张跃辉-矩阵理论与应用 前第四章答案

16,22,23501.:(1)(cosxsinx−sinxcosx)n;(2)(11−11)n;(3)a1a1a1a1an.:(1)(cosnxsinnx−sinnxcosnx);(2)(√2)n(cosnπ4sinnπ4−sinnπ4cosnπ4)n;((1).)2.:nλI.:A=(aij)AEij,AEij=EijA.AEijjAi0EijAiAj0A0aii=ajjA.3.A¡1BCA¡1,A=45023127−3,B=45010231−1279−3,C=450231279−237.:A¡1B=−124−24060−1750−24−4892−10−35−4598,CA¡1=−124−24000−24−296−216−18112−192−22.4.A,B∈Mn,:adj(AB)=adj(B)adj(A).(1)A,B(2)x∈F,x,A−xIB−xI,xadj((A−xI)(B−xI))=adj(B−xI)adj(A−xI).ijxxx=0adj(AB)=adj(B)adj(A).15.:A,r(A¤A)=r(AA¤)=r(A)..αyAA¤=0,(αAA¤)α¤=0(αA)(αA)¤=0,αA0αA=0.yAA¤=0yA=0.6.:nA,r(An)=r(An+1).r(Ai+1)≤r(Ai),A,A2,···,An+12AsAt.st.r(As)=r(As+1)=···=r(At).r(At+1)=r(At).At+1x=0α.AαAtx=0r(At)=r(At¡1)At¡1x=0Atx=0AαAt¡1x=0Atα=0.At+1x=0Atx=0r(At+1)=r(At).r(Ai)=r(Aj),∀i,j≥s.7.ωn(ω=e2πi/n=cos2πn+isin2πn),Fourier111···11ωω2···ωn¡11ω2ω4···ω2(n¡1)............1ωn¡1ω2(n¡1)···ω(n¡1)(n¡1).F.0jn,wjwj=1(ωj)0+(ωj)1+···+(ωj)n¡1=0,FF¤=nI.F¡1=1nF¤.8.A=(aij)n£n.:f(x1,x2,···,xn)=¯¯¯¯¯¯¯¯0x1···xn−x1a11···a1n············−xnan1···ann¯¯¯¯¯¯¯¯A.fxi(−1)(1+i+1)¯¯¯¯¯¯¯¯¯¯−x1a11···a1,i¡1a1,i+1···a1n·····················−xjaj1···aj,i¡1aj,i+1···ajn·····················−xnan1···an,i¡1an,i+1···ann¯¯¯¯¯¯¯¯¯¯jxj−(−1)j+1Mij,MijAaij.,fxixj(−1)i+jMij=AijfAA.9.Frobenius:r(AB)+r(BC)≤r(B)+r(ABC).(BABC)→(BABABC)→(B−BCAB0).210..Aα1,···,αnk1α1+···+knαn=0.AB.PB=PA.BPα1,···,Pαn.k1Pα1+···+knPαn=0..11.An,06=x∈FnAx6=x,I−A.Ax6=xA1,I−A.Aρ(A)1,(I−A)¡1=I+A+A2+···+Am+···().12.nA,xyn.(A+xy¤)¡1,Sherman-Morrison:(A+xy¤)¡1=A¡1−A¡1xy¤A¡11+y¤A¡1x.AA(A+xy¤)¡1=I−xy¤A¡11+y¤A¡1x.(A+xy¤)A¡1=(I−xy¤A¡11+y¤A¡1x)¡1.I+xy¤A¡1=(I−xy¤A¡11+y¤A¡1x)¡1.(y¤A¡1x).13.nA,B,C,Dn×m,m×n,m×m.¯¯¯¯ABCD¯¯¯¯=|A||D−CA¡1B|.(I0−CA¡1I)(ABCD)=(AB0D−CA¡1B),.14.