got a two-part question here.
Given (symmetric, positive definite) matrix $A$ = $$\begin{bmatrix} 2 & -1 \\ -1 & 2 \\ \end{bmatrix} $$ solve the problem: $arg max$ $g(x) = (Ax,x)$ subject to $\lVert x\rVert^2 = 1$.
Use two methods:
(1) Use the fact that the unit circle $S_1$ $\equiv$ $\{x: \lVert x\rVert = 1\}$ can be parametrized using a single variable $t$. This gives a function of one variable to be maximized.
(2) Enforce the constraint using Lagrange multipliers.
I know that the parametrization of $S_1$ using $t$ should be $x(t) = [cos(t), sin(t)]^T$, but I'm not sure how to proceed from there to finding the value that maximizes $g(x)$. As for the Lagrange multipliers, I'm not clear on how to find them from the constraint, nor on how to apply them to find the argmax. How is this done?
Any help on how to proceed is greatly appreciated!