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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
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