(1)A,C,(AB0C).(2)A,D−CA¡1B,(ABCD).(1)(AB0C)¡1=(A¡1−A¡1BC¡10C¡1).3(2)13.13(I0−CA¡1I)(ABCD)=(AB0D−CA¡1B),(1)(G=D−CA¡1B)(ABCD)¡1=((I0−CA¡1I)¡1(AB0G))¡1=(AB0G)¡1(I0−CA¡1I)=(A¡1−A¡1BG¡10G¡1)(I0−CA¡1I)=(A¡1+A¡1BG¡1CA¡1−A¡1BG¡1−G¡1CA¡1G¡1).15.AA−BC,A,A¡1,B,C(A−BC)¡1.(:(ABCI).)X=A−BC.(ABCI)=(A−BCB0I)(I0CI),(ABCI)¡1=(I0−CI)(A−BCB0I)¡1=(I0−CI)(X¡1−X¡1B0I)=(X¡1−X¡1B−CX¡1I+CX¡1B).(2)X¡1=A¡1+A¡1BG¡1CA¡1G=I−CA¡1B(G).16.Ω=(0In−In0)2n.2nMMTΩM=Ω.:(1)2n,,;(2)1.(:.)(1).(2)MTΩM=Ω|M|2=1,|M|=±1.|M|=117.:(1)A=1230021−11021;(2)A=1−111−11−1−1−1−11111−1−1.(1)A=100−111(12300−2−11);(2)A=1−1−11−1−111(100001−1−1);.18.(I+aEij,i6=j,a6=0).,?4(I+aEij,i6=j,a6=0)Eij(i6=j)E12.i=1Eij2j(E12)2j0E12.Eij(i6=j)E12.aE12(a6=0E12.P=I+(a−1)E22..19.AA2−A+2I=0,A??.(A−I/2)2+7I/4=0,(A−I/2+bI)(A−I/2−bI)=0(b2=−7/4)A.(A−xI)(A−yI)=0,A.20.:(1)Hermite,.(2)HermiteA⇐⇒LA=LL¤.(1)A¤=Aa∈C,α.Aα=aα,αTAT=aαT,α¤A¤=¯aα¤.α¤A¤α=¯aα¤α.=α¤Aα=a(α¤α),α¤α6=0,a=¯a.a.A¤=Ab6=a∈C,bβ,Aβ=bβ.aα¤β=(Aα)¤β=α¤A¤β=α¤(A¤β)=α¤(Aβ)=bα¤β,(a−b)α¤β=0,a−b6=0,α¤β=0αβ.21.V={},F=R.Vx⊕y=xy();VFk•x=xk().(V,⊕,•)..22.V=C\{−1}.V¦:a¦b=a+b+ab.¦.,V♥,(V,¦,♥)R..(a¦b)¦c=(a+b+ab)¦c=a+b+ab+c+(a+b+ab)c=a+b+c+ab+ac+bc+abc.00¦a=0+a+0a=a.a6=−1x,0=a¦x=a+x+ax,x=−a/(1+a)(−1).¦..523.V=C\{−1},♣,a♣b=a+b+xab,x.♣♠,(V,♣,♠)..,CR?.24.A={a1,a2,···,an}.(1):dimFFA=n;(2)FA;(3)FAA.(1)(2)fi∈FAfi(aj)=δij,f1,···,fnFAdimFFA=n.(3)(2)FA∼=HomF(Fn,F).AFA∼=HomF(|A|,F),FjAjA.25.:J={α1,α2,···,αs}K={β1,β2,···,βt}nV,J.αj∈JK,s≤t;Ksα1,α2,···,αs,K.αi=a1iβ1+···+atiβt,1≤i≤s,(α1,···,αs)=(β1,···,βt)A(J=KA)A=(Aij)t£s∈Ft£s.JJx=00KAx=0Ax=00s≤t.L={α1,α2,···,αs,β1,β2,···,βt}.Lsα1,α2,···,αsβjK.26...27..J,KA,BK=JA,J=KB.K=K(BA).BA=IAB.28.1,x−1,(x−1)2,···,(x−1)nR[x]n+1,f(x)=a0+a1x+a2x2+···+anxn.Taylorf(x)=a0+a1x+a2x2+···+anxn=n∑i=0f(i)i!(x−1)i.R[x]n+11,x−1,(x−1)2,···,(x−1)nn+1=.29.:(α,β)1(α,β)2V,(α,β)=(α,β)1+(α,β)2V.V.6..30.x=(x1,x2)T,y=(y1,y2)T,(x,y)=ax1y1+bx1y2+bx2y1+cx2y2.(x,y)R2⇐⇒a0,acb2.(abbc).⇐⇒(abbc)⇐⇒a0,acb2.31.V={acost+bsint,a,b}.f,g∈V,(f,g)=f(0)g(0)+f(π2)g(π2).(f,g)V,h(t)=3cos(t+7)+4sin(t+9).(f,g)V.h(t)=3cos(t+7)+4sin(t+9)5.acost+bsint√a2+b2.32.R[x]2(f,g)=1∫¡1f(x)g(x)dx.(1)1,t,t2;(2)f(x)=1−x+x2g(x)=1−4x−5x2.1G=202/302/302/302/5;(2)0.33.ai,1≤i≤n,xi,yi,(n∑i=1aixiyi)2≤(n∑i=1aix2i)(n∑i=1aiy2i).x=(x1,···,xn)T,y=(y1,···,yn)T∈Rn,(x,y)=n∑i=1aixiyi(x,y)Rn..34.(1)CR2.C,i1+iC,?(2)R3,e1,e1+e2,e1+e2+e3.e2e3??71.2e1,e1+e2,e1+e2+e3(e1,e1)=1,(e1,e2)=−1,(e1,e3)=0,(e2,e2)=2,(e2,e3)=−1,(e3,e3)=2.((x1,x2,x3),(y1,y2,y3))=x1y1+2x2y2+2x3y3−(x1y2+x2y1+x2y3+x3y2).35.34..36.Cauchy-Schwarz.Cauchy-Schwarzα=0α6=0.t(β−tα,β−tα)≥0,(β,β)−2(β,tα)+t2(α,α)≥0.t=(β,α)/(α,α).(α+β)2=α2+β2+2(α,β)≤α2+β2+2||α||||β||(Cauchy-Schwarz).37.R4,α1=(1,0,1,1)T,α2=(2,1,0,−3)Tα3=(1,−1,1,−1)T.()β1=(1,0,1,1)T/√3,β2=(7,3,1,−8)T/√123,β3=(3,54,−23,20)T/√3854.38.Vαβd(α,β)=||α−β||.d(α,β)V,:(d1):d(α,β)=d(β,α);(d2):d(α,β)≥0,⇐⇒α=β;(d3):d(α,β)+d(β,γ)≥d(α,γ)..39.2Vα1,α2,A=(5445).Vα1,α2.α1,α2β1=α1+α2,β2=α1−α2γ1=(α1+α2)/√18,β2=(α1−α2)/√2.α1,α2γ1=α1/√5,β2=(4α1−5α2)/√45.40.nVα1,α2,···,αn,A.α,β∈Vxy.(1)(α,β)=xTA¯y.,V,(α,β)=xTAy.(2)(1).81B=(α1,α2,···,αn),α=Bxβ=By(α,β).(2)β1,β2,···,βnVG=((βi,βj)).α,βuv.α1,α2,···,αnβ1,β2,···,βnP.x=Pu,y=Pv.βi=BPi,1≤i≤n,PiPi.(βi,βj)=(BPi,BPj)=PTiA¯Pj,1≤i,j≤n.G=((βi,βj))=PTA¯P.(α,β)=uTG¯v=uTPTA¯P¯v=(Pu)TA¯Pv=xTA¯y.(1).41.V=Mn(R)Mn(C).A=(aij),B=(bij)∈V.(1)(A,B)=tr(AB¤)V;(2)(1),A??(3):Eij,1≤i,j≤nV;(4)M2(R)