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x=x1e1+···+xqeq. e
1= [1,0,...,0]T, e2= [0,1,...,0]T, ...,ep= [0,0,...,1]T
e=Iq.???? ?? [a(x)]f= [a]fe[x]e.????? [b◦a]ge= [b]g f[a]fe.????? f ????x?E?? ? [x]e?= [id]e? e[idE]e=P-1[x]e.????? [a]f? e ?= [idF]f? f[a]fe[idE]ee?=Q-1[a]feP.????? [a]e? e ?= [idE]e? e[a]ee[idE]ee?=P-1[a]eeP.????? q
D= diag(λ1Im1,...,λpImp).
A= diag(A11,...,App).
????? ??? ??????? ??????? ?? ???????q(q+ 1)/2? ?? ??? ???? ?????? ??? ??? ?????? ??? -1P(x)dx?? ?? ??? ???Rek??fk??? e k(x) =e(2k-n)x, fk(x) = (sinhx)k(coshx)n-k. k=0ckek ?1 0 ca 1?? a0
0d-bca
1ba 0 1? =?a b c d? ?Ip0 C 1Iq?? A0 0D1?? IpB1 0Iq? =?A B C D? ?Ip0 C 1Iq? C D? ??[b]f= [b1,...,bp]T.??e?= (e?1,...,e?q)??f?= (f?1,...,f?p)???? ??? ????? ?????? ??? e [a]ee=?A B 0C? m ik=p? j=1a ijbjk ?? ???? ?????(AB) =?q i=1? p
A=?1 0
1 1? , B=?1 0 0 2? , C=?1 1 0 1? ????? ??E? ????a?L(E).???????det[a]ee= det[a]e? e ?,???????[a]ee=?????[a]e? e ???? ?????? ????? ? ???? ???Aij= 0??i > j.????? detA= detA11detA22...detAnn. 0A22?
σ?Sp?(σ)f(σ)????
f(σ) =p 1? i=1a iσ(i)p i=p1+1a iσ(i). ??????(σ1σ2) =?(σ1)?(σ2)?? ?????? ? detA= (?
1?(σ1)p
1? i=1a iσ1(i))(?
2?(σ2)p?
i=p1+1a iσ2(i)) = detA11detA22. A=? ??A
11A12... A1n
0A22... A2n
0 0... Ann?
0B22? B 11=? ??A
11A12... A1,n-1
0A22... A2,n-1
0 0... An-1,n-1?
????? ??? ????? ??2q?????? ????? ? det(A+B) =?
T?{1,...,q}detATdetBT?.
det(DA+B) =?
σ?Sq?(σ)q?
i=1c iσ(i)=?
σ?Sq?(σ)q?
i=1(λiaiσ(i)+biσ(i)).????? det(DA+B) =?
T?{1,...,q}c
T? j?Tλ j ?? ???cT???? ????K.???? ?????? ??????? ???cT= detATdetBT?.?? ??????? ????? ...=λq= 0.?? ??????? ????? ??????? det(DA+B) =?
σ?Sq?(σ)k?
i=1(λiaiσ(i)+biσ(i))q? i=k+1b iσ(i).
1...λk?????k
i=1(λiaiσ(i)+biσ(i))?q S k i=1a iσ1(i)k i=k+1b iσ2(i).
1?Sk?(σ1)k?
i=1a iσ1(i)?
2?Sq-k?(σ2)q?
i=k+1b iσ2(i)) = detA{1,...,k}detB{k+1,...,q} det ?A B C D? = det(A)det(D-CA-1B). det?A B C D? = det(AD-CB). p??q.??????? ???XTAY=?????(AY XT). ?????? ???b?→?????b). det ?A-B B A? =|det(A+iB)|2. ?IqiIq iI qIq?? A-B B A?? IqiIq iI qIq? -1 P kXPk=xkkPk??f(XPk) =f(PkXPk) =xkkf(Pk). U kP1U-1 k.???? ????c=f(P1).??????? ? ?????? ?? ?? ???c=f(Pk)???? ????k= 1,...,n. f(X) =c?????X. ???|T|= 0,1,≥2. ???? ????? ? ?? ?? ???????detD?= 0.??????? ? ?????? ?? ?? ??????? ?? ??????? ??? det(D-Jq) =μ1···μq?
