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:Z:j;<Z:>:1:" }{TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 314 32 "CHAPITR
E 8: UTILISATION DE MAPLE" }{TEXT 316 0 "" }{TEXT 317 1 "\n" }{TEXT 
315 19 "EN ALGEBRE LINEAIRE" }}{PARA 19 "" 0 "" {TEXT -1 39 "\251 Appr
endre Maple. Alain Le Stang. 2006" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 18 "FONCTION
S DE BASE:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Les
 vecteurs et les matrices peuvent \352tre d\351finis comme des tableau
x rectangulaires, par la " }}{PARA 0 "" 0 "" {TEXT -1 9 "fonction " }
{TEXT 257 5 "array" }{TEXT -1 48 ", rencontr\351e dans le chapitre sur
 les tableaux :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 44 "Vecteur-ligne d'ordre 3 sans initialisation:" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 17 "A:=array(1..3);  " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%&arrayG6$;\"\"\"\"\"$7\"" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 45 "Vecteur-ligne d'ordre 3 avec initialisation: " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "B:=array([x,y,z]); " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'vectorG6#7%%\"xG%\"yG%\"zG" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Vecteur-colonne d'ordre 3 sans i
nitialisation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "U:=array(
1..3,1..1);print(U);    " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG-%&a
rrayG6%;\"\"\"\"\"$;F)F)7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matr
ixG6#7%7#&%\"UG6$\"\"\"F+7#&F)6$\"\"#F+7#&F)6$\"\"$F+" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 46 "Vecteur-colonne d'ordre 3 avec initialisa
tion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "V:=array([[x],[y],
[z]]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%'matrixG6#7%7#%\"xG
7#%\"yG7#%\"zG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 45 "Matrice carr\351e d'ordre 2 sans initialisation:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "M:=array(1..2,1..2);print(M
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%&arrayG6%;\"\"\"\"\"#F(7
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$&%\"MG6$\"\"\"F+
&F)6$F+\"\"#7$&F)6$F.F+&F)6$F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
45 "Matrice carr\351e d'ordre 2 avec initialisation:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "N:=array([[a,b],[c,d]]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"NG-%'matrixG6#7$7$%\"aG%\"bG7$%\"cG%\"dG" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 259 43 "O
p\351rations sur les vecteurs et les matrices" }{TEXT -1 20 " , avec l
a fonction " }{TEXT 258 5 "evalm" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" 
{TEXT -1 57 "Somme de vecteurs ou de matrices de m\352me dimension ave
c +" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalm(A+U); " }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'matrixG6#7%7#,&&%\"AG6#\"\"\"F,&%\"UG6$F,F,F,7#
,&&F*6#\"\"#F,&F.6$F4F,F,7#,&&F*6#\"\"$F,&F.6$F;F,F," }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 75 "Multiplication de vecteurs ou de matrices de di
mensions compatibles avec &*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "evalm(M &* N); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7
$7$,&*&&%\"MG6$\"\"\"F-F-%\"aGF-F-*&&F+6$F-\"\"#F-%\"cGF-F-,&*&F*F-%\"
bGF-F-*&F0F-%\"dGF-F-7$,&*&&F+6$F2F-F-F.F-F-*&&F+6$F2F2F-F3F-F-,&*&F<F
-F6F-F-*&F?F-F8F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Calcul de \+
