Welcome to the Shard!
So, it has been a while, but an eigenbasis is a situation where every vector is an eigenvector, correct? And eigenvectors helps make linear transformations easier. Eigenvectors and eigenvalues put together, in nonlinear motion dynamics, can help us better understand data by transforming and representing the data in manageable sets. They can be used to decouple three phase systems through symmetrical component transformation.
It has been a while, and that is all I remember, was I close?