Harj. 6 AV

21.10.01 HA

1.

> restart: with(linalg): with(LinearAlgebra):

Warning, the name changecoords has been redefined

Warning, the protected names norm and trace have been redefined and unprotected

Warning, the assigned name GramSchmidt now has a global binding

> A:=<<1,5>|<1,-3>>;

> IdentityMatrix(2,2);

> A:=matrix(2,2,[1,1,5,-3]);

> E:=IdentityMatrix(2,2);

> p:=det(A-lambda*E);

> lam:=solve(p=0,lambda);

> ALI1:=A-lam[1]*E;ALI2:=A-lam[2]*E;

> eigenvectors(A);

> V:=<<1, -5>|<1, 1>>;

> b:=<0,1>;

> c:=LinearSolve(V,b);

> V.c;

>

3.

> restart: with(linalg): with(LinearAlgebra):

Warning, the name changecoords has been redefined

Warning, the protected names norm and trace have been redefined and unprotected

Warning, the assigned name GramSchmidt now has a global binding

> A:=<<2,-1>|<10,4>>;

> E:=IdentityMatrix(2,2):

> p:=det(A-lambda*E);

> lam:=solve(p=0,lambda);

> ALI1:=A-lam[1]*E;#ALI2:=A-lam[2]*E;

> gaussjord(ALI1);

> w:=<1-3*I,1>;

Tarkistus:

> eigenvectors(A);

> lambda:=lam[1]:

> alpha:=Re(lambda);beta:=Im(lambda);

> u:=map(Re,w);v:=map(Im,w);

> y:=exp(alpha*t)*((a*cos(beta*t)+b*sin(beta*t))*u+(b*cos(beta*t)-a*sin(beta*t))*v);

> eval(subs(t=0,y));

> V:=<u|v>;

a)

> LinearSolve(V,<1,0>);

> ya:=subs(a=%[1],b=%[2],y);

b)

> LinearSolve(V,<0,1>);

> yb:=subs(a=%[1],b=%[2],y);

> evalm(ya);

> yya:=exp(3*t)*(-1/3*sin(3*t)+cos(3*t)), -1/3*exp(3*t)*sin(3*t);

> plot([yya,t=-5..1]);

> evalm(yb);

> yyb:=10/3*exp(3*t)*sin(3*t), exp(3*t)*(cos(3*t)+1/3*sin(3*t)):

> plot([yyb,t=-5..1]);

>

4.

> restart: with(linalg): with(LinearAlgebra):

Warning, the name changecoords has been redefined

Warning, the protected names norm and trace have been redefined and unprotected

Warning, the assigned name GramSchmidt now has a global binding

> A:=(1/50)*<<-4,4>|<1,-4>>;

> eigenvectors(A);

> om:=Eigenvectors(A);

> lam:=om[1]:ov:=om[2]:

> y:=C1*exp(lam[1]*t)*Column(ov,1)+C2*exp(lam[2]*t)*Column(ov,2);

> eval(subs(t=0,y));

> CC:=LinearSolve(ov,<25,0>);

> Y:=subs(C1=CC[1],C2=CC[2],y);

> evalm(Y);

> y1:=25/2*exp(-1/25*t)+25/2*exp(-3/25*t);y2:=25*exp(-1/25*t)-25*exp(-3/25*t);

> plot([y1,y2],t=0..20);

> t0:=solve(y1=y2,t);

> evalf(%);

> plot([y1,y2],t=20..50);

> plot([y1,y2],t=50..100);

>

5.

> restart: with(linalg): with(LinearAlgebra):

Warning, the name changecoords has been redefined

Warning, the protected names norm and trace have been redefined and unprotected

Warning, the assigned name GramSchmidt now has a global binding

> A:=<<8,3>|<-14,-4>>;

> E:=IdentityMatrix(2,2):

> p:=det(A-lambda*E);

> lam:=solve(p=0,lambda);

> ALI1:=A-lam[1]*E;

> gaussjord(ALI1);

> om:=Eigenvectors(A);

> lambda:=(om[1])[1];w:=Column(om[2],1);

> alpha:=Re(lambda);beta:=Im(lambda);

> u:=map(Re,w);v:=map(Im,w);

> y:=exp(alpha*t)*((a*cos(beta*t)+b*sin(beta*t))*u+(b*cos(beta*t)-a*sin(beta*t))*v);

>