la matrice inverse, lorsque N est inversible, avec ^(-1)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalm(N^(-1)); " }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'matrixG6#7$7$*&%\"dG\"\"\",&*&%\"aGF*F)F*F**&%\"bG
F*%\"cGF*!\"\"F1,$*&F/F*F+F1F17$,$*&F0F*F+F1F1*&F-F*F+F1" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 24 "Puissance n-i\350me avec ^n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalm(N^3);    " }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'matrixG6#7$7$,&*&,&*$)%\"aG\"\"#\"\"\"F/*&%\"bGF/%
\"cGF/F/F/F-F/F/*&,&*&F-F/F1F/F/*&F1F/%\"dGF/F/F/F2F/F/,&*&F*F/F1F/F/*
&F4F/F7F/F/7$,&*&,&*&F2F/F-F/F/*&F7F/F2F/F/F/F-F/F/*&,&F0F/*$)F7F.F/F/
F/F2F/F/,&*&F>F/F1F/F/*&FBF/F7F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 38 "Multiplication par un scalaire avec * " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "evalm(2*M-5*N);    " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7$7$,&&%\"MG6$\"\"\"F,\"\"#*&\"\"&F,%\"aGF,
!\"\",&&F*6$F,F-F-*&F/F,%\"bGF,F17$,&&F*6$F-F,F-*&F/F,%\"cGF,F1,&&F*6$
F-F-F-*&F/F,%\"dGF,F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 260 29 "LES OUTILS DU PACKAGE LINALG:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Le package " }
{TEXT 261 6 "linalg" }{TEXT -1 60 " contient plus d'une centaine d'out
ils en alg\350bre lin\351aire :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7^r%.BlockDiagonalG%,GramSchmidtG%,JordanBlockG%)LUdeco
mpG%)QRdecompG%*WronskianG%'addcolG%'addrowG%$adjG%(adjointG%&angleG%(
augmentG%(backsubG%%bandG%&basisG%'bezoutG%,blockmatrixG%(charmatG%)ch
arpolyG%)choleskyG%$colG%'coldimG%)colspaceG%(colspanG%*companionG%'co
ncatG%%condG%)copyintoG%*crossprodG%%curlG%)definiteG%(delcolsG%(delro
wsG%$detG%%diagG%(divergeG%(dotprodG%*eigenvalsG%,eigenvaluesG%-eigenv
ectorsG%+eigenvectsG%,entermatrixG%&equalG%,exponentialG%'extendG%,ffg
ausselimG%*fibonacciG%+forwardsubG%*frobeniusG%*gausselimG%*gaussjordG
%(geneqnsG%*genmatrixG%%gradG%)hadamardG%(hermiteG%(hessianG%(hilbertG
%+htransposeG%)ihermiteG%*indexfuncG%*innerprodG%)intbasisG%(inverseG%
'ismithG%*issimilarG%'iszeroG%)jacobianG%'jordanG%'kernelG%*laplacianG
%*leastsqrsG%)linsolveG%'mataddG%'matrixG%&minorG%(minpolyG%'mulcolG%'
mulrowG%)multiplyG%%normG%*normalizeG%*nullspaceG%'orthogG%*permanentG
%&pivotG%*potentialG%+randmatrixG%+randvectorG%%rankG%(ratformG%$rowG%
'rowdimG%)rowspaceG%(rowspanG%%rrefG%*scalarmulG%-singularvalsG%&smith
G%,stackmatrixG%*submatrixG%*subvectorG%)sumbasisG%(swapcolG%(swaprowG
%*sylvesterG%)toeplitzG%&traceG%*transposeG%,vandermondeG%*vecpotentG%
(vectdimG%'vectorG%*wronskianG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Les fonctions " }{TEXT 262 6 "matr
ix" }{TEXT -1 4 " et " }{TEXT 263 6 "vector" }{TEXT -1 59 " permettent
 de d\351finir respectivement une matrice (tableau " }}{PARA 0 "" 0 "
" {TEXT -1 76 "rectangulaire d'objets) ou un vecteur (tableau d'objets
 \340 une seule ligne) :" }}{PARA 0 "" 0 "" {TEXT -1 31 "\nD\351finiti
on d'une matrice 2x4 :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "P:=matrix
([[s,t,u,v],[w,x,y,z]]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'
matrixG6#7$7&%\"sG%\"tG%\"uG%\"vG7&%\"wG%\"xG%\"yG%\"zG" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 75 "D\351finition d'un vecteur-ligne d'ordre \+
4 et d'une matrice-colonne d'ordre 3:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "V:=vector([a,b,c,d]); " }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"VG-%'vectorG6#7&%\"aG%\"bG%\"cG%\"dG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "C:=matrix([[1],[2],[3]]);  " }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"CG-%'matrixG6#7%7#\"\"\"7#\"\"#7#\"\"$" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Une matrice a le type " }{TEXT 
