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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 
257 8 "Matrices" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "1. Un changem
ent de base" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(
linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names \+
norm and trace have been redefined and unprotected\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 62 "f1:=vector([1,1,1]):f2:=vector([1,-1,2]):
f3:=vector([-1,1,1]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "A:
=matrix([f1,f2,f3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matri
xG6#7%7%\"\"\"F*F*7%F*!\"\"\"\"#7%F,F*F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "rank(A);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Ainsi, les trois v
ecteurs constituent une famille de rang 3, donc une base de l'espace.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "T:=matrix([[2,1,0],[0,2
,0],[0,0,-1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG-%'matrixG6#7
%7%\"\"#\"\"\"\"\"!7%F,F*F,7%F,F,!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 36 "Ben oui : voir la d\351finition de u..." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 16 "P:=transpose(A);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"PG-%'matrixG6#7%7%\"\"\"F*!\"\"7%F*F+F*7%F*\"\"#F*
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "C'est la matrice de passage d
e la base canonique vers la base des fi" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "A:=evalm(P&*T&*inverse(P));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%#\"\"\"\"\"##F+\"\"'#\"\"%\"\"$7%
#F1F,#\"\"(F.#!\"#F17%F3#!\"&F.F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "I3:=Matrix(3,3,shape=identity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#I3G-%'RTABLEG6$\"*'>$*[8-%'MATRIXG6#7%7%\"\"\"\"\"!F
/7%F/F.F/7%F/F/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "kernel
(A-2*I3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'vectorG6#7%\"\"\"F(F
(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Forc\351ment..." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "kernel(A+I3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<#-%'vectorG6#7%!\"\"\"\"\"F)" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 91 "Pareil... Ces deux derniers sous-espaces ne sont pas su
ppl\351mentaires : voir les dimensions." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "seq(evalm(T^k),k=1..10);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6,-%'matrixG6#7%7%\"\"#\"\"\"\"\"!7%F*F(F*7%F*F*!\"\"-F$6
#7%7%\"\"%F2F*7%F*F2F*7%F*F*F)-F$6#7%7%\"\")\"#7F*7%F*F9F*F,-F$6#7%7%
\"#;\"#KF*7%F*F@F*F4-F$6#7%7%FA\"#!)F*7%F*FAF*F,-F$6#7%7%\"#k\"$#>F*7%
F*FMF*F4-F$6#7%7%\"$G\"\"$[%F*7%F*FTF*F,-F$6#7%7%\"$c#\"%C5F*7%F*FenF*
F4-F$6#7%7%\"$7&\"%/BF*7%F*F\\oF*F,-F$6#7%7%Ffn\"%?^F*7%F*FfnF*F4" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "Ca ne serait pas du genre triangu
laire, avec 2^k et (-1)^k sur la diagonale, et k*2^(k-1) dessus ???" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "evalm(matrix([[2^k,k*2^(k-
1),0],[0,2^k,0],[0,0,(-1)^k]])&*T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'matrixG6#7%7%,$)\"\"#%\"kGF*,&F)\"\"\"*(F*F-F+F-)F*,&F+F-F-!\"\"F-
F-\"\"!7%F2F(F27%F2F2,$)F1F+F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 
"Gagn\351 (pour la preuve par r\351currence, un calcul \340 la main va
 plus vite...)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "An:=evalm
(P&*matrix([[2^n,n*2^(n-1),0],[0,2^n,0],[0,0,(-1)^n]])&*inverse(P));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AnG-%'matrixG6#7%7%,&)\"\"#%\"nG#
\"\"\"F,*&F.F/)!\"\"F-F/F/,(F+#F/\"\"'*&#F/\"\"$F/*&F-F/)F,,&F-F/F/F2F
/F/F2*&#F/F5F/F1F/F2,(F9#F/F8*&F?F/F+F/F/*&#F/F8F/F1F/F27%,&F+F.*&#F/F
,F/F1F/F2,(F+#\"\"&F5*&#F/F8F/F9F/F2*&F4F/F1F/F/,(F9F?*&#F/F8F/F+F/F2*
&F?F/F1F/F/7%FD,(F+#F2F5*&#F/F8F/F9F/F2*&F4F/F1F/F/,(F9F?*&#F,F8F/F+F/
F/*&F?F/F1F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(n=50
,evalm(An));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%#\"1DE
