= Distance extrema on a central plane section of an ellipsoid
{title2=$d_{\min}^2=\lambda_{\max}(U^TAU)^{-1},\quad d_{\max}^2=\lambda_{\min}(U^TAU)^{-1}$}
Let $A$ be a real <positive-definite matrix> and let the columns of $U$ be an <orthonormal basis> of a two-dimensional <linear subspace> in three-dimensional <Euclidean space>. Points in that central plane are $x=Uq$, with $|x|^2=|q|^2$. Restricting the <ellipsoid> $x^TAx=1$ gives $q^T(U^TAU)q=1$. The compressed <matrix> $U^TAU$ is a <positive-definite matrix>, and its extreme <eigenvalues> give the distance formulas in the title by the <Rayleigh quotient>. The two extremal axes are the images under $U$ of the corresponding <eigenvectors>. This restriction reduces a problem with two <Lagrange multipliers> to a two-dimensional <quadratic form>.
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