264 6 "matrix" }{TEXT -1 24 " , un vecteur a le type " }{TEXT 265 6 "v
ector" }{TEXT -1 1 "." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT 318 27 "Op\351rations sur les matrices" }{TEXT 
-1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
26 "A la place de la fonction " }{TEXT 266 5 "evalm" }{TEXT -1 53 ", o
n peut utiliser les outils sp\351cifiques du package " }{TEXT 267 6 "l
inalg" }{TEXT -1 1 ":" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 73 "Multiplication de vecteurs ou de matrices
 de dimensions compatibles avec " }{TEXT 319 9 "multiply " }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "mult
iply(M,N); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,&*&&%
\"MG6$\"\"\"F-F-%\"aGF-F-*&&F+6$F-\"\"#F-%\"cGF-F-,&*&F*F-%\"bGF-F-*&F
0F-%\"dGF-F-7$,&*&&F+6$F2F-F-F.F-F-*&&F+6$F2F2F-F3F-F-,&*&F<F-F6F-F-*&
F?F-F8F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Calcul de la matric
e inverse,lorsque N est inversible, avec" }{TEXT 320 8 " inverse" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(N);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$*&%\"dG\"\"\",&*&%\"aGF*F)F*F**&%
\"bGF*%\"cGF*!\"\"F1,$*&F/F*F+F1F17$,$*&F0F*F+F1F1*&F-F*F+F1" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Multiplication par un scalaire " }
{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 6 " avec " }{TEXT 321 
9 "scalarmul" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "scalarmul(P
,lambda);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7&*&%'lam
bdaG\"\"\"%\"sGF**&F)F*%\"tGF**&F)F*%\"uGF**&F)F*%\"vGF*7&*&F)F*%\"wGF
**&F)F*%\"xGF**&F)F*%\"yGF**&F)F*%\"zGF*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 37 "Test d'\351galit\351 de deux matrices avec " }{TEXT 322 
5 "equal" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "equal(M,N);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 20 "Etude d'une
 matrice:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
73 "D\351terminant et trace d'une matrice carr\351e (somme des \351l
\351ments diagonaux):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "N:=matrix(
[[1,2],[4,-3]]);det(N);trace(N);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"NG-%'matrixG6#7$7$\"\"\"\"\"#7$\"\"%!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 19 "Rang d'une matrice:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 11 "rank(N);   " }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 233 "kernel(A);  ou nullsp
ace(A); d\351termine une base du noyau de A\ncolspace(A); d\351termine
 une base du sous-espace engendr\351 par les vecteurs-colonnes de A\nr
owspace(A); d\351termine une base du sous-espace engendr\351 par les v
ecteurs-lignes de A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A:=m
atrix([[2,-2],[-1,1]]);kernel(A); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"AG-%'matrixG6#7$7$\"\"#!\"#7$!\"\"\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<#-%'vectorG6#7$\"\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "colspace(A);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-
%'vectorG6#7$\"\"\"#!\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "rowspace(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'vectorG6#
7$\"\"\"!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 
"" {TEXT 269 24 "Matrice caract\351ristique:" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "lambda*I-A" "6#,&*&%'lambdaG\"\"\"%\"IGF&F&%\"AG!\"\"" 
}{TEXT -1 3 " , " }{TEXT 270 30 "polyn\364me caract\351ristique de A:
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "det(lambda*I-A)" "6#-%$detG6#,&*&%'l
