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

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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,

Respuestas (1)

Torsten
Torsten el 24 de Abr. de 2017
Editada: Torsten el 24 de Abr. de 2017
vol=pi/sqrt(det(A))
Best wishes
Torsten.
  5 comentarios
Torsten
Torsten el 25 de Abr. de 2017
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
nadia nadi el 25 de Abr. de 2017
Editada: nadia nadi el 25 de Abr. de 2017
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

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