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142007O151.21A1673999X200704001405..122×.(1)()(2)()()(3)()()()()().mn+22×00mmnnIII+⎛⎞=⎜⎟⎝⎠.12(1)00nmII⎛⎞⎜⎟⎝⎠(2)00,00mnPIIQ⎛⎞⎛⎞⎜⎟⎜⎟⎝⎠⎝⎠(3)10,0mmnnIRIISI⎛⎞⎛⎞⎜⎟⎜⎟⎝⎠⎝⎠,PQmn1,mnnmRRSR××∈∈.(1)10000nmmnIIII−⎛⎞⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠20070726200712Dec.20074JournalofYiliNormalUniversityNaturalScienceEditionNo.4415(2)11110000,0000mmnnPIIPIQQI−−−−⎛⎞⎛⎞⎛⎞⎛⎞==⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎝⎠(3)111100,00mmmmnnnnIRIRIIIISISI−−−⎛⎞⎛⎞⎛⎞⎛⎞==⎜⎟⎜⎟⎜⎟⎜⎟−⎝⎠⎝⎠⎝⎠⎝⎠.(1)10000000000nnmnmmmnmnIIIIIIIIII−⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞==⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠(2)110000000000mnnnnnPPPIPIIIII−−⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞==⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠110000000000mmmmmnIIIIIIQQQQ−−⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞==⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠(3)11111000000mmmmmnnnnnIRIRIRIRIIIIII−−⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞==⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠1000000mmmmmnnnnnIIIIISISISISII−⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞==⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟−⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠...22×ABMCD⎛⎞=⎜⎟⎝⎠()()M().ABMCD⎛⎞=⎜⎟⎝⎠(1)00nmIABCDICDAB⎛⎞⎛⎞⎛⎞=⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠(2)0000mnPABPAPBIABABICDCDQCDQCQD⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞==⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠(3)1110mnIRABARCBRDICDCD++⎛⎞⎛⎞⎛⎞=⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠0mnIABABSICDCSADSB⎛⎞⎛⎞⎛⎞=⎜⎟⎜⎟⎜⎟++⎝⎠⎝⎠⎝⎠...2n()()()()(,,,),rrrnrnrrnrnrABMARBRCRDRCD××−−×−×−⎛⎞=∈∈∈∈⎜⎟⎝⎠M()()()()()02,()0rrrnrrnrnrnrInnMIII×+−+−−×−⎛⎞×=⎜⎟⎝⎠()()1,rnrIM+−11MM−=.M1212ssnMPPPPPPI==iP(1,2,,is=)1122()()112200rrssnrnrssIABABABABICDCDCDCD×−×−⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞=⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠(1,2,,)iiiiiABPisCD⎛⎞==⎜⎟⎝⎠M1620071111()()1100rrssnrnrssIABABABICDCDCD−−×−×−⎛⎞⎛⎞⎛⎞⎛⎞=⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎝⎠111111()()1100rrssnrnrssIABABABICDCDCD−−−×−×−⎛⎞⎛⎞⎛⎞⎛⎞=⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎝⎠212M()()00rrnrnrII×−×−⎛⎞⎜⎟⎝⎠()()00rrnrnrII×−×−⎛⎞⎜⎟⎝⎠M1M−.M()()00rrnrnrII×−×−⎛⎞⎜⎟⎝⎠.M2nn×()(),rnrMI+−()1(),rnrIM−+−.nm+ABMCD⎛⎞=⎜⎟⎝⎠nnnmAR,BR,××∈∈,)mnmmCRDR××∈∈,1,,ADCABM−−1M−.110000nnmmIIABABICAICDDCAB−−⎛⎞⎛⎞→⎜⎟⎜⎟−−⎝⎠⎝⎠####111110000nnmmIABABIPDCABCAIIPPCAP−−−−−⎛⎞⎛⎞=−→⎜⎟⎜⎟−−⎝⎠⎝⎠##JJJJJJJJJJJJJJJJG##1111111111111110000nnmmAIIBPCABPAABPCAABPIIPCAPPCAP−−−−−−−−−−−−−−−⎛⎞⎛⎞+−+−→→⎜⎟⎜⎟−−⎝⎠⎝⎠####11111111111ABAABPCAABPMCDPCAP−−−−−−−−−−−⎛⎞+−⎛⎞==⎜⎟⎜⎟−⎝⎠⎝⎠1111111111111()()()()AABDCABCAABDCABDCABCADCAB−−−−−−−−−−−−−⎛⎞+−−−=⎜⎟−−−⎝⎠.,BC(1)1111100ABAABCCC−−−−−⎛⎞−⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠(2)1111100AACDDCAD−−−−−⎛⎞⎛⎞=⎜⎟⎜⎟−⎝⎠⎝⎠.2nm+ABQCD⎛⎞=⎜⎟⎝⎠,,,nmnnmmmnARBRCRDR××××∈∈∈∈,C1,BACDQ−−1Q−.11000000nmmmnnIIIABCDCDIIIACCDABBACD−−⎛⎞⎛⎞⎛⎞→→⎜⎟⎜⎟⎜⎟−−⎝⎠⎝⎠⎝⎠######111110000mmnnICDCDIPBACDIACIPPPAC−−−−−⎛⎞⎛⎞=−→⎜⎟⎜⎟−−⎝⎠⎝⎠##JJJJJJJJJJJJJJJJG##1111111111111110000mmnnCIDPIDPACCDPCCDPACIIPPACPPAC−−−−−−−−−−−−−−−⎛⎞⎛⎞−+−+→→⎜⎟⎜⎟−−⎝⎠⎝⎠####41711111111111ABCDPCCDPACQCDPPAC−−−−−−−−−−−⎛⎞−+⎛⎞==⎜⎟⎜⎟−⎝⎠⎝⎠1111111111111()()()()CDBACDCCDBACDACBACDBACDAC−−−−−−−−−−−−−⎛⎞−−+−=⎜⎟−−−⎝⎠.,AD(1)1111100BCDBCCDB−−−−−⎛⎞−⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠(2)1111100ABCCBBAC−−−−−⎛⎞⎛⎞=⎜⎟⎜⎟−⎝⎠⎝⎠..3A,Bnm×mn×nIAB+nmIBA+.nm+0nmIAMIBA−⎛⎞=⎜⎟+⎝⎠200000nnnnnnmmmmmmIAIIAIIABIABAIBAIBIBIBIBI−−++⎛⎞⎛⎞⎛⎞→→⎜⎟⎜⎟⎜⎟+⎝⎠⎝⎠⎝⎠######nIAB+n,0nnmmIABIABABIBI++⎛⎞⎜⎟⎝⎠##11100()00()00()nnnnnmmnmmnIABIABAIIIABAIIBIABAIIBIABA−−−++⎛⎞+⎛⎞→→⎜⎟⎜⎟−+−+⎝⎠⎝⎠####111()0()nnmnIIABAMIBIABA−−−⎛⎞+=⎜⎟−+⎝⎠.mIBA+11()()mmnIBAIBIABA−−+=−+.A,BnIAB+IBA+11()()IBAIBIABA−−+=−+.4An,αβn110TAβα−+≠TAαβ+1()TAαβ−+.1()TAαβ−+1[()]TTnAAIAαβαβ−+=+A1()TnIAαβ−+110TAβα−+≠1()TnIAαβ−+1111111[()]()[1]1TTTTnnnTAIAIAAIAαβαβαβαββα−−−−−−−+=−+=−+1()TAαβ−+=11[()]TnIAαβ−−+1A−=1A−1111TTAAAαββα−−−−+1Sherman-Morrison.5A,Bnm×mn×mn≥mnmIABλλ−−=nIBAλ−.182007ArmPnQ000rIPAQ⎛⎞=⎜⎟⎝⎠121134BBQBPBB−−⎛⎞=⎜⎟⎝⎠1Brr×1121130000BBBPABPQBAQB−−⎛⎞⎛⎞==⎜⎟⎜⎟⎝⎠⎝⎠1210rmrmrmrIBBIABIBIλλλλλ−−−−−==−1130rnrnrnrIBIBAIBBIλλλλλ−−−−==−−mnmnIABIBAλλλ−−=−.,mnnmARBR××∈∈ABBA.ABABBA()(),trABtrBA=1()trPAPtrA−=..22×.·ElementaryTransformationandApplicationofBlockMatrixGAOBai-jun(Dept.ofMath.,IliNormalUniversity,Yining835000,Xinjiang,China)Abstract:Duetotheimportanceofthemethodandtheapplicationofpartitioningofmatricesinlinearalgebra,thethoughtandthemethodofelementarytransformationareappliedinpartitioningofmatrices.Theauthorproposesthenatureofelementarytransformationofblockmatrix,anditsapplicationinthesolutionofinverseofamatrixandcharacteristicpolynomialofamatrix.Keywords:blockmatrix;elementarytransformation;blockelementarymatrixANoteaboutLaplaceTransformTIANBao,TIANHong-gen(SchoolofMaths-physicsandInformationSciences,XinjiangNormalUniversity,Urumqi830054,Xinjiang)Abstract:Thisarticlehasstudiedoneofmethodsinthesolutionsimultaneousdifferentialequation,whichmakesuptheinsufficiencyinusingtheLaplacetransformationtosolvetheproblem.Keywords:homogeneous;non-homogeneous;Laplacetransformation