1-1μ
1- ··· -1μ
q? s?Sas.???? C= [c1,...,cq]??? ??????? ?????? ???????q????? ???cj? {a,b}???? ????j.???? ????
S={j;cj=a}??S?={j;cj=b}.??????? ??? ??λ?K?????
det(λIq+C) =λq+ (aS+bS?)λq-1+ (aSbS?-aS?bS)λq-2.
T;|T|=2detCT.
??????? ???????q A det(AB) =?
T?TdetATdetBT.
T?TcT?
j?Tλj.??????? ???cT= det(AB)S×R=?
T?TdetAS×TdetBT×R.
???ker(a-λidE)?={0} ? ?x?E; (a-λidE)(x) = 0? ?x?E;a(x) =λx.?? ????? x
1=...=xp= 0.???? ?? ??? ?? ???? ??????F??E???? ?F=F1+···+Fp??? ??????
L j(X) =? i?=j(X-λi)? i?=j(λj-λi). ???P(λj) =aj.??????? ? ?????P=a1L1+···+apLp:?? ?????? ?? ??????? ??????? ???P
A= [a]ee?????[ak]ee=Ak.
P(a) =c0idE+c1a+c2a2+···+cnan.
?? ???? ????? ??A= [a]ee????? [P(a)]ee=P(A) =c0Iq+c1A+c2A2+···+cnAn. R=? ????0 1 0...0
0 0 1...0
0 0 0...1
1 0 0...0?
????R2=? ????0 0 1...0
0 0 0...0
1 0 0...1
0 1 0...0?
[P(r)]ee=P(R) =a0Iq+a1R+···+aq-1Rq-1=? ????a
0a1a2... aq-1
a q-1a0a1... aq-2 a
2a3a4... a1
a
1a2a3... a0?
x a
2(xj) =a(a(xj)) =a(λjxj) =λja(xj) =λ2jxj.
0 =P(a)(0) =P(a)(x1+···+xp) =P(a)(x1)+···+P(a)(xp) =P(λ1)(x1)+···+P(λp)(xp).
x
F=Eλ1? ···Eλp.
dimE≥dimF= dimEλ1+···+ dimEλp≥p. xy ???????1,j,j2???? ??? ??????? ??????? ??a. ??????0n0 0...0 0 0
1 0n-1 0...0 0 0
0 2 0n-2...0 0 0
0 0 0 0... n-1 0 1
0 0 0 0...0n0?
A=12 (A+AT) +12 (A-AT). ????a?L(E,F)??b?L(F,E).??? ?? ??????? ???(idF-ab)-1??????? ??????? ??? (id P a(X) = det? ??a
11-X a12... a1q
a
21a22-X ... a2q
a q1aq2... aqq-X? a??? ?? ?????q??Pa(X) = (-1)qXq+ (-1)q-1?????(a)Xq-1+···+ deta.?? P a(X) = detB=?
σ?Sq?(σ)q?
i=1b iσ(i). deg?q ???q i=1bii=?q ??? ?? ????? ??? ????e??E????? ???(e1,...,ek)???? ??? ???? ??Eλ.????? ?? ??????? [a]ee??????? ??? ????? [a]ee=?λIkA 0B? ??e??? ??? ???? ??E? ??A= [a]ee?? ??Pa(X) =c0+c1X+···+ (-1)qXq,????? ???? ??? ?? ??????? ?????? ???????q?????? ???c0Iq+c1A+···+(-1)qAq??? ?? ??????? ?????? ????? ??A=?5 2 -2-1? ??? ????Pa(X) =X2-4X-1.?????A2=?21 8 -8-3? A
2-4A-I2=?21 8
-8-3? -4?5 2 -2-1? +?1 0 0 1? =?0 0 0 0? A
2=tA-dI2, A3=tA2-dA= (t2-d)A-tdI2, A4= (t2-2dt)A-(t2d-d2)I4.
