nonsymmetric border perturbations:
\[
A = \left[ \begin{array}{cl}
    B & -\eta \\
             \eta^t & t 
\end{array} \right]
\]
eigenvalues, eigenvectors

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direct connections with perturbed quadratic minimization
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trust region problems
\[  \min_{x^tx=1} (\max)~ \mu (x) = x^tBx -2 \eta^tx \]
Lagrange multipliers, stationary points