ambdaG\"\"\"%\"IGF)F)%\"AG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "charmat(A,lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'matrixG6#7$7$,&%'lambdaG\"\"\"\"\"#!\"\"F+7$F*,&F)F*F*F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "charpoly(A,lambda);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%'lambdaG\"\"#\"\"\"F(*&\"\"$F(F&
F(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Valeurs propres de A (
ce sont les racines du polyn\364me caract\351ristique): " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvals(A); " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"\"!\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Un " 
}{TEXT 324 14 "vecteur propre" }{TEXT -1 14 " associ\351 \340 la " }
{TEXT 325 13 "valeur propre" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "lambda" 
"6#%'lambdaG" }{TEXT -1 16 " est un vecteur " }{TEXT 323 1 "X" }{TEXT 
-1 17 " non nul tel que " }{XPPEDIT 18 0 "A*X = lambda* X" "6#/*&%\"AG
\"\"\"%\"XGF&*&%'lambdaGF&F'F&" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "eigenvects(A); " }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$7%\"\"!\"\"\"<#-%'vectorG6#7$F%F%7%\"\"$F%<#-F(6#7$!\"#F%" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Le
 r\351sultat affich\351 signifie que " }{XPPEDIT 18 0 "lambda=0" "6#/%
'lambdaG\"\"!" }{TEXT -1 59 " est une valeur propre d'ordre 1 de vecte
ur propre associ\351 " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "X=(1,1)" "6#/%
\"XG6$\"\"\"F&" }{TEXT -1 8 " et que " }{XPPEDIT 18 0 "lambda=3" "6#/%
'lambdaG\"\"$" }{TEXT -1 59 " est une valeur propre d'ordre 1 de vecte
ur propre associ\351 " }{XPPEDIT 18 0 "X=(-2,1)" "6#/%\"XG6$,$\"\"#!\"
\"\"\"\"" }{TEXT -1 1 "." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT 271 27 "Modification d'une matrice:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(P);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7&%\"sG%\"tG%\"uG%\"vG7&
%\"wG%\"xG%\"yG%\"zG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "" 0 "" {TEXT 272 8 "row(P,k)" }{TEXT -1 81 " permet d'extraire la k
-i\350me ligne de la matrice P. Le r\351sultat est un vecteur.  " }}
{PARA 0 "" 0 "" {TEXT 273 8 "col(P,k)" }{TEXT -1 81 " permet d'extrair
e la k-i\350me colonne de la matrice P. Le r\351sultat est un vecteur.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "row(P,2),col(P,3);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6$-%'vectorG6#7&%\"wG%\"xG%\"yG%\"zG-F$6#7$%\"uGF
)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
275 9 "rowdim(P)" }{TEXT -1 4 " et " }{TEXT 276 9 "coldim(P)" }{TEXT 
-1 61 " retournent le nombre de lignes et de colonnes de la matrice " 
}{TEXT 274 1 "P" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "
rowdim(P),coldim(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#\"\"%" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 277 21 "
augment(M1,M2,...,Mn)" }{TEXT -1 42 " permet de concat\351ner horizont
alement les " }{TEXT 279 1 "n" }{TEXT -1 10 " matrices " }{TEXT 281 
12 "M1,M2,...,Mn" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"augment(A,P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7(\"\"
#!\"#%\"sG%\"tG%\"uG%\"vG7(!\"\"\"\"\"%\"wG%\"xG%\"yG%\"zG" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 278 25 "stackma
trix(M1,M2,...,Mn)" }{TEXT -1 40 " permet de concat\351ner verticaleme
nt les " }{TEXT 280 1 "n" }{TEXT -1 10 " matrices " }{TEXT 282 12 "M1,
M2,...,Mn" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "stackm
atrix(A,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7$\"\"#!
