矩阵论讲义(西工大版)

Whythematrixanalysis1.92.93.1.Jordan2.3.ToeplitzHankelHilbert4.™\20048\20049\“MatrixAnalysis”,RogerA.Horn\G.H.™\MatlabC„„„aS∈aS∉::„„„S,aSaS1.{1,3,5,7,9}N=2.MM={a|a}22{(,)|1}Nabab=+=0{|isintegral}Nnn=BA⊇or,aAaBAB∀∈∃∈⇒⊆„AB⊂AB⊃„AB=„{}|,ABxxAxB=∈∈∩„{}|,ABxxAxB=∈∈∪„{}|,ABxyxAyB=∈∈„ZZZ90,aZaZ∀∈∃−∈9901,1/aPaP∀∈∃∈99:C;R;Q9KQ,,,,klK∈∈xyzVVKVV+=+xyyx()()++=++xyzxyz0+=xx()+−=0xxV()kkk+=+xyxy()klkl+=+xxx()()klkl=xx1=xxVKKVKVV8[]1010{|,,,}nnnnPxaxaxaaaa=+++∈R320yyy′′′−+=2{|,}xxYaebeab=+∈Rnnn×Rn10201122=+=00001.2.1xx2x11xx=+012()xxx=++12()xxx=++12()xxx=++2x=+02x=1.2.,0,1kK∈,,V−∈0xx3.k=0x=0x0k=0=0x()1−=−xxk=00K12,,,rkkkx12,,,rxxx1122rrkkk+++=0xxx12,,,(1)rr≥xxxV1122rrkkk=+++xxxx12,,,rxxxx12,,,rkkk12,,,rxxx()()()121,0,,0,0,1,,0,0,0,,1.n===εεε()()()121,1,,1,1,0,1,,1,1,0,0,,0,1.n′=′=′=εεε1122rrkkk+++=0εεε1122rrkkk′′′+++=0εεε10011101001100001001≠≠nR1234111111,,111111aaaa====AA,AA11223344kkkk+++=0AAAA111111111111aaaa12340kkakk+++=()()313aa=−+12340akkkk+++=12340kakkk+++=12340kkkak+++=1,3aa≠≠−22×R1234,,,kkkk2221204ttttttteeeeee212340ttkekek++=()323tet=−21230tttkekek++=212320ttkekek++=0≠2,,ttteeVR,V2,,tttee123,,kkk1.xx=012,,,rxxx2.12,,,syyyrs≤912,,,rxxx3.12,,,,ryxxxy12,,,rxxxVdimn=VnV„nKnnVn→∞„x12,,,nξξξ1,,(1)rr≥xxKVVr12,,,rxxx1.12,,,rxxx2.Vx12,,,rxxxV(1,2,,)iir=x912,,,nxxxVnVnn∈xV1122nnξξξ=+++xxxx()12,,,Tnξξξ„9K19K12,,,nxxxn∈xV:Vn12,,,nxxxx1i.11112121212122221122nnnnnnnnnnccccccccc=+++=+++=+++yxxxyxxxyxxx()()1112121222121212,,,,,,nnnnnnnnccccccccc=yyyxxx()12,,,nxxxC12,,,nxxx12,,,nyyy„VnC1V,1212,,,;,,,nnxxxyyyC1212,,,,,,nnxxxyyy1212(,,,)(,,,)nnyyyxxxC=,1212,,,,,,nnyyyxxxB1212(,,,)(,,,)nnxxxyyyB=1212(,,,)(,,,)nnyyyyyyBC=1212(,,,)(,,,)nnxxxxxxCB=,1212,,,;,,,nnxxxyyy∵.CBBCI∴==C,C1B.K()ijnnCc×=V12,,,,nxxx1212(,,,)(,,,)nnyyyxxxC=1,1,2,,nijiiycxjn===∑j11212(,,,)(,,,)nnxxxyyyC−=C1212,,,,,,nnxxxyyy∴..12,,,nxxx12,,,nyyyV.12,,,nyyyC.1212,,,,,,nnxxxyyy2C,1212,,,,,,nnxxxyyyC-1.1212,,,,,,nnyyyxxx3C,1212,,,,,,nnxxxyyyB1212,,,,,,nnyyyzzzCB.1212,,,,,,nnxxxzzz1212(,,,)(,,,)nnzzzyyyB=1212(,,,)(,,,)nnyyyxxxC=,1212(,,,)((,,,))nnzzzxxxCB=12(,,,)nxxxCB=VPn12,,,;nxxxV12,,,nyyy1112121222121212(,,,)(,,,)nnnnnnnnccccccyyyxxxccc=x12,,,nxxx12,,,nyyyxV∈12(,,,)nξξξ12(,,,)nηηη1212(,,,)nnxxxxξξξ=1212(,,,)nnxyyyηηη=1112111221222212nnnnnnnncccccccccξηξηξη=11112111221222212nnnnnnnncccccccccηξηξηξ−=(**)(*)(**)xC(*):Vn12,,,nxxx12,,,nyyy12(1,0,,0),(0,1,,0),,(0,,0,1)nxxx===12(1,1,,1),(0,1,,1),,(0,,0,1)nyyy===11222nnnnyxxxyxxyx=+++=++=12,,,nxxx12,,,nyyy.12,,,nyyy12(,,,)naaaα=11212100110(,,,)(,,,)111nnxxxyyy−=1210001100(,,,)01100001nyyy−=−1212100110(,,,)(,,,)111nnyyyxxx=12,,,nxxx12,,,nyyy1000110001100001−−12,,,nxxx12,,,nyyy10011011112(,,,)naaaα=12,,,nxxx12(,,,)naaa12,,,nyyyα12(,,,)nbbb111222111000110001100001nnnnbaabaaabaaa−−−==−−12,,,nyyyα1211(,,,)nnaaaaa−−−:V41234,,,yyyy1234,,,xxxx1234(1,2,1,0)(1,1,1,1)(1,2,1,1)(1,1,0,1)yyyy=−=−=−=−−1234(2,1,0,1)(0,1,2,2)(2,1,1,2)(1,3,1,2)xxxx===−=12(1,0,0,0),(0,1,0,0),ee==34(0,0,1,0),(0,0,0,1)ee==1234123411112121(,,,)(,,,)11100111yyyyeeee−−−−=−11234123411112121(,,,)(,,,)11100111eeeeyyyy−−−−−=−1234123420211113(,,,)(,,,)02111222xxxxeeee−=1234(,,,)xxxx112341111202121211113(,,,)1110021101111222yyyy−−−−−−=−112341111202121211113(,,,)1110021101111222yyyy−−−−−−=−123410011101(,,,)01110010yyyy=1234,,,yyyy1234,,,xxxx100111010111001022R×()()()()1112212210010000,,,;00001001EEEE====()()()()1112212210111111,,,00001011FFFF====1112,21221112,2122,,,,EEEEFFFF.11122122,,,FFFF()3542A−=1111121112211112212211122122FEFEEFEEEFEEEE==+=++=+++∵1112,21221112,212211110111(,,)(,,)00110001FFFFEEEE∴=1112,21223542AEEEE=−+++A1112,2122,,FFFF1234(,,,),aaaa1123411113011150011400012aaaa−−=8122−=11003011050011400012−−−=−A1112,2122,,FFFF(8,1,2,2).−™P26:1011122006-12-1„„„„„V1KV1,∈xyV1+∈xyV1.1,k∈∈xVK1k∈xV2.V1V„„„1∃∈0V11,xVxV∀∈∃−∈V1V10=∈0xVi1,1V∀∈−∈xK()11−=−∈xxVV1V1KV1,,,kl∀∈∀∈xyVK1kl∃+∈xyV„:k=l=111,∀∈⇒+∈xyVxyVl=011,kk∀∈∀∈⇒∈xVKxV„:11,kk∀∈∀∈⇒∈xVKxV11,ll∀∈∀∈⇒∈yVKyV1kl+∈xyV111122121122221122000nnnnsssnnaxaxaxaxaxaxaxaxax+++=+++=+++=nVnRnV„Wn(A)„V111122121122221122000nnnnsssnnaxaxaxaxaxaxaxaxax+++=+++=+++=nn-s11112211,11122222,112,11ssssnnssssnnsssssssnncxcxcxcxcxcxcxcxcxcxcxcx+++++++++=++++=++=++1,,snxx+()1,0,,0()0,1,,0()0,0,,1n-s()11121,,,1,0,,0sγγγ()12,,,,0,0,,1ssssγγγRn11212{(,,,)0,}nniVxxxxxxxR=+++=∈V1V3RnV2Rn.21212{(,,,)1,}nniVxxxxxxxR=+++=∈3121{(,,,,0),1,2,,1}niVxxxxRin−=∈=−Rn.V1n.V1n1120nxxx+++=V1.1(1,1,0,,0),η=−1(1,0,,0,1)nη−=−,2(1,0,1,0,,0),η=−V2,xy1212(,,,),(,,,)nnxxxxyyyy==1122()()()nnxyxyxy++++++1212()()112nnxxxyyy=+++++++=+=1122(,,,)nnxyxyxyxy+=+++2,xyV∴+∉V2Rn.V3RnV3n11213(,,,,0)nkxkxkxkxV−=∈1122113(,,,,0)nnxyxyxyxyV−−+=+++∈3,,,