= Distance extrema on a real quadratic surface
{title2=$x^TAx=1,\quad |x|^2=1/\lambda$}
For a real <symmetric matrix> $A$, every stationary squared distance $x^Tx$ on $x^TAx=1$ occurs in a positive-<eigenvalue> <eigenspace>. The <Lagrange multiplier> equation is $x=\Lambda Ax$; multiplying by $x^T$ gives $\Lambda=|x|^2>0$, hence $Ax=|x|^{-2}x$. If $A$ is a <positive-definite matrix>, the nearest and farthest squared distances are $1/\lambda_{\max}$ and $1/\lambda_{\min}$. With a repeated <eigenvalue>, the stationary set may contain an entire sphere within its <eigenspace>. For an indefinite or singular form the surface may be unbounded, and a stationary point need not be a global extremum.
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