It's awkward, but to be completely honest: I never understood what eigenvectors are all about. The only reasons I can think of why they could be interesting are:
- They are calculated to diagonalize a matrix (which is useful, because one can calculate faster with diagonal matrices?).
- If one has a linear transformation that represents a rotation, an eigenvector would be the rotation axis.
- Everybody says that they are used "everywhere".
But I find all of these reasons unsatisfying. The concept of an eigenvector just seems unnatural to me. I associate this concept with a lot of unmotivated calculations. I would love to hear an honest, down-to-earth explanation of why eigenvectors are interesting.
Note: I am not asking what eigenvectors are, my question is purely about why a pure mathematician (not interested in numerical calculations) should care.