??a. B k,?? ???????B???????
B=B0+XB1+···+Bq-1Xq-1?????
??Pa(X) =c0+c1X+···+cqXq?? ???????B(A-XIq) =Pa(X)Iq?? ??????? B
0A+X(B1A-B0) +X2(B2A-B1) +···+Xq-1(Bq-1A-Bq-2)-XqBq-1
=c0Iq+c1XIq+···+cqXqIq. B
0A=c0IqB0A=c0Iq
B
1A-B0=c1IqB1A2-B0A=c1A
B
2A-B1=c2IqB0A3-B1A2=c0A2
-Bq-1=cqIq-Bq-1Aq=cqAq (aij)? ?A XIq I q0?? B-XIq -IqA? ,?B-XIq -IqA?? A XIq I q0? A=? ?3-2-1 3-1 1
6 3-2?
T?{1,...,q}(-X)|T?|detCT,??|T?|
M=? ??a
1b1a2b1... aqb1
a
1b2a2b2... aqb2
a
1bqa2bq... aqbq?
????? ???M??? ?? ???? ?? det(C-XIq) = (-X)q-kdet(M-XIk). ??????a??E??? ???A= [a]ee???? ???? ? A=? ?5-6-6 -1 4 2
3-6-4?
5x-6y-6z=x
-x+4y+2z=y
3x-6y-4z=z
4x-6y-6z= 0
-x+3y+2z= 0
3x-6y-5z= 0
-1 3? ???= 18.?? ???? ???
4x-6y= 6z
-x+3y=-2z
3e1-e2+ 3e3.
3x-6y-6z= 0
-x+2y+2z= 0
3x-6y-6z= 0
3x= 6y+ 6z.
f
P= [idE]ef=?
?3 2 2 -1 1 0
3 0 1?
[a]f f=D=? ?1 0 0 0 2 0
0 0 2?
P -1= [idE]fe=? ?-1 2 2 -1 3 2
3-6-5?
f[idE]fe=P-1DP?? ?????? A=? ?3 2 2 -1 1 0
3 0 1?
?1 0 0 0 2 0
0 0 2?
?-1 2 2 -1 3 2
3-6-5?
??????a??E??? ???A= [a]ee???? ???? ?
A=?0 1
0 0? [a]ee???? ???? ?
R(θ) =?cosθ-sinθ
sinθcosθ??????? i1? ?? ????P-1= [idE]fe=12 ?1-i -i1? ??[a]f f=D=?eiθ0
0e-iθ?
R(θ) =12
1i i1?? eiθ0
0e-iθ??1-i
-i1? ??????? ??? ??? ?????K=R? ?? ?? ??? ?? ?????? ????θ?≡0 modπ.?? ????? ?? ?????? ??? ????f??E????? ???[a]f q= dimE≥dim(Eλ1+···+ dimEλq)≥q. dimEλj= 1???? ????j. f??????? ??? ????? ? D=?
1Im1...0
0...0
0... λpImp.?
f-XIq). c
0idE+c1a+···+cnan= 0,
????? ? ???? ?????? ???P(X) =c0+c1X+···+cnXn?????? ?? ??????? ??? ??? ??I(a). f-I3)(([a]f f-2I3) = 0??? ([a]ee-I3)(([a]ee-2I3) = 0. ??? ??0????? ??? ?????? ?????? ???a= idE?? ????? ??1????? ??? ?????? ?????? ???a= 0. P r(X) =mr(X) =Xq-1. q??????? ?? ???????e2ikπq ????k= 0,...,q-1. )????k= 0,...,q-1. m a(X) =b0+b1(X-λ) +b2(X-λ)2+...+bp(X-λ)p
0 =b0idE+b1(a-λidE) +b2(a-λidE)2+...+bp(a-λidE)p,
(a-λidE)[-b1b 0id E-b2b
0(a-λidE) +...-bpb
0(a-λidE)p-1] = idE.
m
1+···+mp? ?????D= [a]f
f??????? ??? ????? ????? ?? ??????? ????? ?? ?? ??????
