In "An algebraic condition for the separation of two ellipsoids" (https://i.cs.hku.hk/~ykchoi/quadrics/Ellipsoid_Separation.pdf) , the authors offer conditions for ellipsoid intersection that depend on expressing ellipsoids as
$$ \mathcal{A}: X^{T} A X, \\ \mathcal{B}: X^{T} B X, $$
where $X = (x,y,z,w)^{T}$, $A$ or $B$ is a 4 by 4 matrix, and their "characteristic equation" is defined as $$ f(\lambda) = \text{det}(A \lambda + B) = 0. $$ If I want to deal with ellipsoids that are defined more normally, like $$ \mathcal{C}: (X-X_0)^{T} C (X-X_0) $$ with $X = (x,y,z)^{T}$ and now $X_0$ including the center coordinates, so $C$ is 3 by 3, can I still use the same conditions for intersection? I would think the characteristic equation must change (from maximally 4 roots to maximally 3), and I can't use the same conditions. Is there a simple way to convert from the form of $C$ to that of $A$ or $B$? I want to so I can easily rotate the ellipsoid first (I can't apply a 3 by 3 rotation matrix to an ellipsoid equation that uses a 4 by 4 center matrix).