= Quadratic-ball barrier eigenvalue branch
{title2=$\Psi(x,\mu)=x^TAx-\mu\log(1-x^Tx)$}
For symmetric $A$, every nonzero stationary point of this barrier lies in a negative-eigenvalue eigenspace: $Ax=\lambda x$, $\lambda<0$, $\|x\|^2=1+\mu/\lambda$. The origin is also stationary for every positive $\mu$ and must be considered separately. On a fixed nonzero branch, the Hessian has radial eigenvalue $4\lambda^2\|x\|^2/\mu$ and transverse eigenvalues $2(\lambda_i-\lambda)$. A simple lowest negative eigenvalue gives a minimizing branch with a condition number growing as $1/\mu$; repeated lowest eigenvalues produce zero transverse curvature instead. Nevertheless the minimizing radius tends to one and the limiting vectors solve the constrained quadratic problem. The condition number measures numerical difficulty, not failure of the approximation principle.
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