What is an eigenvalue exactly. I have been using it for years, and I am still trying to wrap my mind around it.  I've used it in classical Linear Algebra, and I a have used it in with trigonometric wave functions.  I don't quite understand what I am doing.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.