Given a set of 3 orthogonal vectors, how can I find a minimum volume enclosing ellipsoid expressed in the Cartesian coordinate frame? Generalized Problem
Given values to start the problem:


*

*A 3D orthonormal coordinate frame (we'll call it the 'V' coordinate frame) that is rotated from the global coordinate system (we'll call it 'G') but shares a common origin.

*3 magnitudes representing the semi-major axes of an ellipsoid defined in the 'V' coordinate frame

*All necessary information (angles, etc.) required to fully describe the system.


Desired output:


*

*3 magnitudes representing the semi-major axes of an ellipsoid expressed in the 'G' coordinate frame that completely encompasses the input ellipsoid.


I realize that there may be different solutions depending on the type of ellipsoid-fit. I believe what I desire is a minimum volume enclosing ellipsoid. I don't believe that a mere direction cosine matrix rotation is a sufficient solution to be physically meaningful.
Problem Background
To give context for this problem, I am trying to geolocate a ground feature using Line-of-Bearing measurements (with range measurements) from a quad-copter in the air the stationary ground feature. I am trying to convert the sensor covariances ($\sigma_{psi}$, $\sigma_{theta}$, and $\sigma_{range}$) of the sensor to their minimum equivalent values represented in the global coordinate system (North, East, and Up in this instance).
Given that $\sigma_{psi}$ and $\sigma_{theta}$ are expressed as covariance of an angular measurement, their values can be converted to a distance using the approximation of $s=r\theta$ or any similar equation. $\sigma_{range}$ is already expressed in units of distance.
Depending on the location and altitude of the quad-copter, the alignment of the 'V' coordinate system with 'G' will change.
As I will be tracking the location of the target using an estimator (in the global coordinate frame), I want to know the equivalent covariances in XYZ of each measurement as they depend on the location and slant range of the quad-copter and target.
Please let me know if any further information or explanations are needed. Thanks in advance for any help!
 A: I have a partial solution.
First, to find a sphere enclosing a given ellipsoid with semiaxes $\vec v_i = ((v_i)_j)$, you would need the largest singular value of the matrix $V$ with elements $v_{ij} := (v_i)_j$:
$$r = \sqrt{\max \sigma(V^T V)}$$
Now this also works when the vectors $\vec v_i$ are not orthogonal, i.e., when the ellipsoid is deformed by a linear transform along with its spanning vectors. So you can try different semiaxis ratios by scaling the problem in $(x, y, z)$, finding an enclosing sphere, and scaling back to deform the sphere to an ellipsoid.
If the scaling transform has a determinant of 1, that is, if the scale factors $\alpha_x, \alpha_y, \alpha_z$ in the three axes multiply to 1, then the volume of the ellipsoid found this way is identical to the volume of the sphere and thus proportional (cubically) to
$$\tilde r = \sqrt{\max \sigma(D V^T V D)},$$
where
$$D = \mathrm{diag}(\alpha_x, \alpha_y, \alpha_z), \qquad \alpha_x\alpha_y\alpha_z = 1.$$
So the problem really is to find a constrained minimum of the maximal eigenvalue of $D V^T V D$ where $D$ conforms to the conditions above.
I don't know a trick to do this reduced problem but someone might take it from here. But you can finish it using a direct approach. The characteristic equation is cubic and a priori known to have three positive roots so it's possible in principle to express the maximal eigenvalue analytically and then differentiate it by $\alpha_i$ and use the method of Lagrange coefficients to find the constrained extreme.
Or, since you are going to implement this on a computer anyway, this is an optimization problem with two independent variables and a very quick evaluation, so just use any stochastic optimization strategy. It might be a good idea to encode $\alpha_i = \exp \beta_i$ and convert $\prod\alpha_i = 1$ to $\sum\beta_i = 0$.

Addendum: By the way, the problem becomes very simple if you only ask for the smallest-volume ellipsoid enclosing the three vectors alone, not an ellipsoid defined by them. It's easy to prove that would be the ellipsoid uniquely defined by containing the three vectors, its semiaxes are related to the solutions to the linear system of equations
$$\sum_{j\in\{x,y,z\}} a_j v_{ij}^2 = 1, \quad \forall i \in \{1,2,3\}$$
by $a_j = r_j^{-2}$. Maybe that could be sufficient for your application?