D-λ1Iq=?
??(λ1-λ1)Im1...0 0...0
0...(λp-λ1)Imp?
?? ?? ???? ???(D-λ1Iq)...(D-λ1Iq) = 0.????? ???? ???(X-λ1)...(X-λp)??? ?? P=Q1...Qp??a?L(E).????? ???kerQj(a)???? ?? ????? ??????? ?? kerP(a) = kerQ1(a)? ··· ?kerQp(a). ?????kerma(a) =E?? ???kerQj(a) =Eλj?? ?E=Eλ1? ··· ?Eλp??E??? ?????
1 =P1(X)Q1(X) +P2(X)Q2(X),
id
E=P1(a)Q1(a) +P2(a)Q2(a),
x=P1(a)Q1(a)(x) +P2(a)Q2(a)(x).?????? ker(Q1Q2)(a)? ??????xj=Pj(a)Qj(a)(x).?????x1?kerQ2(a),???????Q2(a)(x1) = Q kerQ1(a).????x=x1+x2??? ??????? ???? ? ker(Q1Q2)(a)?ker(Q1)(a)?ker(Q2)(a). Q
1(a)Q2(a)(x1+x2) =Q2(a)Q1(a)(x1) +Q1(a)Q2(a)(x2) = 0 + 0.
ker(Q1Q2)(a)?ker(Q1)(a)?ker(Q2)(a), (Q1...Qk-1,Qk), R??????Q1...Qk-1.?? ????R??????Q1...Qk-2? ??? ???????Q1, ker(Q1...Qp)(a) = ker(Q1...Qp-1)(a)?kerQp(a). a(b)?= 0. (i,j)?= (k,l)??xkl= 1.????e= (E11,E12,E13,...,Enn)????? ??? ???? ??M.??A??? A
2+A-(d-1)In=J.(?)
????? ??(?)???n=d2+1.??????? ??? ??1= (1,...,1)T?????A1=d1,???(A-dIn)J= 0? ????(nd,nα,nβ)???? ??? ??
D= diag(dInd,αInα,βInβ)
n d+nα+nβ=d2+ 1, dnd+αnα+βnβ= 0, d2nd+α2nα+β2nβ=d(d2+ 1) (2nα-d2)⎷4d-3 =d(2-d) ?? ???s=⎷4d-3???? ???? ?????? ??d?= 2? ?? ?? ??????? ??? ???? ?? ???s?????? ??? A t=?1t 0 1? ??A={a1,...,aN}????Λj?? ??????? ??aj?? ???? (E(j) j)λj?Λj ?? ??????? ??? ??????? ??????? ??aj.????k??? ????{1,...,N}?? ????Λk= Λ1×...×Λk, ???? ?? ??????? ??????? ??? ????λk= (λ1,...,λk).???? ?? ???? F
λk=k?
j=1E(j) j? a j◦ak+1(x) =ak+1◦aj(x) =λjak+1(x), ?? ??? ??? ???? ???ak+1(x)??? ?? ??????? ?????? ??aj???? ?? ?????? ??????λj. ????ak+1(x)?E(j)
0 =m(ak+1),0 =m(ak+1)(x) =mb(x)
F k+1?Λk+1=F∩Bk+1. E=? k?ΛkF
λk.
B k+1. B ??A??? ?????? ????A?={a1,...,aN} ? A??? ???A????? ??? ???? ?? ???? ?????? ?? P a(X) = det(A-XIq) = (a11-X)...(aqq-X). ?????[a]f f??????? ??? ????? [a]f f=?a11B 0A? f A? ??[p◦a1]f? f P a(X) = (aqq-X)det(A-XIq-1).????quotesdbs_dbs48.pdfusesText_48
[PDF] ALGÈBRE 1