Difference between eigenfunctions and eigenvectors of an operator?

Question

What is the difference between the eigenfunctions and eigenvectors of an operator, for example Laplace-Beltrami operator?

Answer 1

An eigenfunction is an eigenvector that is also a function. Thus, an eigenfunction is an eigenvector but an eigenvector is not necessarily an eigenfunction. For example, the eigenvectors of differential operators are eigenfunctions but the eigenvectors of finite-dimensional linear operators are not.