load(eigen);

/*    yl 365 a   */

A: matrix(
[1,1],
[1,1]
);

p: expand(factor(charpoly(A,x)));
sols: solve(p=0,x);
sols: [ev(x,sols[1]),ev(x,sols[2])];

A-sols[1]*ident(2);
%.matrix([x],[y]);
transpose(%);
args(%)[1];
solve(%,[x,y]);
e1: [-1,1];
e1: 1/sqrt(innerproduct(e1,e1))*e1;

A-sols[2]*ident(2);
%.matrix([x],[y]);
transpose(%);
args(%)[1];
solve(%,[x,y]);
e2: [1,1];
e2: 1/sqrt(innerproduct(e2,e2))*e2;

AA: matrix(
[sols[1],0],
[0,sols[2]]
);
V: matrix(e1,e2);


/*    yl 365 c   */

A: matrix(
[11,2,-8],
[2,2,10],
[-8,10,5]
);

p: expand(factor(charpoly(A,x)));
expand((63-5*x)*(180-49*x+x^2)-2*(18-4*x)*(63-5*x)-(18-4*x)*(585-58*x+x^2));
sols: solve(p=0,x);
sols: [ev(x,sols[1]),ev(x,sols[2]),ev(x,sols[3])];

A-sols[1]*ident(3);
%.matrix([x],[y],[z]);
transpose(%);
args(%)[1];
solve(%,[x,y,z]);
e1: [-2,1,2];
e1: 1/sqrt(innerproduct(e1,e1))*e1;

A-sols[2]*ident(3);
%.matrix([x],[y],[z]);
transpose(%);
args(%)[1];
solve(%,[x,y,z]);
e2: [2,2,1];
e2: 1/sqrt(innerproduct(e2,e2))*e2;

A-sols[3]*ident(3);
%.matrix([x],[y],[z]);
transpose(%);
args(%)[1];
solve(%,[x,y,z]);
e3: [1,-2,2];
e3: 1/sqrt(innerproduct(e3,e3))*e3;

AA: matrix(
[sols[1],0,0],
[0,sols[2],0],
[0,0,sols[3]]
);
V: matrix(e1,e2,e3);


/*    yl 365 e   */

A: matrix(
[17,-8,4],
[-8,17,-4],
[4,-4,11]
);

p: expand(factor(charpoly(A,x)));
sols: solve(p=0,x);
sols: [ev(x,sols[1]),ev(x,sols[2]),ev(x,sols[2])];

A-sols[1]*ident(3);
%.matrix([x],[y],[z]);
transpose(%);
args(%)[1];
solve(%,[x,y,z]);
e1: [2,-2,1];
e1: 1/sqrt(innerproduct(e1,e1))*e1;

A-sols[2]*ident(3);
%.matrix([x],[y],[z]);
transpose(%);
args(%)[1];
solve(%,[x,y,z]);
e2: [1,1,0];
e3: [-1,0,2];
e3: gramschmidt(matrix(e2,e3))[2];

e2: 1/sqrt(innerproduct(e2,e2))*e2;
e3: 1/sqrt(innerproduct(e3,e3))*e3;


AA: matrix(
[sols[1],0,0],
[0,sols[2],0],
[0,0,sols[3]]
);
V: matrix(e1,e2,e3);


/*    yl 365 f   */

A: matrix(
[1,1,1,1],
[1,1,-1,-1],
[1,-1,1,-1],
[1,-1,-1,1]
);

p: factor(charpoly(A,x));
sols: solve(p=0,x);
sols: [ev(x,sols[1]),ev(x,sols[2]),ev(x,sols[2]),ev(x,sols[2])];

A-sols[1]*ident(4);
%.matrix([x],[y],[z],[u]);
transpose(%);
args(%)[1];
solve(%,[x,y,z,u]);
e1: [-1,1,1,1];
e1: 1/sqrt(innerproduct(e1,e1))*e1;

A-sols[2]*ident(4);
%.matrix([x],[y],[z],[u]);
transpose(%);
args(%)[1];
solve(%,[x,y,z,u]);
e2: [1,1,0,0];
e3: [1,0,1,0];
e4: [1,0,0,1];
e3: gramschmidt(matrix(e2,e3,e4))[2];
e4: gramschmidt(matrix(e2,e3,e4))[3];

e2: 1/sqrt(innerproduct(e2,e2))*e2;
e3: 1/sqrt(innerproduct(e3,e3))*e3;
e4: 1/sqrt(innerproduct(e4,e4))*e4;


AA: matrix(
[sols[1],0,0,0],
[0,sols[2],0,0],
[0,0,sols[3],0],
[0,0,0,sols[4]]
);
V: matrix(e1,e2,e3,e4);