Consider the objective $$\max_{x,y} x^TAx +y^TBy $$ subject to: $$\|x\|^2 +\|y\|^2 =1 $$
Setting the derivative of the Lagrangian w.r.t to $x$ and $y$ to zero, yields:
$$Ax = \lambda x$$ $$By = \lambda y$$
Is there a possibility to express the solution without using Lagrange multipliers?