{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "MS Sans Serif" 1 8 128 0 128 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 9 128 0 128 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Fo nt 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 8 0 128 128 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Seitenumbruch" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 1 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 4 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 38 "Zu den Beispielen 11 und 12 in \247 14: " }}{PARA 3 "" 0 "" {TEXT 259 42 "Interpolatio n und orthogonale Idempotente," }}{PARA 3 "" 0 "" {TEXT 261 62 "Darste llung von A als direkte Summe mit Hilfe der Idempotenten" }}{PARA 3 " " 0 "" {TEXT 262 60 "und Beispiel zu unserem Beweis des Satzes von Sti ckelberger:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "Es handelt sich um ein absicht lich \"kleines\" und einfaches Beispiel, an dem aber bereits unsere wi chtigsten Ergebnisse in \247 14 verdeutlicht werden k\366nnen." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "Das Arbe itsblatt muss ohne Auslassung von oben nach unten durchlaufen werden, \+ da Zwischenergebnisse weiterbenutzt werden." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart:with(Groebner):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "w ith(Ore_algebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "A:=pol y_algebra(x,y):T:=termorder(A,tdeg(y,x));\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 256 20 "benutzte Prozeduren:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 955 "Kbasis :=\n \n proc(PList,VList,torder)\n local B,Bs,C,G,i,v,t,l,m,pB,r;\n if \+ is_finite(PList,VList) then\n G := gbasis(PList,torder);\n \+ B := [1];\n for v in VList do\n m := degree(u nivpoly(v,G,VList),v);\n C := B;\n for t in C \+ do\n for l to m-1 do\n t := t*v;\n if normalf(t,G,torder) = t then\n \+ B := [op(B),t]\n fi\n od \n od\n od;\n Bs:=B;\n r:=nops(B); \n pB := 0;\n for i to r do pB := pB+B[i] end do;\n \+ i := 0; Bs := []; while pB <> 0 do\n l := Groebner:- leadmon(pB,torder);\n i := i+1;\n Bs := [l[2], o p(Bs)];\n pB := pB-l[2]\n end do;\n B:=Bs;\n \+ RETURN(B)\n else\n ERROR(`Das Ideal ist nicht null- dimensional, und es existiert keine endliche Basis.`)\n fi\n end: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1473 "holMatrix := proc(f,KbasisLis t,GBList,VList,torder)\n local KK,GG,r,ff,fff,XX,i,k,l,Ku, pKu;\n global B,Mf;\n Ku := KbasisList;\n \+ r := nops(Ku);\n XX := VList;\n \+ GG := GBList;\n #\n # zuerst wird \+ \"Ku\" sortiert gem\344\337 \"torder\" :\n #\n \+ pKu := 0;\n for i to r do pKu := pKu+Ku[i] od;\n i := 0;\n KK := [];\n \+ while pKu <> 0 do l := leadmon(pKu,torder); i := i+1; KK := [l[2],op (KK)]; pKu := pKu-l[2] od;\n B:=KK;\n \+ #print(`Die nach der vorgegebenen Monomordnung sortierte K-Basis ist : B = `,B);\n #\n # jetzt wird die Matri x \"Mf\" bestimmt :\n #\n ff := f;\n \+ Mf := matrix(r,r,0);\n for i to r do\n \+ fff := ff*KK[i];\n R||i := normal f(fff,GG,torder);\n while R||i <> 0 do\n \+ l := leadmon(R||i,torder);\n fo r k to r do if KK[k] = l[2] then Mf[k,i] := l[1]; k := r fi od;\n \+ R||i := R||i-l[1]*l[2]\n od\n od;\n #print(`Die Matrix` ,` M`[f],` \+ der Multiplikation mit f` = f,` in `,'K'*XX*`/ I`,\n \+ # ` bezueglich dieser K-Basis ist`);\n evalm(Mf );\n end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 553 "zdimradic al := proc(PList,VList)\n local p,pred,v,RList;\n \+ if is_finite(PList,VList) then\n RLi st := PList;\n for v in VList do\n \+ p := univpoly(v,PList,VList); \n\011\011 pred : = simplify(p/gcd(p,diff(p,v))); \n RList := \+ [op(RList),pred]\n od;\n R ETURN(RList)\n else ERROR(`Das vorgegebene Ideal ist nicht nulldimensional !!!`)\n fi\n en d:" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 257 15 "zu Beispiel 11:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Das Ideal I sei erzeugt von den folgenden Polyn omen " }{XPPEDIT 18 0 "f[1],f[2];" "6$&%\"fG6#\"\"\"&F$6#\"\"#" } {TEXT -1 3 " :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f:=[x*y+x+y, x^2 +2*x+y^2+2*y];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Die L\366sungen lassen sich bei diesm sehr einfachen Beispile leicht bestimmen, z.B. \+ so:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "gbasis(f,plex(x,y)); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "factor(%[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "gbasis(f,plex(x,y));" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "factor(%[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "#solve(\{op(f)\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Die L\366sungen sind demnach:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "v1:=vector([0,0]);v2:=vector([-2,-2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "eine trennende Linearform:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "l:=vector([1,1]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "der Variablenvektor:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "X:=[x,y];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "N ach \247 8 der Vorlesung sind damit (Interpolation zur Gewinnung der \+ " }{XPPEDIT 18 0 "g[i];" "6#&%\"gG6#%\"iG" }{TEXT -1 3 " )" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "s:=(u,v)->dotprod(u,v,orthog onal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "g1:=(s(l,X)-s(l,v 2))/(s(l,v1)-s(l,v2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "g 2:=(s(l,X)-s(l,v1))/(s(l,v2)-s(l,v1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Im Folgenden wird mit der Gr\366bnerbasis bez\374glich de r oben festgelegten Monomordnung T gerechnet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Wie erwartet gilt mit de n soeben bestimmten Polynomen g1, g2 :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "gbasis(zdimradical([op(G),g1,g2],[x,y]),T);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Mit Hilfe der " }{XPPEDIT 18 0 "g[i];" "6#&%\"g G6#%\"iG" }{TEXT -1 41 " sind nun die Idempotenten zu bestimmen:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "#map(sort,[op(G),g1,g2],[x,y],plex) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "#M:=matrix(nops(G)+2,1,%);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "q:=1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "gbasis([op(G),(g1*g2)^q],T)-G;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Es gen\374gt also chon " }{XPPEDIT 18 0 "q = 1;" "6 #/%\"qG\"\"\"" }{TEXT -1 3 " !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Zur Abk\374rzung sei:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 46 "nf:=proc(u,v) global G,T;normalf(u*v,G,T);en d;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Man berechnet:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "matrix(2,2,(i,j)->nf(g||i,g||j));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "In diesem Beispiel sind demnach die " }{XPPEDIT 18 0 "g[i];" "6#&%\"gG6#%\"iG" }{TEXT -1 45 " schon \+ selbst die orthogonalen Idempotenten." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "B:=Kbasis(G,[x,y],T);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Basisbestimmung f\374r A1:=g1*A und A2:=g2*A durch Verk \374rzung der Erzeugendensysteme g1*B, g2*B:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "`Erzeugendensystem von A1` =matrix(4,1,[seq(normalf(g 1*B[k],G,T),k=1..4)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "` Erzeugendensystem von A2` =matrix(4,1,[seq(normalf(g2*B[k],G,T),k=1..4 )]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Basen von A1 und A2 sind \+ demnach: " }{XPPEDIT 18 0 "\{g1, h1\};" "6#<$%#g1G%#h1G" }{TEXT -1 6 " und " }{XPPEDIT 18 0 "\{g2, h2\};" "6#<$%#g2G%#h2G" }{TEXT -1 7 " mit " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "h1:=x^2+3*x-y;h2:=x^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "wie wird multipliziert in A1 un d A2 ??" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nf(g1,h1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nf(h1,h1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nf(g2,h2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nf(h2,h2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "or thogonal ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nf(h1,g2);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nf(h2,g1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Sind die \+ beiden Algebren A1 und A2 mit den Einselementen g1 bzw. g2 isomorph ? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Ich beantworte diese Frage mit Hilfe der Minimalpolynome von Eleme nten aus A1 und A2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Das Minima lpolynom von " }{XPPEDIT 18 0 "h1 = x^2+3*x-y;" "6#/%#h1G,(*$%\"xG\" \"#\"\"\"*&\"\"$F)F'F)F)%\"yG!