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.
I am an undergrad studying Math and Physics, in the third year of my degree. So, while I'm sure there are many other reasons to care about eigenvectors, the best ones are have seen so far are as follows;
In solving systems of differential equations
When studying differential equations, you often come across situations where multiple differential equations need to be satisfied simultaneously. In order to solve them, you can put them into a matrix, and find the eigenvectors. (I only have a basic understanding of this topic. More info can be found here)
In quantum mechanics
This is probably the most interesting reason to care about eigenvectors that I have come across. In quantum mechanics, anything you can measure directly has an operator associated with it. If you represent this operator with a matrix, and the state of the system you are studying with a vector, then we can know the quantity we want to measure exactly if and only if the state vector is an eigenvector of the matrix. And it turns out that the eigenvalue is the quantity we were originally measuring.
In theoretical computer science
I am currently taking a course on logic and computation where we learn about how you can represent a set of bits in a given state as a vector, and a logical operator as a matrix. Eigenvectors and eigenvalues can reveal some cool properties of logical operators.