How can you get eigen- vectors/values/functions in several different contexts? In differential equations, you can find the eigenvectors/values of a square matrix: A v = \lambda v, where A is a square matrix. You can also find eigen functions of a differential operator, D, in the differential equation D f = \lambda f And finally, you can find eigen- values/vectors/functions for a Sturm-Liouville equation. From what I understand, an eigen vector is one that maps to itself after going through the indicated function. Why is it possible to obtain eigenvectors in three different contexts? Is there possibly some hidden connection between the three?

This is because operators, and functionals can be considered as (possibly) infinite dimensional matrices.

Of we solve $Av=\lambda v$, the eigenvalue(s) we seek, are the values that make the mapping $Av-\lambda v$ non injective, so we cannot invert the map.

For a matrix this corresponds to a matrix being singular, I.e uninvertible.

The eigenvectors are the elements in the kernel of this non invertible mapping.