\"\"" }{TEXT -1 8 " in A2:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 6 "g1,h1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "nf(h1^2,g1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "das Minim alpolynom ist demnach:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "mh 1:=t^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "nf(subs(t=h2,mh1),g1);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 24 "Das Minimalpolynom von " }{XPPEDIT 18 0 "h2 = x^2;" "6 #/%#h2G*$%\"xG\"\"#" }{TEXT -1 8 " in A2:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 6 "g2,h2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "nf(h2^2-8*h2+16*g2,g2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "das M inimalpolynom ist demnach:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "mh2:=t^2-8*t+16;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "nf(subs(t=h2,mh2),g2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Gibt es in A2 ein Element mit Minimalpoly nom " }{XPPEDIT 18 0 "mh2 = t^2;" "6#/%$mh2G*$%\"tG\"\"#" }{TEXT -1 3 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "k:=(a*g2+b*h2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "A:=poly_algebra(x,y,a,b,rational=a,rational=b):T:=ter morder(A,tdeg(y,x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nf(k ,1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Durch Subtrahieren erh \344lt man:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "nf(k^2-2*(a+4*b)*k+4 *(1/4*a^2+2*a*b+4*b^2)*g2,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "mk:=t^2-2*(a+4*b)*t+4*(1/4*a^2+2*a*b+4*b^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "nf(subs(t=k,mk),g2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Durch Wahl von a erreicht man das Polynom t^2:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(b=-1/4*a,mk);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "w:=factor(subs(b=-1/4*a,k));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "W\344hle am besten a=4:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "k2:=subs(a=-4,w);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Probe:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "nf(k2,k2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "nf(k2,g2) - k2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 260 15 "zu Beispiel 12:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Die orthogonalen Idempotenten waren:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 6 "g1;g2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 " Die alte Basis war:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "'B'=B ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Die neue Basis ist " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Bn:=map(sort,[g1,h1,g2,h2],[x,y],td eg,ascending);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Die Komponenten darstellungen der Basispolynome aus B sind" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "matrix(4,2,(i,j)->nf(B[i] ,g||j));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "Wie sieht die Basiswechselmatrix aus f\374r den Wechsel \+ von der Standardmonombasis zur neuen Basis \{g1, h1, g2, h2\} aus ??? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "v1:=vector([1,1/4,1/4,0]):v2:=vector([0,3,-1,1]):w1:=vector([0,-1 /4,-1/4,0]): w2:=vector([0,0,0,1]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "P:=concat(v1,v2,w1,w2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Wa hl von h:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "h:=x-y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "#nf(h,g1 ),nf(h,g2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Matrix von Mh bez \374glich B:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Mh:=holMatrix(h,B,G ,X,T);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Damit erh\344lt man f \374r die Matrix MBh bez\374glich der neuen Basis:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "MBh:=evalm(P^(-1)&*Mh&*P);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "charpoly(MBh,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "MBh muss sich trigonalisieren lassen:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "U:=submatrix(MBh,3..4,3..4):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(U):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 " Q:=evalm(diag(matrix(2,2,[1,0,0,1]),matrix(2,2,[2,-4,2,1]))&*diag(1,1/ 2,1,-5)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalm(Q^(-1)&*MBh&*Q); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0 1" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }