矩阵特征值的估计及其应用

:2008-11-14:(1982-),,,.Email:[e-mail]:1671-9352(2009)12-0048-04薛建明,邹黎敏(,404000):讨论矩阵特征值估计及其在稳定性理论中的应用证明了矩阵的所有特征值都位于一个圆盘中,给出了定常线性系统在平衡位置渐近稳定的一个充分条件,并给出了数值算例:特征值;F函数;范数;估计;稳定性:O1512:AEstimationforeigenvaluesanditsapplicationXUEJian-ming,ZOUL-imin(CollegeofMathematicsandComputerScienceofChongqing,ThreeGorgesUniversity,Chongqing404000,China)Abstract:Thepurposeofthispaperistodiscusstheestimationforeigenvaluesofmatricesanditsapplicationinstabilitytheory.Weprovethatalltheeigenvaluesofanycomplexmatrixarelocatedinonedisk.Afterthat,wepresentasufficientconditionthatalineartime-invariantsystemisasymptoticallystableinequilibriumposition.Somenumericalexamplesaregiven.Keywords:eigenvalues;Ffunction;norm;estimation;stability,,,,,;,Hermite,[1-4],dxdt=f(x),dxdt=Ax,A,Routh-Hurwitz,,,A,,Cnnnn,A=(aij)Cnn,AF=tr(A*A)AF-,A*A1矩阵特征值的估计11f:CnnR+{0},f4412Vol.44No.12()JournalofShandongUniversity(NaturalScience)200912Dec.2009(1)A,|trA|2f(A),A;(2)C,f(I-A)=n-1n(|tr(I-A)|2-|trA|2)+f(A);(3)f(A)n-1n|trA|2,fF,Ff11ACnn,ff,A:zC:z-trAnnf(A)-(n-1)|trA|2nA,H=I-A,F,|trH|2f(H),Ff(I-A)=n-1n(|tr(I-A)|2-|trA|2)+f(A),|tr(I-A)|2n-1n(|tr(I-A)|2-|trA|2)+f(A),|tr(I-A)|2nf(A)-n-1n|trA|2,n2-trAn2nf(A)-(n-1)|trA|2,-trAnnf(A)-(n-1)|trA|2n12MCnn:M=AkkBk(n-k)C(n-k)kD(n-k)(n-k),(1kn-1)f(M)=(n-1)M2F-max1kn-1Bk(n-k)F-C(n-k)kF2,f(M)f(1)T(M)=|trM|2-(n-1)M2F-max1kn-1Bk(n-k)F-C(n-k)kF2,|trM|2f(M),T(M)0,Ms,rankMs,|trM|2=si=1i2ssi=1|i|2rankMsi=1|i|2M,|trM|2(n-1)si=1|i|2[3]1si=1|i|2M2F-max1kn-1(Bk(n-k)F-C(n-k)kF)2,|trM|2(n-1)M2F-max1kn-1(Bk(n-k)F-C(n-k)kF)2,(1)T(M)0,,M12,:49(2)C,f(I-M)=(n-1)I-M2F-maxkkn-1Bk(n-k)F-C(n-k)kF2I-M2F=1n[|tr(I-M)|2-|trM|2]+M2F,f(I-M)=n-1n(|tr(I-M)|2-|trM|2)+f(M)(3)f(M)-n-1n|trM|2=(n-1)M2F-max1kn-1Bk(n-k)F-C(n-k)kF2-n-1n|trM|2,f(M)-n-1n|trM|2=(n-1)M2F-max1kn-1Bk(n-k)F-C(n-k)kF2-|trM|2n,(1)f(M)n-1n|trM|2f(M)f11MCnn:M=AkkBk(n-k)C(n-k)kD(n-k)(n-k)Cnn,(1kn-1)f(M)=(n-1)M2F-max1kn-1Bk(n-k)F-C(n-k)kF2,M:zC:z-trMnnf(M)-(n-1)|trM|2n11111212MCnn:M=AkkBk(n-k)C(n-k)kD(n-k)(n-k)Cnn,(1kn-1),f(M)=(n-1)M2F-max1kn-1Bk(n-k)F-C(n-k)F2,=a+b-1M,RetrMn-nf(M)-(n-1)|trM|2naRetrMn+nf(M)-(n-1)|trM|2n13dxdt=Ax,A,RetrAn-nf(A)-(n-1)|trA|2n,x=0注11F,f(A)=(n-1)A2F,12,f(A)f,[5]11F50()442数值算例21A=5-13-12-23-23A,11,:zC:z-103143,-1333380000,Gerschgorin,A:{zC:|z-5|4}{zC:|z-2|3}{zC:|z-3|5},-2000090000A1=00610,2=20881,3=78509,11A22x=0[6]dxdt=dx1dtdx2dtdx3dtdx4dt=-531-2-1-401-12-411-20-3xRetrA4=trA4=-40000,f(A)=252,-4f(A)-3|trA|24=-38730,RetrA4-4f(A)-3|trA|24,13,x=0:[1]HORNRA,JOHNSONCR.Matrixanalysis[M].Cambridge:CambridgeUniversityPress,1985.[2].[M].:,2008.[3].()[J].:,1982(04):416-422.[4],.[M].:,2007.[5].[J].,1994(04):501-511.[6].[J].,1988(01):67-74.(编辑:陈丽萍)12,:51