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0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 313 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 270 "" 0 "" {TEXT 263 13 "Calcul Formel" }{TEXT -1 18 " MASS 1 2003-2004\n" }{TEXT 264 9 "Feuille 7" }{TEXT -1 22 " \+ Alg\350bre lin\351aire (" }{TEXT 265 26 "calcul_mass1_2003_f7.mws )" }}{PARA 271 "" 0 "" {TEXT -1 52 "J'ai tout copi\351 sur la feuille 5 d e Math\351matiques..." }}}{SECT 0 {PARA 272 "" 0 "" {TEXT -1 40 "Valeu rs \"propres\" , vecteurs \"propres\" " }{TEXT 266 24 "d'une matrice \+ carr\351e 2x2" }}{PARA 273 "" 0 "" {TEXT -1 14 "les fonctions " } {TEXT 267 38 "eigenvalues, eigenvectors, entermatrix" }}{SECT 0 {PARA 274 "" 0 "" {TEXT -1 30 "Fonctions disponibles en Maple" }}{EXCHG {PARA 275 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 276 "> " 0 "" {MPLTEXT 1 0 25 "A:=matrix(2,2,[1,6,3,4]):" }}}{EXCHG {PARA 277 "" 0 "" {TEXT -1 56 "Les valeurs propres de A sont calcul\351es pa r la fonction " }{TEXT 268 11 "eigenvalues" }{TEXT -1 30 " ( valeurs p ropres en anglais)" }}}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 1 0 32 "val eurs_propres:=eigenvalues(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0va leurs_propresG6$!\"#\"\"(" }}}{EXCHG {PARA 279 "" 0 "" {TEXT -1 34 "Qu elles sont les valeurs propres? " }}{PARA 280 "" 0 "" {TEXT -1 53 "On \+ calcule les vecteurs propres grace \340 la fonction " }{TEXT 269 14 " eigenvectors" }}}{EXCHG {PARA 281 "> " 0 "" {MPLTEXT 1 0 19 "L:=eige nvectors(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG6$7%!\"#\"\"\"<# -%'vectorG6#7$F'F(7%\"\"(F(<#-F+6#7$F(F(" }}}{EXCHG {PARA 282 "" 0 "" {TEXT -1 56 "Comment interpr\351ter la valeur d\351livr\351e par la f onction " }{TEXT 290 13 "eigenvectors " }{TEXT -1 35 ":\nL est une sui te a deux \351l\351ments :" }}}{EXCHG {PARA 283 "> " 0 "" {MPLTEXT 1 0 19 "elem_suite1:= L[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,elem_s uite1G7%!\"#\"\"\"<#-%'vectorG6#7$F&F'" }}}{EXCHG {PARA 284 "" 0 "" {TEXT 270 11 "elem_suite1" }{TEXT -1 20 " est une liste dont " }} {PARA 300 "" 0 "" {TEXT -1 4 " " }{TEXT 257 44 " le premier \351l \351ment est une valeur propre, " }}{PARA 285 "" 0 "" {TEXT -1 61 " \+ le deuxi\350me est la \"multiplicit\351\" de la valeur propre." }} {PARA 286 "" 0 "" {TEXT -1 122 " le troisi\350me est un ensemble \+ ( \340 cause des accolades) , l'ensemble des vecteurs propres associ \351s \340 la valeur propre." }}{PARA 287 "" 0 "" {TEXT -1 0 "" }} {PARA 288 "" 0 "" {TEXT 271 11 "ATTENTION: " }{TEXT -1 83 "on vous a d it et je le r\351p\350te, dans le cas 2x2, si une valeur propre est \+ \"double\" " }}{PARA 307 "" 0 "" {TEXT -1 53 "(multiplicit\351 2) et q ue la matrice \351tudi\351e n'est pas " }{TEXT 284 4 "d\351j\340" } {TEXT -1 50 " diagonale, alors elle n'est pas \"diagonalisable\"." }} {PARA 289 "" 0 "" {TEXT 275 14 "Conclusion : " }{TEXT -1 12 "la fonct ion " }{TEXT 276 13 "eigenvectors " }{TEXT -1 67 "fournit une informat ion compl\350te sur la matrice donn\351e en argument." }}{PARA 290 "" 0 "" {TEXT -1 88 "Pour \"r\351cup\351rer\" le (ou les vecteurs ) propr es, on prend le troisi\350me \351l\351ment de elem1 :" }}}{EXCHG {PARA 291 "> " 0 "" {MPLTEXT 1 0 20 "ens:=elem_suite1[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ensG<#-%'vectorG6#7$!\"#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 291 57 "on obtient le premier \351l\351ment de l 'ensemble ens grace \340 :" }}}{EXCHG {PARA 292 "> " 0 "" {MPLTEXT 1 0 14 "v:= op(1,ens);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'vecto rG6#7$!\"#\"\"\"" }}}{EXCHG {PARA 293 "" 0 "" {TEXT -1 12 "C'est fait \+ !" }}{PARA 308 "" 0 "" {TEXT -1 78 "Quand il y a plusieurs vecteurs pr opres, on les obtient tous avec la fonction " }{TEXT 282 2 "op" } {TEXT -1 13 " \351galement : " }}{PARA 309 "" 0 "" {TEXT -1 19 " \+ par exemple, " }{TEXT 283 2 "op" }{TEXT -1 42 "(k,ens) est alors le k \+ \350me \351l\351ment de ens." }}}}{SECT 0 {PARA 294 "" 0 "" {TEXT -1 19 "Du code bien \340 nous" }}{EXCHG {PARA 295 "" 0 "" {TEXT -1 216 "E tant donn\351e une matrice 2 x 2 A , on veut construire une matric e 2 x 2 dont les vecteurs colonnes seront les vecteurs propres de A.On utilise ce qu'on sait sur la fonction eigenvectors pour \351crire les fonctions " }{TEXT 292 16 "matrice_colonnes" }{TEXT -1 4 " et " } {TEXT 293 9 "matrice_P" }{TEXT -1 1 ":" }}{PARA 296 "" 0 "" {TEXT -1 5 " " }{TEXT 277 1 " " }{TEXT 272 16 "matrice_colonnes" }{TEXT -1 74 "(u,v) est la matrice ayant u comme premier vecteur colonne, v comm e second" }}}{EXCHG {PARA 297 "> " 0 "" {MPLTEXT 1 0 27 "matrice_colon nes:=proc(u,v)" }{TEXT -1 42 "# u et v sont des vecteurs \340 2 compos antes" }{MPLTEXT 1 0 92 "\nlocal w;\nw:=[u[1],v[1],u[2],v[2]];\nRETURN (evalm(matrix(2,2,w)));\nend:\n\nmatrice_P:=proc(A) " }{TEXT 256 23 " # A est une matrice 2x2" }{MPLTEXT 1 0 110 " \nlocal L,u,v,w;\nL:=eige nvectors(A);\nu:= op(1,L[1][3]);\nv:= op(1,L[2][3]);\nRETURN(matrice_c olonnes(u,v));\nend:" }}}{EXCHG {PARA 301 "" 0 "" {TEXT 278 30 "Pour v oir ce que cela donne : " }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 30 "matrice_colonnes([1,1],[2,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%'matrixG6#7$7$\"\"\"\"\"#F'" }}}{EXCHG {PARA 257 "> " 0 "" {MPLTEXT 1 0 16 "P:=matrice_P(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'m atrixG6#7$7$!\"#\"\"\"7$F+F+" }}}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 0 14 "Q:=inverse(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG-%'mat rixG6#7$7$#!