\"#7$!\"\"\"\"\"7$F,F(7$\"\"%!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 283 17 "copyinto(M,N,m,n)" }{TEXT -1 9 " \+
: copie " }{TEXT 284 2 "M " }{TEXT -1 5 "dans " }{TEXT 285 1 "N" }
{TEXT -1 12 " , l'\351l\351ment" }{TEXT 326 1 " " }{TEXT 286 6 "M[1,1]
" }{TEXT -1 16 " \351tant copi\351 en " }{TEXT 287 6 "N[m,n]" }{TEXT 
-1 2 " :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "copyinto(N,P,1,2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7&%\"sG\"\"\"\"\"#%\"vG7
&%\"wG\"\"%!\"$%\"zG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "" 0 "" {TEXT 288 14 "swapcol(M,m,n)" }{TEXT -1 4 " et " }{TEXT 289 
14 "swaprow(M,m,n)" }{TEXT -1 51 " permettent d'\351changer les colonn
es (ou les lignes)" }}{PARA 0 "" 0 "" {TEXT 290 1 "m" }{TEXT -1 4 " et
 " }{TEXT 291 1 "n" }{TEXT -1 15 " de la matrice " }{TEXT 292 1 "M" }
{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "swaprow(A,1,2);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$!\"\"\"\"\"7$\"\"#!\"
#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
293 13 "transpose(M) " }{TEXT -1 32 "permet de transposer la matrice \+
" }{TEXT 294 1 "M" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "print(M);transpose(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matri
xG6#7$7$&%\"MG6$\"\"\"F+&F)6$F+\"\"#7$&F)6$F.F+&F)6$F.F." }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$&%\"MG6$\"\"\"F+&F)6$\"\"#F+7$&F
)6$F+F.&F)6$F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 295 15 "addcol(M,m,n
,k)" }{TEXT -1 4 " ou " }{TEXT 296 16 "addrow(M,m,n,k) " }{TEXT -1 40 
"rendent une matrice obtenue \340 partir de " }{TEXT 297 1 "M" }{TEXT 
-1 4 " par" }}{PARA 0 "" 0 "" {TEXT -1 12 "l'op\351ration " }{TEXT 
298 29 "col n := col n + k * col m   " }{TEXT -1 17 "(respectivement  \+
" }{TEXT 299 26 "lig n := lig n + k * lig m" }{TEXT -1 3 ") :" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "pri
nt(N);addcol(N,1,2,mu);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6
#7$7$\"\"\"\"\"#7$\"\"%!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matr
ixG6#7$7$\"\"\",&%#muGF(\"\"#F(7$\"\"%,&F*F-\"\"$!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 300 23 "Matrices particuli\350res:" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 23 "vandermonde([a,b,c,d]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7&7&\"\"\"%\"aG*$)F)\"\"#F(*$)F)\"\"$F(7&F(
%\"bG*$)F1F,F(*$)F1F/F(7&F(%\"cG*$)F7F,F(*$)F7F/F(7&F(%\"dG*$)F=F,F(*$
)F=F/F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "JordanBlock(expr
,4); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&%%exprG\"\"\"
\"\"!F*7&F*F(F)F*7&F*F*F(F)7&F*F*F*F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 336 36 "LES OUTILS DU PACKAGE LinearAlgebra:
" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 63 "A partir de la version 7, existe dans Maple un nouve
au package " }{TEXT 337 14 "LinearAlgebra " }{TEXT 338 0 "" }{TEXT 
339 0 "" }{TEXT 340 4 "avec" }}{PARA 0 "" 0 "" {TEXT 343 40 "des outil
s similaires \340 ceux du package " }{TEXT 341 6 "linalg" }{TEXT 342 
58 " et destin\351 \340 supplanter l'ancien. En voici les fonctions:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#7hq%$AddG%(AdjointG%3BackwardSubstitut
eG%+BandMatrixG%&BasisG%-BezoutMatrixG%/BidiagonalFormG%-BilinearFormG
%5CharacteristicMatrixG%9CharacteristicPolynomialG%'ColumnG%0ColumnDim
ensionG%0ColumnOperationG%,ColumnSpaceG%0CompanionMatrixG%0ConditionNu
mberG%/ConstantMatrixG%/ConstantVectorG%2CreatePermutationG%-CrossProd
