= Solution
The <logarithmic barrier> is
$$
\boxed{\Psi(x,\mu)=x^TAx-\mu\log(1-x^Tx),\qquad\|x\|_2<1,\quad\mu>0.}
$$
Its <gradient> is $2Ax+2\mu x/(1-x^Tx)$. Thus every nonzero stationary point satisfies
$$
\boxed{Ax=\lambda_1x,\qquad\lambda_1=-\frac\mu{1-\|x\|_2^2}<0,\qquad\|x\|_2^2=1+\mu/\lambda_1.}
$$
A fixed negative-eigenvalue branch exists for $0<\mu<-\lambda_1$. The zero point is stationary for every $\mu$ as well, and it is not an <eigenvector>; at zero the displayed scalar $-\mu$ need not be an <eigenvalue> of $A$. Thus the claimed eigenvalue conclusion requires the nonzero-point qualification. This distinction is essential for the following Hessian analysis.
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