Real life examples for eigenvalues / eigenvectors There are already good answers about importance of eigenvalues / eigenvectors, such as this question and some others, as well as this Wikipedia article.

I know the theory and these examples, but now in order to do my best to prepare a course I'm teaching, I'm looking for ideas about good real life examples of usage of these concepts.

Do you know some good simple real-life examples (in economics or data analysis or anything else), in which  the usage of eigen values/vectors is a crucial tool?
Here are just some of the many uses of eigenvectors and eigenvalues:


Using singular value decomposition for image compression. This is a note explaining how you can compress and image by throwing away the small eigenvalues of $AA^T$. It takes an $8$ megapixel image of an Allosaurus, and shows how the image looks after compressing by selecting $1$,$10$,$25$,$50$,$100$ and $200$ of the largest singular values.
Deriving Special Relativity is more natural in the language of linear algebra. In fact, Einstein's second postulate really states that "Light is an eigenvector of the Lorentz transform." This document goes over the full derivation in detail.
Spectral Clustering. Whether it's in plants and biology, medical imaging, buisness and marketing, understanding the connections between fields on Facebook, or even criminology, clustering is an extremely important part of modern data analysis. It allows people to find important subsystems or patterns inside noisy data sets. One such method is spectral clustering which uses the eigenvalues of a the graph of a network. Even the eigenvector of the second smallest eigenvalue of the Laplacian matrix allows us to find the two largest clusters in a network.
Dimensionality Reduction/PCA. The principal components correspond the the largest eigenvalues of $A^TA$ and this yields the least squared projection onto a smaller dimensional hyperplane, and the eigenvectors become the axes of the hyperplane. Dimensionality reduction is   extremely useful in machine learning and data analysis as it allows one to understand where most of the variation in the data comes from. 
Low rank factorization for collaborative prediction. This what Netflix does (or once did) to predict what rating you'll have for a movie you have not yet watched. It uses the SVD, and throws away the smallest eigenvalues of $A^TA$.
The Google Page Rank algorithm. The largest eigenvector of the graph of the internet is how the pages are ranked.
Let me give you a direct answer. In application eigen values can be:
1- Control Field: eigen values are the pole of the closed loop systems, if there values are negative for analogue systems then the system is stable, for digital systems if the values are inside the unit circle also the system is stable.

2- Mechanical system: eigen values are the natural frequency and the eigen values are the mode shapes.