Distance extrema on a real quadratic surface ($x^{T}Ax=1,|x{|}^2=1/\lambda$)

For a real symmetric matrix $A$, every stationary squared distance $x^{T}x$ on $x^{T}Ax=1$ 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.