Magma Instruktionen in der LA II


Q:=RationalField();

A:=Matrix(Q,3,3,[1,2,3,4,5,6,7,8,9]);
A;
Determinant(A);


A:=Matrix(Q,3,3,[1,2,3,4,5,6,7,8,8]);
A;
Determinant(A);
A^-1;
A^10;
Trace(A^1000);

CharacteristicPolynomial(A);
chi<T>:=CharacteristicPolynomial(A);
Factorization(chi);

A:=Matrix(Q,3,3,[1,2,3,4,5,6,7,8,7]);
A;
chi<T>:=CharacteristicPolynomial(A);
Factorization(chi);
Eigenvalues(A);

A:=Matrix(Q,3,3,[1,2,3,4,5,6,7,8,9]);
chi_A<T>:=CharacteristicPolynomial(A);
Factorization(chi_A);
Eigenvalues(A);

L<omega>:=NumberField(T^2 - 15*T - 18);
omega^2;
B:=ChangeRing(A,L);
chi_B<T>:=CharacteristicPolynomial(B);
Factorization(chi_B);
Eigenvalues(B);
JordanForm(B);

p:=11;
F_p:=FiniteField(p);
N:=ScalarMatrix(5,F_p!0);
N[2,1]:=1; N[3,2]:=1; N[5,4]:=1;
N;


chi_N<T>:=CharacteristicPolynomial(N);
mu_N<T>:=MinimalPolynomial(N);
chi_N; mu_N;

print "";
S:=RandomUnimodularMatrix(1000,5);
S:=ChangeRing(S,F_p);
S;
Determinant(S);
S^-1;

print "";
A:=S*N*S^-1;
chi_A<T>:=CharacteristicPolynomial(A);
A;
print "";
JordanForm(A);
Transpose(JordanForm(A));


print "";
for i:=1 to 5 do 
	B:=A^i;
	B; 5-Rank(B);
end for;




Q:=RationalField();
R<T>:=PolynomialRing(Q);
R;
P:=T^7+3*T^4-6*T^2+1;
Q:=T^3+T^2-5*T+3;
Gcd(P,Q);
Factorization(P);
Factorization(Q);

P:=(T-1)*(T-2)*(T^3+7*T^3-8);
G:=GreatestCommonDivisor(P,Q);

ExtendedGreatestCommonDivisor(P,Q);
G,A,B:=ExtendedGreatestCommonDivisor(P,Q);
G eq A*P+B*Q;


