foranym>s. WhenX一A.weknowthatifAisnilpotent,Xisalsonilpotent.Sowewritethatthenilpotency indexofAisn(A)andthenilpotencyindexofXisn(X).It'seasytofindn(A)?n(X)since A'一(X)一一0. Corollary2.7IetA?(C),wheren?{2,3,…}andAbeanilpotentmatrixofnilpotency indexn(A).ThenX一A,wherethenilpotencyindexofXisn(X),hasnom—rootinM((e), pr.videdthat>. ProofAssumebycontradictionthatsuchanXexists,i.e.X一A,for> X'A=A'A=0andX''A一—A'A一?O. Thus()一1)<n(.()一1)+1?n(hencem?. [1] [2] [3] [4] Itcontradictsthefactthatm>n(X)一1 n(A)一1' References n(X),1 n(A),1.Then WintersJL.ThematrixequationXm—A[J].JournalofAlgebra,1980,67:82—87. DennisJE.TraubJFandWeberRP.Algorithmsforsolventsofmatrixpolynomials[J].SIAMJ. Numer. Ana1.,1978,15(3):523—533. MenzaqueFEandPatarraC.Solutionsofmatrixpolynomialequations[J].Comp.App1.Mat h.,15(3): 241—256. HornRAS,JonsonCR.Iopicsinmatrixanalysis[M].London:CambridgeUniversityPress,1 991. 44大学数学第21卷 幂零矩阵的77/次根 张小旺,徐常青 (安徽大学数学与计算科学学院,合肥230039) [摘要]主要研究当A是幂零矩阵时,方程—A的性质.我们可以得到一些关于方程Xm—A无解性与A自身 的特点之间的关系. 关键词:幂零矩阵;无解性;m次根;幂零指数