equation describes either an ellipsoid , or a hyperboloid of $k$ sheets, depending on the number $k$ of negative eigenvalues of $A$, which is an invariant under similarity or change of basis transformations. (If $A\mathbb{x}=\lambda\mathbb{x}$ and $B=TAT^{-1}$ for $T$ invertible then $\mathbb{y}=B\mathbb{x}$ satisfies $B\mathbb{y}=\lambda\mathbb{y}$ .) I have oversimplified because actually the