分块矩阵的初等变换及其在求逆和行列式中的应用

第12卷　第3期Vol.12　No.3重庆工业管理学院学报JournalofChongqingInstituteofTechnologyManagement1998年6月Jun.1998分块矩阵的初等变换及其在求逆和行列式中的应用吴　　云(643000)将矩阵的初等变换概念推广到分块矩阵并建立了计算分块矩阵的逆矩阵和分块方阵的行列式的若干简易方法。分块矩阵　初等变换　初等矩阵1A=AijrSAijrS:A=A11A1sAr1ArsAijminj1,1A:(1)A(rirjij);(2)KA(KriKi);(3)AK(ri+KrjjKi),,,I=IriSS:I=Ir1IrsIriri(1iS)2I,(1)Ii,j()I(i,j);(2)K()Ii()I(i(K));(3)KIjiI(j(k),i),KI:1997-10-23ij1I=Il000Im000InI=Il000Im000Inr1r300In0Im0Il00=I(1,3)I=Il000Im000InKr2Il000K000In=I(2(K))(Km)I=Il000Im000Inr1+Kr3Il0K0Im000In=I(3(K),1)(Kln),1ArS(1)A,Arr;(2)A,ASS2A33:A=A11A12A13A21A22A23A31A32A3333I(1,3)A:I(1,3)A=00Ir30Ir20Ir100A11A12A13A21A22A23A31A32A33=A31A32A33A21A22A23A11A12A13A1,333I(3(K),1)A:AI(3(K),1)=A11A12A13A21A22A23A31A32A33Ir1000Ir2000Ir3=A11A12A13+A11KA21A22A23+A21KA31A32A33+A31KA2A=(Aij)()SS,Aijri(1iS),ASSA():A=A11A12A1s0A22A2s00AssA-111,A-121,,A-1ssA1,2,,s,B=Ir1B12B1s0Ir2B2s00Irs101吴　云:分块矩阵的初等变换及其在求逆和行列式中的应用-BisBSi(i=1,2,S-1),C;-Bis-1CS-1i(i=1,2,,S-2);,A:I=Ir1Irs1,A,A,P1,P2,,PmPmPm-1P2P1A=I(1)A=P1-1P2-1Pm-1,,A2BSSB2A,BA2,(1)PmP2P1I=A-1(2)(1),(2),AI,IA-1,,A:AIr2rAI,AI,IA-1,AIIA-13P=AC0B,A,Bmn,P-1:PI=AC0BIm00InB-1r2AC0InIm00B-1r1+(-C)r2A00InIm-CB-10B-1A-1r1Im00InA-1-A-1CB-10B-1=Ip-1P-1=A-1-A-1CB-10B-13P-1,P-14P=00A0B0C00,A,BCl,mn,P-1:PI=00A0B0C00Il000Im000Inr1r3C000B000A00In0Im0Il00C-1r1,B-1r2,A-1r3In000Im000Il00C-10B-10A-100=IP-1P-1=00C-10B-10A-1002,102重庆工业管理学院学报ISS:I=Ir1IriIrjIrsIriri(1iS),I2,:(1)I(i,j)=(-1),=ri(ri+1++rj)+rj(ri+1++rj-1(ij),,j=i+1,I(i,j)=(-1)rirj;(2)I(i(K))=k,Kri;(3)I(j(K),i)=1,Krirj(1),I(i,j)I(i,j)I,,I(i,j)=(-1)I=(-1)(2),(3)A,A,,4A(1)Ai,j(),(-1)A,=ri(ri+1++rj)+rj(ri+1++rj-1);,A()(ii+1),(-1)riri+1A;(2)riK()Ai(),KA;(3)()A()(),4(2)A()()K,K5P=ABCD,Ar,P:4(3):P=ABCD=AIrA-1BCD=AIrA-1B0D-CA-1B=AD-CA-1BAD,P=AD-ACA-1B;AC(AC=CA),P=AD-CB6D2n=ababcdcd,a0,A:D2n=ababcdcd=ABCDA,C,103吴　云:分块矩阵的初等变换及其在求逆和行列式中的应用D2n=AD-CB=adad-bcbc=(ad-bc)I=(ad-bc)n7A,B,CDn,ABCD=DCBA4(1),ABCD=(-1)n2CDAB=(-1)n2(-1)n2DCBA=DCBA8ABn,ABBAAB-I=BA-I,AI,57AB-I=AIIB=BIIA=BA-I1..:,19912,..:,1995ElementaryTransformationofBlockMatrixandItsApplicationinInversionandDeterminantWuYunABSTRACTByextendingtheconceptofelementarytransformationofmatrixtoblockma-trix,wecancaculatetheinversematrixanddeterminantofblockmatrixmoreeasily.KEYWORDSblockmatrix;elementarytransformation;elementarymatrix(彭　熙)104重庆工业管理学院学报