I'm starting to get a better grasp of eigenvalues, eigenvectors,... but have a few questions
1. Is it correct in saying that eigenvectors and eigenfunctions correspond to the same thing but eigenvectors come from matrices whereas eigenfunctions come from any linear operator such as a differential operator?
2. Why is it in courses, we are rarely taught how to find an eigenfunction of a certain equation like the wave equation?
3. When I have seen in books the eigenfunction of a differential operator, the author simply writes that by plugging in the eigenfunction to the operator one can see it is right. In other words, are there any methods for solving for eigenfunctions or is it more of guess and check based on intuition?
I assume that the answer to these questions is that for any equation or problem, one can break down the problem and solve it using a numerical method which implies the usage of a matrix. Since a matrix is involved eigenvectors are used instead of eigenfunctions. Is this a correct assumption?