= Nearest points to the origin on a positive-definite ellipsoid
{title2=$d_{\min}=\lambda_{\max}^{-1/2}$}
For a <positive-definite symmetric matrix> $A$, the <ellipsoid> $x^TAx=1$ has nearest points to the origin exactly at $x=\lambda_{\max}^{-1/2}u$, where $u$ ranges over <unit vectors> in the largest-<eigenvalue> <eigenspace>. Indeed, the <spectral theorem for real symmetric matrices> gives $x^TAx\leq\lambda_{\max}|x|^2$, with equality precisely in that <eigenspace>. A simple largest <eigenvalue> gives two opposite nearest points; multiplicity $m>1$ gives a scaled $(m-1)$-sphere of nearest points.
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