搜索
:2006-10-13:(1965-),,,,,.:1004-4280(2007)02-0092-03邹本强(,264200):,,,,,:;;:O151.21:AVariouspropertiesofsymmetricmatrixandant-isymmetricmatrixZOUBen-qiang(WeihaiVocationalCollege,Weihai264200,China)Abstract:MatrixservesasakeytoolinthestudyofAdvancedAlgebra.Howeverinstudyingthetransposeofmatrixmuchfocushasbeengivenonthedefinitionofsymmetricmatrixandant-isymmetricmatrixwhiletheirpropertieshavenotbeenfullyexplored.Asspecialformsofmatrixes,symmetricandant-isymmetricmatrixplaysakeyrolenotonlyinthetheoryofmatrixbutalsoinactualapplication.Propertiesofthetwoformsofspecialmatrixesarefrequentlyusedanddiscussedinthestudyofmatrixandrelatedmathematicalknowledge.Thispaperfirstpresentsthedefinitionofsymmetricmatrixandant-isymmetricmatrixandthenmovesonthedisscussionofcertainpropertiesofthem.Keywords:symmetricmatrix;ant-isymmetricmatrix;properties,,,1AT=A,A;AT=-A,A11);2)21);2)2)AB,k,AT=-A,BT=-B,,(A+B)T=AT+BT=-A-B=-(A+B),(A-B)T=AT-BT=-A+B=-(A-B),(kA)T=kAT=-kA31)A,A-1;2)A,A-11)A,(A-1)T=(AT)-1=A-12)A,(A-1)T=(AT)-1=(-A)-1=-A-141)A,A21220076JOURNALOFSHANDONGINSTITUTEOFLIGHTINDUSTRYVol.21No.2Jun.2007A*;2)A,AA*;A,AA*1)An(n\2),(A*)T=(AT)*=A*2)An(n\2),(A*)T=(AT)*=(-A)*=(-1)n-1A*n,(A*)T=A*;n,(A*)T=-A*51)A,;2)A,1)A,BA,P,B=PTAP,BT=(PTAP)T=PTATP=PTAP=B2)A,BA,P,B=PTAP,BT=(PTAP)T=PTATP=PT(-A)P=-PTAP=-B6AB,1)AB+BA;2)AB-BA1)(AB+BA)T=(AB)T+(BA)T=BTAT+ATBT=BA+AB=AB+BA;2)(AB-BA)T=(AB)T-(BA)T=BTAT-ATBT=BA-AB=-(AB-BA)71)AB,ABAB=BA;2)AB,ABAB=BA;3)AB,ABAB=BA1)AB,AB=BA,(AB)T=BTAT=BA=AB;(AB)T=AB,AB=(AB)T=BTAT=BA2)AB,AB=BA,(AB)T=BTAT=(-B)(-A)=AB;(AB)T=AB,AB=(AB)T=BTAT=(-B)(-A)=BA3)AB,AB=BA,(AB)T=BTAT=(-B)A=-AB;(AB)T=-AB,AB=-(AB)T=-BTAT=-(-B)A=BA8|A|X0,1)A*,A-1;2)A*,A-1|A|X0,A-1=A*|A|1)A*,(A-1)T=(A*|A|)T=A*|A|=A-1;2)A*,(A-1)T=(A*|A|)T=-A*|A|=-A-19|A|X0,1)A*,AT;2)A*,AT|A|X0,A*,AA*=|A|E,A=|A|(A*)-1,1)AT=|A|[(A*)-1]T=|A|[(A*)T]-1=|A|(A*)-1=A;2)AT=|A|[(A*)-1]T=|A|[(A*)T]-1=|A|(-A*)-1=-A10A,1)A+ATAAT;2)A-AT1)(A+AT)T=AT+A=A+AT,(AAT)T=(AT)TAT=AAT;2)(A-AT)T=AT-(AT)T=AT-A=-(A-AT)111);2)01)KA,A,AA=KA,A=A,AA=AA=AA=KA,ATAA=A(AA)=AT(ATA)=(AA)TA=(KA)TA=KATA,ATAA=AT(AA)=AT(KA)=KATA,,KATA=KATA,(K-K)ATA=0,AX0,ATA0,K=K,,K2),K+K=0,,K0121)A,;2)A,1)2)K1K2,A1A2K1K2,,AA1=K1A1,AA2=K2A2,AA1=K1A1,A1TAT=-A1TA=K1A1T,-A1TA=K1A1TA2,-A1TAA2=K1A1TA2=-K2A1TA2,,(K1+K2)A1TA2=0,K1+K2X0,A1TA2=0130An(n),AT=-A,,|AT|=|-A|=(-1)n|A|=-|A|,|AT|=|A|,,|A|=|-A|,|A|=0932::[1].[M].:,1984.[2].[M].:,1995.[3].[M].:,1995.[4].[M].:,2004.[5],.[M].:,1996.(89)H2SNH3,,,,,,,,,,,,,,,,,:[1].[J].,1997,16(5):40-43.[2].[J].,1999,21(2):27-29.[3],.[J].,2003,23(6):44-46.[4],.[J].,l998,26(10):25-27.[5],.[J].,1998(4):521.[6],,.)[J].,1997(1):3-7.[7],,.[J].,2001,7(2):l2l-l24.9421