I want to find the unit vector $m$ which minimises the following expression
$m^T Sm$
where $S$ is a real symmetric matrix of shape $n$x$n$. This is not an eigenvalue problem as $m$ is restricted to lie in a subspace of $S$. For example,
$m = \alpha m_1 + \beta m_2$
where $m_1$ and $m_2$ are known but are not one of the four main subspaces of $S$.
I understand that the solution will involve finding the principle axes of an ellipse on the subspace.
Ideally, I want a solution which does not involve interations or finding the eigenvectors of large matrices.