Semi-axis ($a_i>0$)

Half the length of one principal diameter of an ellipsoid. For a centred quadratic form $x^{T}Sx=C$ with $S$ symmetric positive definite and $C>0$, its semi-axis lengths are $\sqrt{C/{\lambda }_i}$, where ${\lambda }_i$ are the eigenvalues of $S$. The shortest centre-to-surface distance corresponds to the largest eigenvalue.