I'm struggling to find a geometric, or at least some intuitive understanding of eigenvalues and eigenfunctions in Sturm-Liouville problems (which I've been looking at in a PDE course).
For instance, for the Sturm-Liouville problem:
with boundary conditions of course, I struggle to see why this rather arbitrarily placed $\lambda$ deserves the high honour of being called an eigenvalue. And then we solve for $\phi$ and call it an eigenfunction. In linear algebra, eigenvalues and eigenvectors of a transformation have a number of nice geometric interpretations, and frankly I feel quite comfortable in seeing their importance in that setting, but I'm unsure why we can prescribe this terminology here.
In the course I'm taking we're making a great deal of fuss about non-negativity of eigenvalues, the orthogonality of eigenfunctions, etc., etc., but while I can follow the various proofs algebraically I must admit I feel quite lost without this basic understanding.