Find critical points of $\langle Av,v\rangle$ 
Let $A$ be an $n \times n$ real symmmetric matrix. Find the critical points of the function $\langle Av,v\rangle$ restricted to the unit sphere in $\mathbb{R}^n$.

I would think you just use Lagrange multipliers, and $\nabla\langle Av,v\rangle=2Ax$, since $A$ is symmetric. But this only gives the possible locations for the local max/min. So the gradient is zero at these points, but doesn't necessarily include all points were the gradient is zero. Is there something I'm missing here?

Actually the restriction of a function to the unit sphere does not have directional derivates perpendicular to the sphere surface. This means that the relevant gradient is the gradient of the extended function projected onto the tangent plane of the surface that is $\nabla A - \langle \nabla A, \hat n\rangle$:
$$2Ax - \langle 2Ax, x\rangle x = 0$$
or
$$(A-\langle Ax, x\rangle I)x = 0$$
which only have non-trivial solutions when $\langle Ax, x\rangle$ is an eigenvale of $A$ with the solutions being eigenvectors with that eigenvalue.
