how to find volume of the ellipse not in the origin by using the eigenvalues

dear all,

i have matrix A has the information of an ellipse not in the origin. i need to find the volume of the ellipse using the eigenvalues but i got wrong volume. in 2-dimension i have points x enclosed by the ellipse that its information is A and b.

x = [ 0.55  0.0  ;
      0.25   0.35  
      -0.2   0.2     
      -0.25 -0.1   
      -0.0  -0.3  
      0.4  -0.2 ]';
  A=[ 2.4282   -0.0557
     -0.0557    2.9680]; 
  b=[-0.3393; -0.0562];
ei=eig(A)
vol=pi*sqrt(ei(1))*sqrt(ei(2));

the eigenvalues are ei=[ 2.4225, 2.9737 ]; the volume is 8.4320 which i think is wrong. can anyone help me to move the ellipse to the center to find the eigenvalues then the volume please. i tried to do that but its not correct.

many thanks,

Nadia,

Answer 1

[Torsten]

vol=pi/sqrt(det(A))

Best wishes

Torsten.

[nadia nadi]

Dear Torsten,

thanks for replying, i except the volume to be about 0.5132 or less using the the formula pi*a*b, but with yours its 1.1705 still large. see the figure please. i used also

vol=pi/(log(det(A)))

but still 1.5910.

many thanks, Nadia,

[Torsten]

Your figure looks incorrect, or I misunderstood your notation:

http://www.wolframalpha.com/input/?i=2.4282*x%5E2%2B2.968*y%5E2%2B2*(-0.0557)*x*y-0.3393*x-0.0562*y-1%3D0

Best wishes

Torsten.

[nadia nadi]

Dear Torsten,

thanks for the link, it seems the same matrix A but i have different center. this matrix result from the problem of finding the minimum volume ellipsoid (described by matrix A and vector b) that covers all of the points x by solving the optimization problem

 [n,m]= size(x)
  % Create and solve the model
  cvx_begin
      variable A(n,n) symmetric
      variable b(n)
      maximize( det_rootn( A ) ); %%or (n*log(det_rootn(A)))% minimize( - log_det( A ) )
      subject to
          norms( A * x + b * ones( 1, m ), 2 ) <= 1;
  cvx_end
  scatter ( x(1,:), x(2,:) ); 
    hold on
    k = convhull ( x(1,:), x(2,:) );
    plot ( x(1,k), x(2,k),'k' );
    noangles = 200;
    angles   = linspace( 0, 2 * pi, noangles );
    ellipse  = A \ [ cos(angles) - b(1) ; sin(angles) - b(2) ];
    plot( x(1,:), x(2,:), 'ko', ellipse(1,:), ellipse(2,:), 'b-' );
    axis on
     hold off

you cant run the cvx because it is a package work with Matlab but you can run the plot and see that it is different from the plot in that link.

thank you.

[Torsten]

In your problem formulation, you'll have to work with A^2 instead of A. If you do this, you get an approximate volume of 0.436.

Best wishes

Torsten.

[nadia nadi]

Dear Torsten,

that's great thanks, it gives good approximation. I also used this

ei=eig(A);
VOL=pi*(1/(ei(1)))*(1/(ei(2)))

and it gives me the same from your formula, now for three dimension I will write the two formulas like this

vol=(4/3)*pi/sqrt(det(A^2))
VOL=4/3*pi*(1/(ei(1)))*(1/(ei(2)))*(1/(ei(3)))

and for high dimension I need to rewrite this formula so i write this. do you think is correct?.

m=>4
mm=((m/2)+1)-1;%or(m/2)
mm=ceil(mm);
vol=1/sqrt(det(A^2))*((pi^(m/2))/factorial(mm))

many thanks,

Nadia