dimension of space of origin-symmetric ellipsoids

Question

I wonder how I can compute the dimension of the space of all origin-symmetric ellipsoids? What is the best approach?

Answer 1

All the ellipsoid with center at the origin and axis parallel to the coordinate axis, are symmetric with respect the origin and, in $n$ dimensional space, are represented by a quadratic form $\vec x^T A \vec x=1$ where $A$ is a diagonal matrix with all diagonal elements positive (the reciprocals of the squares of the semi-axes).

Any rotation $R$ in $SO(n)$ transform an ellipsoid of this kind to another ellipsoid symmetric with respect the origin, represented by the equation $$ \vec x^T (RAR^{-1}) \vec x=1 $$

Now, the space of the diagonal matrices $A$ has dimension $n$, The dimension of $SO(n)$ is $\frac{n(n-1)}{2}$, so the dimension of the space of origin-symmetric ellipsoid is $d=n+\frac{n(n-1)}{2}=\frac{n(n+1)}{2}$.