sol54.mw

 > with(LinearAlgebra):

54

 > A:=Matrix([[10,-53],[10,12]]);

 > E:=Eigenvectors(A);

 > M:=E[2];

 > C:=M^(-1).A.M;

 > N:=1/2*Matrix([[1,-I],[1,I]]);

 > M2:=M.N;

 > C2:=M2^(-1).A.M2;

 > C2E:=exp(11*t)*Matrix([[cos(23*t),sin(23*t)],[-sin(23*t),cos(23*t)]]);

 > AE:=M2.C2E.M2^(-1);

 > AE.Matrix([[23],[23]]);

 >

 >

56

 > A:=Matrix([[-9,108,-92],[10,-90,79],[12,-124,107]]);

 > E:=Eigenvectors(A);

 > M:=E[2];

 > N:=Matrix([[1/2,-1/2*I,0],[1/2,1/2*I,0],[0,0,1]]);

 > M2:=M.N;

 > C:=M2^(-1).A.M2;

 > CE:=Matrix([[exp(2*t),0,0],[0,exp(3*t)*cos(4*t),exp(3*t)*sin(4*t)],[0,-exp(3*t)*sin(4*t),exp(3*t)*cos(4*t)]]);

 > AE:=M2.CE.M2^(-1);

 >

 >

57

 > A:=Matrix([[-723,280,2744],[-91,40,343],[-182,70,691]]);

 > g:=Matrix([[exp(5*t)],[3*t+7],[20]]);

 > E:=Eigenvectors(A);

 > M:=E[2];

 > C:=M^(-1).A.M;

 > h:=M^(-1).g;

 > s1:=solve(dsolve(diff(x(t),t)=5*x(t)+h[1,1]),x(t));

 > s2:=solve(dsolve(diff(x(t),t)=5*x(t)+h[2,1]),x(t));

 > s3:=solve(dsolve(diff(x(t),t)=-2*x(t)+h[3,1]),x(t));

 > y:=Matrix([[s1],[subs(_C1=c2,s2)],[subs(_C1=c3,s3)]]);

 > M.y;

 >

58

 > A:=Matrix([[1,3,0,0,0,0,-1,0],[0,18,-5,0,6,-25,0,-2],[0,64,-18,0,24,-100,0,-8],[-2,6,0,2,0,-1,-2,0],[0,116,-34,0,43,-170,0,-14],[0,0,0,0,0,2,0,0],[0,45,-13,0,15,-65,1,-5],[0,322,-94,0,114,-470,0,-37]]);

 > M:=JordanForm(A,output='Q');

 > J:=M^(-1).A.M;

 > Je:=MatrixExponential(J*t);

 > Ae:=M.Je.M^(-1);

 > Ae.Matrix([[1],[0],[1],[0],[0],[0],[0],[0]]);

 >