uctG%-DeleteColumnG%*DeleteRowG%,DeterminantG%/DiagonalMatrixG%*Dimens
ionG%+DimensionsG%+DotProductG%,EigenvaluesG%-EigenvectorsG%&EqualG%2F
orwardSubstituteG%.FrobeniusFormG%4GaussianEliminationG%2GenerateEquat
ionsG%/GenerateMatrixG%2GetResultDataTypeG%/GetResultShapeG%5GivensRot
ationMatrixG%,GramSchmidtG%-HankelMatrixG%,HermiteFormG%3HermitianTran
sposeG%/HessenbergFormG%.HilbertMatrixG%2HouseholderMatrixG%/IdentityM
atrixG%2IntersectionBasisG%+IsDefiniteG%-IsOrthogonalG%*IsSimilarG%*Is
UnitaryG%2JordanBlockMatrixG%+JordanFormG%(LA_MainG%0LUDecompositionG%
-LeastSquaresG%,LinearSolveG%$MapG%%Map2G%*MatrixAddG%.MatrixInverseG%
5MatrixMatrixMultiplyG%+MatrixNormG%5MatrixScalarMultiplyG%5MatrixVect
orMultiplyG%2MinimalPolynomialG%&MinorG%)MultiplyG%,NoUserValueG%%Norm
G%*NormalizeG%*NullSpaceG%3OuterProductMatrixG%*PermanentG%&PivotG%0QR
DecompositionG%-RandomMatrixG%-RandomVectorG%%RankG%6ReducedRowEchelon
FormG%$RowG%-RowDimensionG%-RowOperationG%)RowSpaceG%-ScalarMatrixG%/S
calarMultiplyG%-ScalarVectorG%*SchurFormG%/SingularValuesG%*SmithFormG
%*SubMatrixG%*SubVectorG%)SumBasisG%0SylvesterMatrixG%/ToeplitzMatrixG
%&TraceG%*TransposeG%0TridiagonalFormG%+UnitVectorG%2VandermondeMatrix
G%*VectorAddG%,VectorAngleG%5VectorMatrixMultiplyG%+VectorNormG%5Vecto
rScalarMultiplyG%+ZeroMatrixG%+ZeroVectorG%$ZipG" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "On d\351finit une \+
matrice A dont on calcule le d\351terminant, la matrice et le polyn
\364me caract\351ristique," }}{PARA 0 "" 0 "" {TEXT -1 59 "ainsi que l
es valeurs propres et vecteurs propres associ\351s:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 168 "A:=Matrix([[1,-1,-1],[-2,2,3],[2,-2,-3]]
);\nDeterminant(A);\nCharacteristicMatrix(A,lambda);\nfactor(Character
isticPolynomial(A,lambda));\nEigenvalues(A);\nEigenvectors(A);\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6$\"*[)\\\"\\\"-%'MATR
IXG6#7%7%\"\"\"!\"\"F/7%!\"#\"\"#\"\"$7%F2F1!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"
*WB:\\\"-%'MATRIXG6#7%7%,&%'lambdaG!\"\"\"\"\"F/F.F.7%!\"#,&F-F.\"\"#F
/\"\"$7%F3F1,&F-F.F4F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&%'lambda
G\"\"\"F&!\"\"F&F%F&,&F%F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%
'RTABLEG6$\"*#4<\"\\\"-%'MATRIXG6#7%7#\"\"\"7#\"\"!7#!\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6$\"*S9;\\\"-%'MATRIXG6#7%7#!\"\"7
#\"\"!7#\"\"\"-F$6$\"*Gt;\\\"-F(6#7%7%F.F0F07%F0F0F,7%F,F.F0" }}}
{PARA 256 "" 0 "" {TEXT -1 19 "Exercice corrig\351 8:" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Rechercher les \+
matrices carr\351es " }{TEXT 301 1 "X" }{TEXT -1 23 " d'ordre 2 telles
 que  " }{XPPEDIT 18 0 "X^2 + X = A" "6#/,&*$%\"XG\"\"#\"\"\"F&F(%\"AG
" }{TEXT -1 8 "  , o\371  " }{XPPEDIT 18 0 "A := MATRIX([[1, 1], [1, 1
]])" "6#>%\"AG-%'MATRIXG6#7$7$\"\"\"F*7$F*F*" }{TEXT -1 2 " ." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "with(linalg):A:=matrix(2,2,1
);X:=matrix(2,2):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6
#7$7$\"\"\"F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Y:=evalm
(X^2+X-A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG-%'matrixG6#7$7$,*
*$)&%\"XG6$\"\"\"F0\"\"#F0F0*&&F.6$F0F1F0&F.6$F1F0F0F0F-F0F0!\"\",**&F
-F0F3F0F0*&F3F0&F.6$F1F1F0F0F3F0F0F77$,**&F5F0F-F0F0*&F;F0F5F0F0F5F0F0
F7,*F2F0*$)F;F1F0F0F;F0F0F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "Y:=\{seq(seq(Y[i,j],j=1..2),i=1..2)\}; " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"YG<&,**$)&%\"XG6$\"\"\"F,\"\"#F,F,*&&F*6$F,F-F,&F*6
