Quadric Surfaces Worksheet
A simple worksheet to help you explore the relationship of
eigenvalues to graphs of quadratic forms. Try and predict the
nature of the picture before issuing the plot3d command !
> with(plots):
Warning, the name changecoords has been redefined
> with(linalg):
Warning, the protected names norm and trace have been redefined and unprotected
> setoptions3d(axes=boxed);
> v := vector(2,[x,y]);
> A1:=matrix(2,2,[1,4,4,1]);
> f1 := expand(dotprod(v,evalm(A1&*v))) + 3*x + 2*y -6;
> map(evalf,{eigenvals(A1)});
> plot3d(f1,x=-5..5,y=-5..5,view=-25 .. 25);
> A2:=matrix(2,2,[1,1,1,1]);
> f2 := expand(dotprod(v,evalm(A2&*v)));
> map(evalf,{eigenvals(A2)});
>
plot3d(f2,x=-5..5,y=-5..5,view=-25 .. 25);
> A3:=matrix(2,2,[4,1,1,1]);
>
> f3 := expand(dotprod(v,evalm(A3&*v)));
> map(evalf,{eigenvals(A3)});
>
plot3d(f3,x=-5..5,y=-5..5,view=-25 .. 25);
> A4:=matrix(2,2,[1,4,4,1]);
> f4 := expand(dotprod(v,evalm(A4&*v))) + 3*x + 2*y -6;
> map(evalf,{eigenvals(A4)});
>
plot3d(f4,x=-5..5,y=-5..5,view=-25 .. 25);
> A5:=matrix(2,2,[-4,1,1,-4]);
> f5 := expand(dotprod(v,evalm(A5&*v))) + 3*x + 2*y -6;
> map(evalf,{eigenvals(A5)});
>
plot3d(f5,x=-5..5,y=-5..5,view=-25 .. 25);
> A6:=matrix(2,2,[-4,4,4,-4]);
> f6 := expand(dotprod(v,evalm(A6&*v))) + 3*x + 2*y -6;
> map(evalf,{eigenvals(A6)});
>
plot3d(f6,x=-5..5,y=-5..5,view=-25 .. 25);
>
>