%o!***e7\"\"\"##!2x&)GNa4p^&\"\"'#\"2B#3zd(Rt#H\"\"$7%#\"1BE%o!***e7\"
F*#!2z!=z!e\\l1&F-#\"2xHAkxf@q#F07%F2#!2BQ(*[_*3UdF-#\"2\\3v%[(H*RIF0
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(A^50);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%#\"1DE%o!***e7\"\"\"##!2x&)G
Na4p^&\"\"'#\"2B#3zd(Rt#H\"\"$7%#\"1BE%o!***e7\"F*#!2z!=z!e\\l1&F-#\"2
xHAkxf@q#F07%F2#!2BQ(*[_*3UdF-#\"2\\3v%[(H*RIF0" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 5 "%-%%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "A v\351rifier tout de meme" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalm(A^51-subs(n=51,evalm
(An)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!F(F(F'
F'" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 "2. Une matrice avec un pa
ram\350tre" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(l
inalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names n
orm and trace have been redefined and unprotected\n" }}}{SECT 0 {PARA 
3 "" 0 "" {TEXT 258 30 "Voila comment j'ai construit A" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "A:=matrix([[a-1515,0,0],[0,a-1,0],[
0,0,a-2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%,
&%\"aG\"\"\"\"%::!\"\"\"\"!F/7%F/,&F+F,F,F.F/7%F/F/,&F+F,\"\"#F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A:=addrow(A,2,1,20):A:=addro
w(A,3,1,500);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7
%,&%\"aG\"\"\"\"%::!\"\",&F+\"#?F0F.,&F+\"$+&\"%+5F.7%\"\"!,&F+F,F,F.F
57%F5F5,&F+F,\"\"#F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "A:=
addcol(A,2,1,1789):A:=addcol(A,3,1,-40);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"AG-%'matrixG6#7%7%,&%\"aG\"&\"y:\"%0F\"\"\",&F+\"#?F0!\"\",&
F+\"$+&\"%+5F17%,&F+\"%*y\"F7F1,&F+F.F.F1\"\"!7%,&F+!#S\"#!)F.F9,&F+F.
\"\"#F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A:=addcol(A,3,1,
-31):A:=addcol(A,2,3,-25);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%
'matrixG6#7%7%,&%\"aG\"$\"G\"&0P$\"\"\",&F+\"#?F0!\"\"!$+&7%,&F+\"%*y
\"F5F1,&F+F.F.F1,&F+!#D\"#DF.7%,&F+!#r\"$U\"F.\"\"!,&F+F.\"\"#F1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A:=addrow(A,1,3,-1):A:=addco
l(A,3,2,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%
,&%\"aG\"$\"G\"&0P$\"\"\",&!&?+\"F.*&\"#?F.F+F.F.!$+&7%,&F+\"%*y\"F6!
\"\",&F+!$*\\\"$*\\F.,&F+!#D\"#DF.7%,&F+!$_$\"&jN$F7\"%!)**,&\"$)\\F.F
+F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Et pour la passer \340 mon
 \351diteur :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "latex(%);" 
}}{PARA 6 "" 1 "" {TEXT -1 58 " \\left[ \\begin \{array\}\{ccc\} 281\\
,a+33705&-20020+20\\,a&-500" }}{PARA 6 "" 1 "" {TEXT -1 54 "\\\\\\noal
ign\{\\medskip\}1789\\,a-1789&-999\\,a+999&-25\\,a+25" }}{PARA 6 "" 1 
"" {TEXT -1 66 "\\\\\\noalign\{\\medskip\}-633\\,a-67268&39960&998+a\\
end \{array\} \\right] " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 "V
\351rifications pour A" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ran
k(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 15 "En fait, non..." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "det(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"aG
\"\"#\"\"\"!%=:*$)F&\"\"$F(F(*&\"%ZXF(F&F(F(\"%II!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*(,&%\"aG\"\"\"\"%::!\"\"F&,&F%F&F&F(F&,&F%F&\"\"#F(F&
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Lorsque a ne vaut ni 1, ni 2,
 ni 1515, det(A)<>0, donc A est inversible, donc de rang 3." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(a=1,evalm(A));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"&')R$!&++\"!$+&7%\"
\"!F,F,7%!&:R$\"%!)**\"$*\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "rank(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "subs(a=2,evalm(A));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%\"&nU$!%!)**!$+&7%\"%*y\"!$*\\!#D7%!&nU
$\"%!)**\"$+&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rank(%);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "subs(a=1515,evalm(A));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7%7%\"'?%f%\"&!G?!$+&7%\"(Y&3F!''[b(!&]y$7%!'Voc\"%!
