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);

>

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