I think it's mostly because mathematics is taught as is with little or no attention to practical applications of the theory. For instance, till date, I could not see how eigen values/vectors are useful.
One nice application is that the eigenvectors of a stochastic matrix correspond to stationary states of the Markov chain. That leads naturally to a conversation about PageRank, which is super useful because it made billions of dollars for some people. Then you could talk about the use of eigenvectors for principal component analysis in data science, which is making a lot of money for other people right now. If these examples are not useful, then I don't know what is.
Interesting. Eigenvalues and eigenvectors are one of the most fundamental applications of linear algebra. In physics, quantum mechanics is a giant eigenvalue-eigenvector problem.
One application I really like is in machine learning: the eigenface algorithm.
That might all be the case. I would simply like a thoughtful introduction to the topic about where this abstraction comes from. What is the idea behind it and what was the motivation for its invention?
After I understand the plain English concept, then give me the math notation and proof and applications.
The way it works today is often to simply leave out the first part and I believe this is why many people find it hard to develop intuition and a real understanding of the concept.
For man it's just there, you memorize it, you apply it, you take the result and simply have faith that it is what you need because some book/ prof said to solve problem X use eigenvalues.
The simple idea is that while linear maps(matrices) are generally a combination of rotations, scaling and other stuff in higher dimensions, sometimes they will act on a vector simply by scaling it.
Since scaling is a really easy operation to understand, the space generated by these vectors will be really easy to understand. Note that different vectors can be scaled by different amounts.
Now it often turns out that the space generated like this is actually the whole of the space under consideration and this really simplifies the linear map we started with. Hope that helps.