matrix-analysis-w7

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1、實變函數論─應用數學系吳培元老師MatrixAnalysis第七週課程講義Nonderogatorymatrix:Thm.Tnn,Thenthefollowingareequiv.:(1)()exactlyoneJordanblockinJordanformof;TT(2)TMp()forsomepoly.;p(3)minimalpoly.of=chTarac.poly.of;T(4)dimAlgTn;(5)dimTn;(6)dimker(-TI)1;nn2-n1(7)xxV,Tx,T

2、x,...,Tx(iscycliC);T(8)TT=;(9)Tisabelian;(10)EverymatrixcommutingwithisapolynomialinT.Ta11*a(11)Ta21,with0j.jj,-10aann-1nnDef:isnonderogatoryifitsatisfiesoneoftheabove.TPf:(3)(1):Jordanform:()ii()charac.poly=(-x)kk1...niii()imin.poly=(-

3、xk)k1(1)()ii...k()0i(3)in2ii(3)(2):Rationalform:charac.poly=pp(3)...p1(2)jn2jmin.poly=p1(3)(4):(4)deg.ofmin.poly.=n(3)(minpoly.

4、charac.poly).()ii()()iii()()()i(5)(1):(5)kkn12+3+...kkk122+3+...=ki3=..=0(1).iiii(1)(6):no.ofJordanblocksat=

5、dimker(-TI)=geom.multiofatT()T(1)(),dimker(-TTI)=1n(),dimker(-TTI)=0n1實變函數論─應用數學系吳培元老師(2)(7):0-a01-aTMpMp()()Tn1if()pxxaxn-1...axan-11001-an-110Letx.0000012n-1ThenMpx()TT,Mp()x1,...,Mp()Tx

6、000010TMp()iscyclic.But"cyclic"preservedundersimilaritycyclic.Tnn-1(7)(2):xTx,,...,Txbasisof.00*10*represented,,thisbasisasTwit0100001*0*011*TM01()forsomepoly..pp*011*2實變函數論─應

7、用數學系吳培元老師(11)(7):10Letx.0***a21*Tx0,Tx22...,Txaa*32210aann132a210nn1VxTx,,,Tx.(1)(8):()ii()()ii()()i(8)kk123...k12ik=k3=...=0i(1)iiin-1(7)(11):xTx,,...,Txindep.nGram-Schmidtxx12,,...,x

8、no.n.basisof,wherexbxb,011111xbxbTxb,022122222xbxbTxbTxb,0331323333n-1xbxbTx...bTxb,0nn12nnnnn11b21TxbTxbx-x1111121bbb2222112TxbTxbTx+221221bb3132=...+bxxT223--xbbb333333,xxx,xx21121TTxT12,xT,...,xn**b11*b22=b022b33

9、0003實變函數論─應用數學系吳培元老師(8)(9):Let,ABTThenATABBA.(9)(8):TT.LetAT.ThenBT