)**\"%8?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rank(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 17 "Construction de B" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "
On part d'une matrice diagonale qui r\351pond aux conditions de rang :
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "B:=matrix([[a+1,0,0],[0
,a+1,0],[0,0,a-3]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matri
xG6#7%7%,&%\"aG\"\"\"F,F,\"\"!F-7%F-F*F-7%F-F-,&F+F,\"\"$!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "B:=addcol(B,2,1,2):B:=addcol
(B,3,1,2):B:=addrow(B,3,2,-1):B:=addrow(B,1,3,-1);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7%7%,&%\"aG\"\"\"F,F,\"\"!F-7%\"\")
F*,&F+!\"\"\"\"$F,7%,&F+F,\"\"(F1F-,&F+F,F2F1" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "B:=addrow(B,2,1,2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"BG-%'matrixG6#7%7%,&\"#<\"\"\"%\"aGF,,&F-\"\"#F/F,,
&F-!\"#\"\"'F,7%\"\"),&F-F,F,F,,&F-!\"\"\"\"$F,7%,&F-F,\"\"(F7\"\"!,&F
-F,F8F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(B);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,**$)%\"aG\"\"#\"\"\"!\"\"*&\"\"&F(F&F(F)\"
\"$F)*$)F&F,F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "factor(
det(B));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"aG\"\"\"\"\"$!\"\"F
&),&F%F&F&F&\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rank
(subs(a=3,evalm(B)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "rank(subs(a=-1,evalm(B)));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 5 "Gagn\351" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "3. Une
 matrice stochastique" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "res
tart:with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protec
ted names norm and trace have been redefined and unprotected\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A:=matrix([[1/6,0,1/3],[1/2,
1/2,1/3],[1/3,1/2,1/3]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'
matrixG6#7%7%#\"\"\"\"\"'\"\"!#F+\"\"$7%#F+\"\"#F1F.7%F.F1F." }}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 42 "Caract\351risation des matrices s
tochastiques" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalm(vector
([1,1,1])&*A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"\"
F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Forc\351ment... cela trad
uit exactement le fait que la somme sur chaque colonne vaut 1." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "B:=evalm(A^50):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalm(vector([1,1,1])&*B);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"\"F'F'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 28 "Ainsi, A^50 est stochastique" }}}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 19 "A-I3 non inversible" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 165 "La relation de la question pr\351c\351dente, une \+
fois transpos\351e, nous dit que t(A)-I n'est pas injective donc par i
nversible. Il en est donc de m\352me pour t(t(A)-I)=A-I..." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "I3:=Matrix(3,3,shape=identity);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I3G-%'RTABLEG6$\"*GaqM\"-%'MATRIXG6
#7%7%\"\"\"\"\"!F/7%F/F.F/7%F/F/F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 10 "Ben oui..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "kernel
(A-I3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'vectorG6#7%\"\"\"#\"\"
)\"\"$#\"\"&\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "colspa
ce(A-I3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'vectorG6#7%\"\"!\"\"
\"!\"\"-F%6#7%F)F(F*" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 14 "A^100 e
t A^200" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "map(evalf,evalm(A
^100));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%$\"+A;i@;!#
5F(F(7%$\"+CVKCVF*F,F,7%$\"+aS0aSF*F/F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "map(evalf,evalm(A^200));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%$\"+A;i@;!#5F(F(7%$\"+CVKCVF*F,F,7%$\"+
aS0aSF*F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "Tiens tiens...\nA
u fait, comment Maple fait-il pour calculer A^100 ? Plus pr\351cis\351
ment, il fait combien de multiplications matricielles ?" }}}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 22 "Un polynome annulateur" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 29 "B:=evalm(A^3+a*A^2+b*A+c*I3);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7%7%,*#\"#N\"$;#\"\"\"*&#\"\"&
\"#OF.%\"aGF.F.*&#F.\"\"'F.%\"bGF.F.%\"cGF.,&F5F.*&F5F.F3F.F.,(#\"#<\"
$3\"F.*&F5F.F3F.F.*&#F.\"\"$F.F7F.F.7%,(#\"#J\"#sF.*&#\"\"%\"\"*F.F3F.
