### Eigenthings

Gould's accessibility index in a network - The process uses a modified adjacency matrix and the components of the eigenvector associated with the dominant eigenvalue. Students find this approachable and adaptable. Applications to historical geography, air traffic.**Discrete dynamical systems**- Using linear algebra to study discrete dynamical systems comes in several flavors. Here are some projects that students find interesting and that differ from each other enough that they feel they are not repeating someone else's project.

*Difference equations and the Fibonacci sequence*- Using eigenvalues to write the product of the*n*th power of a diagonalizable matrix and an initial vector allows one to write a closed form for a recursive formula. Matrices of size 2x2 are needed to write the closed form of the*n*th Fibonacci number, but students can easily move from there to the closed form of 3rd and 4th order difference equations. This project is always chosen by some student even though it is not applied to a real-world situation.