What is the volume ratio of the minimal volume ellipsoid?

Transforming the homogeneous coordinates, read column wise, of the original points $\bm{o}$ to the homogeneous coordinates, also read column wise, of the vertices of the regular tetrahedron circumscribed by the unit radius ball $\bm{n}$ by solving the matrix equation: $\bm{a}.\bm{o}=\bm{n}$ , availing $\bm{a}^{-1}$ to go from the unit ball to the ellipsoid, computing the singular values decomposition of $\bm{a}^{-1}$ , computing the product of the scaling matrix therefrom gives the volume ratio: $0.974278579257494$ .

$\bm{a}.\bm{o}=\bm{n}$

$\bm{a}.\left( \begin{array}{cccc} -\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ -\frac{3}{4} & -\frac{3}{4} & -\frac{3}{4} & \frac{9}{4} \\ -\frac{1}{4} & \frac{3}{4} & -\frac{1}{4} & -\frac{1}{4} \\ 1 & 1 & 1 & 1 \\ \end{array} \right)=\left( \begin{array}{cccc} 1 & -\frac{1}{3} & -\frac{1}{3} & -\frac{1}{3} \\ 0 & \frac{2 \sqrt{2}}{3} & -\frac{\sqrt{2}}{3} & -\frac{\sqrt{2}}{3} \\ 0 & 0 & -\sqrt{\frac{2}{3}} & \sqrt{\frac{2}{3}} \\ 1 & 1 & 1 & 1 \\ \end{array} \right)$

$\bm{a}.\bm{o}.\bm{o}^{-1}=\bm{n}.\bm{o}^{-1} \Rightarrow \bm{a}=\bm{n}.\bm{o}^{-1}$

$\bm{a}=\left( \begin{array}{cccc} -1.33333 & 0. & -1.33333 & 0. \\ -0.471405 & 0. & 0.942809 & 0. \\ -0.816497 & 0.544331 & 0. & 0. \\ 0. & 0. & 0. & 1. \\ \end{array} \right)$

$\bm{a}^{-1}=\bm{o}.\bm{n}^{-1}$

$\bm{a}^{-1}=\left( \begin{array}{cccc} -0.5 & -0.707107 & 0. & 0. \\ -0.75 & -1.06066 & 1.83712 & 0. \\ -0.25 & 0.707107 & 0. & 0. \\ 0. & 0. & 0. & 1. \\ \end{array} \right)$

The singular values decomposition if the affine matrix of $\bm{a}^{-1}=\left( \begin{array}{ccc} -0.5 & -0.707107 & 0. \\ -0.75 & -1.06066 & 1.83712 \\ -0.25 & 0.707107 & 0. \\ \end{array} \right)$ gives:

$\left( \begin{array}{ccc} 0.241984 & -0.659683 & 0.711521 \\ 0.961534 & 0.261277 & -0.0847705 \\ -0.129983 & 0.704665 & 0.697533 \\ \end{array} \right), \\ \left( \begin{array}{ccc} 2.32845 & 0. & 0. \\ 0. & 0.839641 & 0. \\ 0. & 0. & 0.498337 \\ \end{array} \right), \\ \left( \begin{array}{ccc} -0.34772 & -0.0503582 & -0.936245 \\ -0.55096 & 0.818937 & 0.160577 \\ 0.758639 & 0.57167 & -0.312506 \\ \end{array} \right)$

The product of the diagonal elements of the scaling (center) matrix is the scaling factor requested: 0.974278579257494.

The computation was done in double precision and reported that way. The solution above was truncated to 6 places.