F.*&#F.\"\"#F.F7F.F.,*FEF.*&#F1\"#7F.F3F.F.*&FMF.F7F.F.F8F.,(#\"#ZF>F.
*&FIF.F3F.F.*&FAF.F7F.F.7%,(#\"#6\"#FF.*&FQF.F3F.F.*&FAF.F7F.F.,(#\"#H
FGF.*&FQF.F3F.F.*&FMF.F7F.F.,*FenF.*&#\"\"(\"#=F.F3F.F.*&FAF.F7F.F.F8F
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eq:=seq(seq(B[i,j],i=1
..3),j=1..3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#eqG6+,*#\"#N\"$;#
\"\"\"*&#\"\"&\"#OF*%\"aGF*F**&#F*\"\"'F*%\"bGF*F*%\"cGF*,(#\"#J\"#sF*
*&#\"\"%\"\"*F*F/F*F**&#F*\"\"#F*F3F*F*,(#\"#6\"#FF**&#F-\"#7F*F/F*F**
&#F*\"\"$F*F3F*F*,&F1F**&F1F*F/F*F*,*F6F**&FEF*F/F*F**&F>F*F3F*F*F4F*,
(#\"#HF8F**&FEF*F/F*F**&F>F*F3F*F*,(#\"#<\"$3\"F**&F1F*F/F*F**&FHF*F3F
*F*,(#\"#ZFWF**&F:F*F/F*F**&FHF*F3F*F*,*FAF**&#\"\"(\"#=F*F/F*F**&FHF*
F3F*F*F4F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(\{eq\},
\{a,b,c\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"bG#\"\"\"\"#O/%\"
cG#!\"\"F(/%\"aGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "minpo
ly(A,X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*#!\"\"\"#O\"\"\"*&#F'F&F
'%\"XGF'F'*$)F*\"\"#F'F%*$)F*\"\"$F'F'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 68 "Cette fonction magique sera expliqu\351e... l'ann\351e pr
ochaine en maths." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "P:=X^3
-X^2+X/36-1/36:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalm(su
bs(X=A,P));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!F(
F(F'F'" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 13 "Calcul de A^n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(P);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,$*&,&%\"XG\"\"\"F'!\"\"F',&*$)F&\"\"#F'\"#OF'F'F'#F
'F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "Le reste dans la division
 de X^n par P est de la forme dX^2+eX+f : on trouve d e et f en \351va
luant cette relation en les racines de P :" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 9 "solve(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"^
##F#\"\"'^##!\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "S:=s
olve(\{1=d+e+f,(I/6)^n=-d/36+e*I/36+f,(-I/6)^n=-d/36-e*I/6+f\},\{d,e,f
\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"SG<%/%\"fG,,)^##!\"\"\"\"'
%\"nG#\"#O\"$f##\"\"\"\"#PF3*&^#F/F3)^##F3F-F.F3F3*&^##!#OF1F3F)F3F3*&
#\"$;#F1F3F7F3F3/%\"eG*&^##F=\"\"(F3,&F7F3F)F,F3/%\"dG,,#F0F4F3*&^##\"
%'H\"F1F3F7F3F3*&^##!%'H\"F1F3F)F3F3*&#F0F1F3F)F3F,*&#F@F1F3F7F3F," }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalm(d*A^2+e*A+f*I3);" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%,.#\"\"'\"#P\"\"\"*&^##\"$;#\"$
f#F,)^##F,F*%\"nGF,F,*&^##!$;#F1F,)^##!\"\"F*F5F,F,*&#\"#JF1F,F:F,F,*&
#\"$'=F1F,F2F,F,*&^##!\"'\"\"(F,,&F2F,F:F=F,F,,,F)F,F-F,F6F,*&#F*F1F,F
:F,F=*&#\"#OF1F,F2F,F=,.F)F,F-F,F6F,*&#F*F1F,F:F,F=*&#FOF1F,F2F,F=*&^#
#!#7FHF,FIF,F,7%,.#\"#;F+F,*&^##\"$w&F1F,F2F,F,*&^##!$w&F1F,F:F,F,*&#F
fnF1F,F:F,F=*&#\"#'*F1F,F2F,F=*&^##!#=FHF,FIF,F,,.FenF,FgnF,F[oF,*&#\"
\"$F+F,F:F,F,*&#\"#=F+F,F2F,F,FdoF,,.FenF,FgnF,F[oF,*&#FfnF1F,F:F,F=*&
#FcoF1F,F2F,F=FUF,7%,.#\"#:F+F,*&^##\"$S&F1F,F2F,F,*&^##!$S&F1F,F:F,F,
*&#FgpF1F,F:F,F=*&#\"#!*F1F,F2F,F=FUF,,.FfpF,FhpF,F\\qF,*&#FgpF1F,F:F,