\"\"\"\"$#\"\"\"F,7$F-#\"\"#F," }}}{EXCHG {PARA 259 "" 0 " " {TEXT -1 89 "On constate alors ( en fait c'est un th\351or\350me d ans votre Cours de math\351matiques ) que " }}{PARA 298 "" 0 "" {TEXT -1 62 "la matrice Diag:=evalm(Q&*A&*P) a une structure particuli \350re :" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 21 "Diag:=evalm(Q &*A&*P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%DiagG-%'matrixG6#7$7$! \"#\"\"!7$F+\"\"(" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 103 "Quand on \+ a calcul\351 P, Q, Diag, on dit qu'on a \"diagonalis\351\" A i.e. on a A = P &*Diag*& Q , la preuve" }}}{EXCHG {PARA 262 "> " 0 "" {MPLTEXT 1 0 33 "Z:= evalm(P &*Diag&* Q);evalm(A);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"ZG-%'matrixG6#7$7$\"\"\"\"\"'7$\"\"$\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"\"\"\"'7$\"\"$\"\" %" }}}}{SECT 0 {PARA 302 "" 0 "" {TEXT -1 11 "Les donn\351es" }} {EXCHG {PARA 303 "> " 0 "" {MPLTEXT 1 0 409 " B:=matrice_colonnes([0,1 ],[-1,0]):\nC:=matrice_colonnes([1,0],[1,2]):\nA1:=matrice_colonnes([4 ,1 ],[4,4]):\nA2:=matrice_colonnes([3,2 ],[-2,1]):\nA3:=matrice_colonn es([1,2 ],[2,1]):\nM1:=matrice_colonnes([-1,2 ],[6,0]):\nM2:=matrice_c olonnes([-3,0 ],[-1,0]):\nM3:=matrice_colonnes([1,3 ],[2,6]):\nM4:=mat rice_colonnes([3,0 ],[1,3]):\nS:=matrice_colonnes([2,a ],[1,4]);\nalia s(Id2=evalm(matrice_colonnes([1,0],[0,1]))):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG-%'matrixG6#7$7$\"\"#\"\"\"7$%\"aG\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvalues(S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&\"\"$\"\"\"*$-%%sqrtG6#,&F%F%%\"aGF%\"\"\"F%, &F$F%F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eigenvector s(S)[1][3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#-%'vectorG6#7$*&,&!\" \"\"\"\"*$-%%sqrtG6#,&F+F+%\"aGF+\"\"\"F+F2F1!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eigenvectors(S)[2][3];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<#-%'vectorG6#7$*&,&!\"\"\"\"\"*$-%%sqrtG6#,&F+F+%\"a GF+\"\"\"F*F2F1!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "e igenvectors(S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%,&\"\"$\"\"\"*$-% %sqrtG6#,&F&F&%\"aGF&\"\"\"F&F&<#-%'vectorG6#7$*&,&!\"\"F&F'F&F-F,!\" \"F&7%,&F%F&F'F5F&<#-F06#7$*&,&F5F&F'F5F-F,F6F&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "matrice_P(S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$*&,&!\"\"\"\"\"*$-%%sqrtG6#,&F+F+%\"aGF+\"\"\"F+F2F 1!\"\"*&,&F*F+F,F*F2F1F37$F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(det(matrice_P(S))=0,a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "det(matrice_P(S ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*$-%%sqrtG6#,&\"\"\"F*%\"aG F*\"\"\"F,F+!