$F-F,F,F,F)F,F,!\"\",*F.F,*$)&F*6$F-F-F-F,F,F7F,F,F3,**&F)F,F/F,F,*&F/
F,F7F,F,F/F,F,F3,**&F1F,F)F,F,*&F7F,F1F,F,F1F,F,F3" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 108 "solutions:=\{solve(Y)\}:\nfor k to nops(so
lutions) do\n  X:=matrix(2,2):assign(solutions[k]):print(X);\nend do: \+
 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"!\"\"\"7$F)F(
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$#!\"$\"\"##!\"\"F*
7$F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$#\"\"\"\"\"#
F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$!\"\"F(F'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Donc 4 matrices solutions." }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 17 "Travail
 dirig\351 8:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "
" {TEXT 303 7 "TD 8.1:" }}{PARA 0 "" 0 "" {TEXT -1 21 "Ecrire une proc
\351dure " }{TEXT 304 12 "pcd(L::list)" }{TEXT -1 40 " retournant \340
 partir d'une liste donn\351e " }{TEXT 305 1 "L" }{TEXT -1 21 " la lis
te obtenue par" }}{PARA 0 "" 0 "" {TEXT -1 54 "permutation circulaire \+
vers la droite des \351l\351ments de " }{TEXT 313 1 "L" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 306 8 "Exemple:" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 21 "pcd([a,b,c,d,e,f,g]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7)%\"gG%\"aG%\"bG%\"cG%\"dG%\"eG%\"fG" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 307 7 "TD 8.2:
" }}{PARA 0 "" 0 "" {TEXT -1 21 "Ecrire une proc\351dure " }{TEXT 308 
28 "circulante_droite(n::posint)" }{TEXT -1 33 " retournant \340 parti
r de l'entier " }{TEXT 309 2 "n>" }{TEXT -1 1 "0" }}{PARA 0 "" 0 "" 
{TEXT -1 27 "une matrice carr\351e d'ordre " }{TEXT 310 1 "n" }{TEXT 
-1 10 " telle que" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 67 "         - sa premi\350re ligne est constitu\351e par les
 entiers de 1 \340 " }{TEXT 311 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 16 "         - pour " }{TEXT 312 2 "k>" }{TEXT -1 70 "1 , s
a k-i\350me ligne est d\351duite de la ligne pr\351c\351dente par perm
utation" }}{PARA 0 "" 0 "" {TEXT -1 53 "           circulaire vers la \+
droite de ses \351l\351ments." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 302 8 "Exemple:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "circulante_droite(5);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7'7'\"\"\"\"\"#\"\"$\"\"%\"\"&7'F,F(F)F*F+7'F+F,F(F)
F*7'F*F+F,F(F)7'F)F*F+F,F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 1 "" {TEXT 327 7 "TD 8.3:" }{TEXT 335 1 " " }{TEXT -1 0 "" 
}{TEXT 328 29 "Test d'\351galit\351 de 2 matrices:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "2 matrices " }{TEXT 329 
1 "M" }{TEXT -1 4 " et " }{TEXT 330 1 "N" }{TEXT -1 69 " sont \351gale
s si et seulement si elles ont m\352me nombre de lignes, m\352me" }}
{PARA 0 "" 0 "" {TEXT -1 47 "nombre de colonnes, et les m\352mes coeff
icients ." }}{PARA 0 "" 0 "" {TEXT -1 83 "Sans avoir recours \340 la f
onction pr\351existante equal de Maple, \351crire une proc\351dure " }
}{PARA 0 "" 0 "" {TEXT 331 31 "egal(M: : matrix , N: : matrix)" }
{TEXT -1 32 " retournant la valeur bool\351enne " }{TEXT 332 4 "true" 
}{TEXT -1 4 " si " }{TEXT 333 4 "M=N " }{TEXT -1 3 "et " }{TEXT 334 5 
"false" }{TEXT -1 7 " sinon." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
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