F=*&#FdqF1F,F2F,F=FdoF,,.FfpF,FhpF,F\\qF,*&#\"#AF1F,F:F,F,*&#\"$K\"F1F
,F2F,F,FUF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalm(subs(n
=50,%)-A^50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!
F(F(F'F'" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 29 "Limite de A^n quand
 n->infini" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "d,e,f;" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6%,,#\"#O\"#P\"\"\"*&^##\"%'H\"\"$f#F')^
##F'\"\"'%\"nGF'F'*&^##!%'H\"F,F')^##!\"\"F0F1F'F'*&#F%F,F'F6F'F9*&#\"
$;#F,F'F-F'F9*&^##!#O\"\"(F',&F-F'F6F9F',,F6#F%F,#F'F&F'*&^#FFF'F-F'F'
*&^##FBF,F'F6F'F'*&#F>F,F'F-F'F'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "limit(d,n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%&limitG6$,,#\"#O\"#P\"\"\"*&^##\"%'H\"\"$f#F*)^##F*\"\"'%\"nGF*F*
*&^##!%'H\"F/F*)^##!\"\"F3F4F*F**&#F(F/F*F9F*F<*&#\"$;#F/F*F0F*F</F4%)
infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Tsss..." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "d_inf:=36/37:e_inf:=0:f_inf:=1/37:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "B:=evalm(d_inf*A^2+f_in
f*I3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7%7%#\"\"'
\"#PF*F*7%#\"#;F,F.F.7%#\"#:F,F1F1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "map(evalf,B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'m
atrixG6#7%7%$\"+A;i@;!#5F(F(7%$\"+CVKCVF*F,F,7%$\"+aS0aSF*F/F/" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Voir la troisi\350me question..." 
}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "Propri\351t\351s de la limite
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(B^2);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%#\"\"'\"#PF(F(7%#\"#;F*F,F,7%#
\"#:F*F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Ainsi, B est un pro
jecteur." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalm(A&*B),eva
lm(B&*A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'matrixG6#7%7%#\"\"'\"#
PF(F(7%#\"#;F*F,F,7%#\"#:F*F/F/F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
93 "B commute bien avec A. On peut le montrer en passant \340 la limit
e la relation A^(n+1)=A.A^n..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "colspace(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'vectorG6#
7%\"\"\"#\"\")\"\"$#\"\"&\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
224 "C'est le noyau de A-I. D\351j\340, si AX=X, alors A^nX=X pour tou
t n, puis BX=X, donc X est dans l'image du projecteur. R\351ciproqueme
nt, si X est dans l'image de B, alors BX=X, donc  ABX=AX, mais AB=B, d
onc ABX=BX=X, et ainsi AX=X." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 10 "kernel(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'vectorG6#7%!
\"\"\"\"\"\"\"!-F%6#7%F(F*F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 185 "
C'est l'image de A-I. Preuve ? D\351j\340, au niveau des dimensions, l
e th\351or\350me du rang appliqu\351 \340 B et \340 A-I nous assure qu
e \347a marche. Il reste \340 montrer une inclusion : \340 vous de jou
er ! " }}}}}}{MARK "3 9 1 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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