Suppose $H$ is a symmetric positive definite matrix, and $h$ a vector. I want to solve the following optimization problem with linear objective and quadratic equality constraint:
$$\underset{v^THv = 1} {\mathrm{argmin}} \; h^Tv.$$
I am thinking of equipping the space with the inner product $\langle\cdot, \cdot \rangle: x, y \to x^THy$. Then the new normed vector space is still Euclidean, and we are minimizing essentially
$$\underset{||v||_{new}^2= 1} {\mathrm{argmin}} \; \langle v, H^{-1}h \rangle$$
This is because $h^Tv = h^T H^{-1}H v = (H^{-T}h)^T H v = \langle v, H^{-1}h \rangle$. Therefore, the best $v$ is at the direction of $-H^{-1}h$, we normalize it to the equality constraint. Thus, the optimizer $v$ is $-\frac{H^{-1}h}{h^TH^{-1}h}.$ I'm wondering if the reasoning above is correct.
(Note that if we just use Lagragian multiplier and have that $-h = 2\lambda H v$, then $v = -\frac{1}{2}\lambda^{-1}H^{-1}h$. With proper choice of $\lambda$ for the constraint, we similarly have $-\frac{H^{-1}h}{h^TH^{-1}h}$).