Support function of a centred ellipsoid (${\max }_{x^{T}Sx=C}{q}^{T}x=\sqrt{C{q}^T{S}^{-1}q}$)

Let $S$ be a real symmetric positive definite matrix, $C>0$ and $q\ne 0$. The Cauchy-Schwarz inequality in the $S$-weighted inner product gives $q^{T}x=(S^{-1}q{)}^{T}Sx\le \sqrt{q^T{S}^{-1}q}\sqrt{x^{T}Sx}$. Equality is attained by a positive multiple of $S^{-1}q$. Thus the displayed maximum holds. For a translated ellipsoid, add $q^{T}a$ for its centre $a$. This computes a coordinate extremum without constructing a rotation matrix.