I'm given the following:
$$\begin{align} \min &\qquad x^TQx\\ \mathrm{s.t}&\qquad x^TAx <= 1 \end{align}$$
where $A$ is a positive definite.
I'm not sure if and or how this would change the original optimization with the original constraint of $g(x) = x^Tx = 1$ since here we could use the Identity matrix $I$ for A since $I$ is PD? Would the answer be the same and the constraint be binding at optimality? At optimality for the equality constraint the answer is the smallest eigenvalue $\lambda$ of Q and its associated normal eigenvector stationary point.