restart: with(LinearAlgebra): 6.1: Definerer A og A-\316\273I som v\303\246rende K: A:=<<0|1/3|1/2>,<1/2|1/3|0>,<1/2|1/3|1/2>>;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKnNsIlE7

K:=A-lambda*IdentityMatrix(3); Finder determinanten af K, som er det karakteristiske polynomium, jeg derefter definerer som p (\316\273): Determinant(K); p:=lambda->Determinant(K); L\303\270ser for p(\316\273)=0 for at finde egenv\303\246rdierne: solve(p(lambda)=0, lambda); Der er alts\303\245 tre egenv\303\246rdier, der alle er komplekse. I Maple kan jeg finde det karakteristiske polynomium direkte ved kommandoen CharacteristicPolynomial() og egenv\303\246rdierne kan findes med Eigenvalues(): CharacteristicPolynomial(A,lambda); Eigenvalues(A); NullSpace(A-1*IdentityMatrix(3)); NullSpace(A-(-1/12+(1/12)*sqrt(13))*IdentityMatrix(3)); NullSpace(A-(-1/12-(1/12)*sqrt(13))*IdentityMatrix(3)); Med kommandoen Eigenvectors() kan jeg finde egenvektorerne for matricen og s\303\246tte disse sammen til matricen X: L,X:=Eigenvectors(A);

NiQtJkknVmVjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJ2NvbHVtbkdGKTYjL0kkJWlkR0YpIiprUkR2Ii1JJ01hdHJpeEdGJjYjL0YuIiovOVd2Ig==

simplify(X);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKmNRYXYi

simplify(X^(-1).A.X);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKmc8R3gi

Det var D, nu kan jeg g\303\245 videre til at finde hvad der sker n\303\245r jeg ta'r potenser: Dn:=<<1^n|0|0>,<0|(-(1/156)*sqrt(13)*(-13+sqrt(13)))^n|0>,<0|0|(-(1/156)*sqrt(13)*(13+sqrt(13)))^n>>;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKid6UyJ5Ig==

e[3]:=<0|0|1>; e[1]:=<1,0,0>; e[2]:=<0,1,0>;

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LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqVzkyeSI=

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqbzk8XyI=

simplify((e[3]).X.Dn.X^(-1).(e[1]));

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Dette er korrekt og jeg har alts\303\245 eftervist mit udtryk. Den tilsvarende for e_2 i stedet for e_1 bliver: simplify((e[3]).X.Dn.X^(-1).(e[2]));

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N\303\245r de to her g\303\245r imod uendelig for n f\303\245r jeg: limit(6/13-(3/13)*(-1/12+(1/12)*sqrt(13))^n-(3/13)*(-1/12-(1/12)*sqrt(13))^n,n=infinity);

IyIiJyIjOA==

limit(6/13-(1/13)*sqrt(13)*(-1/12+(1/12)*sqrt(13))^n-(3/13)*(-1/12+(1/12)*sqrt(13))^n+(1/13)*sqrt(13)*(-1/12-(1/12)*sqrt(13))^n-(3/13)*(-1/12-(1/12)*sqrt(13))^n,n=infinity);

IyIiJyIjOA==

6.2 (SKAL laves i h\303\245nden, dette er bare til tjek): Definerer A: A:=<<0|1|0>,<-1|0|-2>,<0|0|2>>;

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L,Y:=Eigenvectors(A);

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

Y^(-1);

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Y^(-1).A.Y;

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A.Y;

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