[PDF] Endomorphismes des espaces euclidiens - Xiffr



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[PDF] abc est un triangle les points mn et p sont tels q

[PDF] solution vectorielle sans repere

[PDF] decomposer mn sur les vecteurs ab et ac

[PDF] solution analytique dans le repere (a b c)

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[PDF] toupty maths cm1

[PDF] toupty maths cm2

[PDF] toupty maths 4ème

[PDF] exercices corrigés de microéconomie s2 pdf

[PDF] exercices corrigés sur le comportement du producte

????A2On(R)? ?? ??????? ??? ? ????? ??? ?????? ?????? ??A?

1p+ 1(In+A++Ap)

???????p!+1? ':A2 An(R)7!(InA)(In+A)1?

2GOn(R)1=2Sp(

U=0 B BB@ 2n+1 M??? n+2

1nn+11

C

CCA2On+1(R)???????

M=0 B 1(0) (0)n1 C A ?? M2GLn(R)? ??????? ????? ??????U2On(R)??S2 Sn(R)?????? ??? M=US? tMM? ????A2 Mn(R)? N

1(A) =X

1i;jnjai;jj?

??????? ???N1????? ?? ???????mn?? ?? ???????Mn???On(R)? ??? ????? ??Mnnpn?? ???M3<3p3? f(V?)(f(V))??

8x;y2E;(xjy) = 0 =)(f(x)jf(y)) = 0?

8x2E; f(x) x ????f2=Id? ': (A;B)2 M27!trtAB?

2 M???? ???M7!

M ????'2O(E)? ?? ????M(') = Im('IdE)??F(') = Ker('IdE)? ???? '2O(E)? ??????? ???M(')?F(') =E? ?? (u1;:::;uk)??? ?????? ??????? ?

M(su1 suk) = Vect(u1;:::;uk)?

?? ???? ???(u1;:::;uk)?????? ??????v1;:::;vk2En f0g???? ??? s u1 suk=sv1 svk? ??????? ???(v1;:::;vk)??? ?????? u= (u1;:::;up)2(Rn)p?? ???? M u= ((uijuj))1i;jp? M u=Mv? ??????? ????? ??????f2O(Rn)????? ???f(ui) =vi???? ????i? ??? ??????? Kerv= (Imv)?? u n=1n n1X k=0u k? un(x) ?? ? ???v=uId? ???????

Kerv= (Imv)??

x=x1+v(y)? 1N N1X k=0u k(x) =x1+1N (uN(y)y)?

8x2E;limN!+1

p(x)1N N1X k=0u k(x) = 0? tAA=tBB?

8x2Rq;hf(x);f(y)i=hg(x);g(y)i?

8(i;j)2 f1;:::;rg2;h"i;"ji=h"0i;"0ji?

n u2 L(E)8x2E; u(x) kxko f6=g??u=12 (f+g)? ??????? ???u =2O(E)? ??????? ????? ??????(f;g)22???? ??? f6=g??u=12 (f+g)? ???????(i;j;k)??? A=13 0 @2 2 1 12 2 2121
A

B= (i;j;k)?

????f2 L(E)???? ?? ??????? ???? ?? ????B??? 12 0 @1p2 1 p2 0p2 1p2 1 1 A v;w2P:x+z= 0? ???A=14 0 @3 1p6 1 3p6 p6 p6 2 1 A ???A=19 0 @7 4 4 4 81 4 181

A???A=19

0 @8 4 1 4 7 4 1 481 A ??????(a;b)2R2?? A=0 @a b b b a b b b a1 A

6= 0 [(2)]?

g(u) =u??g(u) =u?

R(i) =j??R(ij+k) =ij+k?

Rot k;=2Rotcosi+sinj;? X=X? t X X ????A2On(R)? ?? ????X= Re(Z)??Y= Im(Z)? ??????? ???Vect(X;Y)??? ?????? ???A?

Vect(X;Y)?

(uId)2=~0? ???? ????k6=1? f(x) =x+k(xja)a (PjQ) =Z 1 0

P(t)Q(t)dt?

u(P)(x) =Z 1 0 (x+t)nP(t)dt (Imp+ Kerq)?= Imq\Kerp? ????8x2E; p(x) kxk? (Imp+ Kerq)?= Imq\Kerp? hx;yi=x1y1+ 5x2y22(x1y2+x2y1)? M=2a 0 0 f a(x) =x+ahx;uiu

8x2R3;

fa0(x) =kxk?

8x2E;(f(x)jx) = 0?

k= sup

2Sp(u)jj?

8x2E; u(x) k x

S=x2Ekxk= 1?

p= minV2Vpmaxx2S\V(f(x)jx)? H u=x2E(u(x)jx) = 1? ????A2GLn(R)????? ???tA=A2? A

3=AtA?

M2+M+In= 0

tMM=MtM? ??????? ???M??Mn(R)?????? ???tM=M2?? ???M????? ?????? ?????? ????A2 Mn(R)????? ???tAA=AtA? ?? ??????? ????? ??????p2N??? ??? A p= 0? tAA= 0? ????A??? ??????? ?????? ?????? ???????n? tAA=AtA? min= minSpA??max= maxSpA?

81in;minai;imax?

(detA)1=n1n trA? ???B=AQ? ??M2 Sn(R)??????Mp=In????p2N? ??? ????M2? ??????? ??? ?? ???? ??A2 Mn(R)??? ???? ?? ?????? ?? ??????? ??????? ??? tAA? ??????m;n2N??M2 Mm;n(R)?

8(i;j)2 f1;:::;mg f1;:::;ng;i6=j=)Ni;j= 0?

A=0 B BBB@a

1b1(0)

c

1a2???

??????bn1 (0)cn1an1 C

CCCA2 M

n(R) D

1AD2 Sn(R)?

??????A;B2 Sn(R)??p2N? ?? ??????? ???A2p+1=B2p+1? ??????? ??? A=B?

9P2On(R);t

(tAA) =AtA? Sp tAAAtAR+? kAk1nptrA?

B=A3+A+In?

B=A2+A+In?

???A=a b b c 2 M

2(R)??B=2 1

1 2 ????? ???A=PBtP?

P2O2(R)???? ???A=PBtP?

P2O2(R)???? ???A=PBtP?

???A=a b c d 2 M

2(R)??B=2 1

1 2

P2GL2(R)????? ???A=PBP1?

P2GL2(R)???? ???A=PBP1?

P2GL2(R)???? ???A=PBP1?

max

P;Q2On(R)detPAtP+QBtQ?

1 2 ??A=12 21
?? A;B2 Mn(R)? sup

P;Q2GLn(R)detPAP1+QBQ1

max P

1;:::;Pk2O2(R)detP1A1tP1++PkAktPk

A= Mate(x1;:::;xn)??B= Mate(y1;:::;yn)?

tAB? ???? (x1;:::;xn)??? ???? ??E? ??????? ????? ?????? ??? ?????? ??????? (y1;:::;yn)??E????? ???

8(i;j)2 f1;:::;ng2;hyi;xji=i;j?

?? ?? ????(x1;:::;xn)? ?? ????(y1;:::;yn)? ?????? ?? ?? ???????

M= (hxi;xji)1i;jn?

8i2 f1;:::;ng;kxik= 1;8(i;j)2 f1;:::;ng2;i6=j=) hxi;xji<0

9v2E;8i2 f1;:::;ng;hxi;vi>0?

??? ????? ???? ??????? (x1;:::;xn)??? ??? ???? ??E? ?? ? ???M= (hxi;xji)1i;jn2 Mn(R)??S=InM=n? ???? (y1;:::;yn)?????? ??(x1;:::;xn)????? ???? ??? ???????

8(i;j)2 f1;:::;ng2;hyi;yji 0?

??????A??B????S2(R)?????? ???A2=A??B2=B?

M=ABT+BAT?

tXAX=tXBX???? ????? ???????X??Mn;1(R)? ??? A=0 @122 2 12 22 11
A A=0 @22 1 2 12 1221
A tPAP=D? 0 B

BBB@a bb

b a bb a1 C CCCA? = suphX;MXiX2 Mn;1(R);kXk= 1 ?????? ?? ??? ?????? ?????? ??M?

8x;y2E;(f(x)jy) =(xjf(y))?

AX=X? tXAX? ??????? 2iR?

8x2E;(u(x)jx) = 0?

(u(x)jy) =(xju(y))? tXAX? ??????? ???2iR? ????A2 Mn(R)????? ???tA=A? B=C0 0 0

8(x;y)2E2;(f(x)jy) =(xjf(y))?

???? x2Van f0Eg? ??????? ??? (s(x)jx) =akxk2= f(x) 2 ??????? ???F= Vect(x;f(x))??F????? ??????? ???f? 0b b0

8x;y2E;(xjf(y)) =(f(x)jy)?

A=0 B

B@0 0 1 i

0 0i 1

1 i 0 0

i1 0 01 C CA? ????n2N??M2 Mn(R)?

M=A+S;tA=A;tS=S?

??????? ???n??? ???? ?? ????? ??????P2On(R)??(a1;:::;ap)2(R+)p???? D

1;:::;Dp??

D i=0ai a i0 M ??????? ???? ???? ????x2E? kxk4 hu(x);xihu1(x);xi? M=0 @a:a a:b a:c b:a b:b b:c c:a c:b c:c1 A ??????? ??? ???? ????X2 Mn;1(R)? t

XAX2R+?

????A2 Sn(R)? ???????

8X2 Mn;1(R);tXAX0()SpAR+?

????A2 S+n(R)? ?? ???? ??????? ????? ?????? ??? ?????? ???????B2 S+n(R)????? B 2=A? Ker

BpIn= KerAIn?

KerB= KerA?

????A2 S+n(R)? ?? ???? ??????? ????? ?????? ??? ?????? ???????B2 S+n(R)????? B 2=A? ??????? ??? ?? B2 S+n(R)??????B2=A????? ???? ????2SpA?

Ker(BpI

n)Ker(AIn)

Ker(BpI

n) = Ker(AIn)? A 2=S? B=A? ????A2 S+n(R)? ??????? ????? ?????? ??? ?????? ???????B2 S+n(R)????? ??? B ????M2 Mn(R)? ??????? ???A=tMM2 Sn(R)??SpAR+?

M2 Mn(R)????? ???A=tMM?

????A2 Mn(R)????? ??? ???? ????U2On(R)?tr(AU)trA? ???? B2 An(R)? ??????? ??? ???? ????x2R?exp(xB)2On(R)? ??? ????? ??A2 S+n(R)?

U2On(R)?????? ???M=SU?

???? >0? ?? ???? S =M2 Sn(R)SpMR+??det(M)? inf

M2Str(AM) =n(det(A))1=n?

????? ?????? ???? ??????? ?? ??? A=In?

A=tPP????P??????? ?????? ?? ??????n?

??????A2 Sn(R)????SpAR+??B2 Mn(R)? ?? ???????

AB+BA= 0?

???????AB=BA= 0? ????A2GLn(R)? tAA2 S++n(R)?

8A2GLn(R);9(O;S)2On(R) S++n(R);A=OS?

A=tTT?

U p=1p+ 1(In+A++Ap)? (IA)Up=1p+ 1(InAp+1)!0quotesdbs_dbs42.pdfusesText_42