\"\"\"\"#" }}}}{SECT 0 {PARA 263 "" 0 "" {TEXT -1 26 "Et \+ maintenant, Au boulot !" }}{PARA 0 "" 0 "" {TEXT 286 67 "Traiter les q uestions dans l'ordre sans en sauter une seule. merci." }}{SECT 0 {PARA 264 "" 0 "" {TEXT 258 2 "Q1" }{TEXT -1 2 " " }{TEXT 280 428 "(i ) Calculer les valeurs propres et les vecteurs propres des matrices B \+ et C; v\351rifiez avec maple vos calculs.\n (ii) Les matrices M1 , M2, M3 sont-elles inversibles? Calculer l'inverse de celle(s) qui l 'est (le sont).\n (iii) Calculer les valeurs propres et les vecte urs propres des matrices A1, A2, A3, M4 et les diagonaliser quand c'es t possible;\n \+ " }{TEXT -1 13 "R\351ponses ICI!" }}}{SECT 0 {PARA 265 "" 0 "" {TEXT -1 0 "" }{TEXT 259 2 "Q2" }{TEXT -1 1 " " }{TEXT 279 152 "( i) Calculer les valeurs propres et les vecteurs propres de S\n (ii ) Pour quelles valeurs de a ( a est un r\351el ) , les valeurs propres sont r\351elles ?" }}{PARA 299 "" 0 "" {TEXT -1 65 " (iii) Pour q uelles valeurs de a , S est-elle diagonalisable?" }}{PARA 0 "" 0 "" {TEXT 287 8 "r\351ponse:" }}}{SECT 0 {PARA 266 "" 0 "" {TEXT -1 18 "So it M la matrice " }}{EXCHG {PARA 267 "> " 0 "" {MPLTEXT 1 0 34 "M:= ma trice_colonnes([2,3],[1,0]):" }}}{EXCHG {PARA 268 "" 0 "" {TEXT 260 2 "Q3" }{TEXT -1 29 " Calculer M^n pour n =1..20\n" }{TEXT 261 2 "Q4" } {TEXT -1 81 " Diagonaliser M , M^2, M^3 comme cela a \351t\351 fait p our A au d\351but de la feuille ;" }}{PARA 269 "" 0 "" {TEXT -1 42 " \+ on notera PMk := " }{TEXT 273 9 "matrice_P" } {TEXT -1 15 "(M^k ) et QMk:=" }{TEXT 274 7 "inverse" }{TEXT -1 7 " (PM k)." }}{PARA 0 "" 0 "" {TEXT 262 118 " Quelle constatation pouve z-vous faire et pouvez-vous justifier ce que vous venez (\351ventuelle ment) de constater?" }}{PARA 0 "" 0 "" {TEXT 288 9 "r\351ponse :" }}}} {SECT 0 {PARA 304 "" 0 "" {TEXT -1 17 "Soit N la matrice" }}{EXCHG {PARA 305 "> " 0 "" {MPLTEXT 1 0 33 "N:=matrice_colonnes([0,1],[1,0]): " }}}{EXCHG {PARA 306 "" 0 "" {TEXT 281 2 "Q5" }{TEXT -1 220 " N est-e lle diagonalisable ? \n Calculer N^2; en d\351duire que, pour tou t k entier, N^(2*k+1)=N et que N^(2*k) = Id2 ( d'abord papier-crayon \+ que vous transcrirez dans votre session).\n V\351rifiez-le pour k =1..100 ! " }}{PARA 0 "" 0 "" {TEXT 289 8 "r\351ponse:" }}}}}{SECT 0 {PARA 310 "" 0 "" {TEXT -1 0 "" }{TEXT 285 43 "Si vous avez d\351j \340 termin\351, question bonus:" }}{PARA 312 "" 0 "" {TEXT 294 22 "So it R la matrice 4x4 " }}{EXCHG {PARA 313 "> " 0 "" {MPLTEXT 1 0 49 "R: =matrix(4,4,[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0]):" }}}{EXCHG {PARA 311 " " 0 "" {TEXT 295 3 "Q6 " }{TEXT -1 163 " Calculer les valeurs propres \+ et les vecteurs propres de R (ils ont 4 composantes!) : meme d\351fini tion que pour les matrices 2 x 2\n Calculer R^k pour k=2..15" } }}}}}{MARK "1 4 1 0 0" 105 }{VIEWOPTS 1 1 0 1 1 1803 }